Monday, February 22, 2016

Building Pascal's Triangle with Sequence Numbers

22 Feb 2016




Building Pascal’s Triangle Using Sequence Numbers
David Brooks
SAWB, BE, MS, MS

What is Pascal’s Triangle?

Pascal’s triangle is well described on several sites on the internet.  I don’t think I could do better, so I have listed several websites that do a good job of this.  If you do not already know what Pascal’s triangle is please visit one or more of the sites listed below.
See:




Rows zero through five of Pascal’s Triangle – Wikipedia
This image also shows at least the beginning of the first five diagonals.

What are Sequence Numbers?

They are integers that have a special property.  When you calculate the decimal expansion of the inverse of a Sequence Number you get a recognizable number sequence (many of these sequences are listed in the Online Encyclopedia of Integer Sequences ( www.OEIS.org )
What kind of mathematics do I need to know in order to work with Sequence Numbers?
You need to know how to take the inverse of an integer.  The inverse of 123 is “1 over 123” or “1 divided by 123” (1/123).
You also need to know how to do long division – really long division that you can’t do on your calculator.  But don’t worry – you can do it by hand on paper OR you can get on the internet and go to ( www.wolframalpha.com ) and use this free “super calculator”.  It will take inputs of about 200 digits, and can provide an output of about 3900 digits.
I would not have been able to do these calculations without access of the Wolfram Alpha website.  And I could not check my answers without the OEIS website.  I would recommend you get on these websites and play around with them to learn how to use them.
However if you understand how to create the inverse of an integer, and you understand how to take a fraction and do long division to get its decimal expansion (how to change a fraction into a decimal),       and you learn how to do these computations on the internet, then you will have it made.
What is Pascal’s triangle?
Look it up on the internet or ask your math teacher or math professor.
Wikipedia has a good article article about Pascal’s triangle: (https://en.wikipedia.org/wiki/Pascal's_triangle )

The Numbers in the First Diagonal of Pascal’s Triangle:
OK, this is the easy one.
A Sequence Number that generates the terms in the first diagonal (left or right side) is 9.  This Sequence Number generates terms written in single digit strings.  Now all we have to do is calculate the decimal expansion of the inverse of this number.
(I usually separate these terms with spaces, to make them easier to separate the terms while you are reading them.)
1/9 =
0.
1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  ...

The Numbers in the Second Diagonal of Pascal’s Triangle:
A Sequence Number that produces the terms in the second diagonal of Pascal’s Triangle is 999,998,000,001.  This particular Sequence Number produces terms written in six digit strings.
1/999998000001 =
0.
000000  000001  000002  000003  000004  000005  000006  000007  000008  000009  000010  000011  000012  000013  000014  000015  000016  000017  000018  000019  000020  000021  000022  000023  000024  000025  000026  000027  000028  000029  000030  000031  000032 
(This sequence of numbers are also known as the “counting numbers”.)

The Numbers in the Third Diagonal of Pascal’s Triangle:
A Sequence Number that will produce a list of the terms in the third diagonal of Pascal’s triangle (written in six digit strings) is 999,997,000,002,999,999.
1/999997000002999999 =
0.
000000  000000  000001  000003  000006  000010  000015  000021  000028  000036  000045  000055  000066  000078  000091  000105  000120  000136  000153  000171  000190  000210  000231  000253  000276  000300  000325  000351  000378  000406  000435  000465  000496  000528  ...
(Note: This sequence of numbers is also known as the “Triangular Numbers”, and in also in C (n, 4) or the “number of way’s to select 4 items from a group of n items”.)
Your can compare this with the list of triangular number in the Online Encyclopedia: http://oeis.org/A000217 and http://oeis.org/A000217/b000217.txt.

The Numbers in the Fourth Diagonal of Pascal’s Triangle:
A Sequence Number that will produce a list of the terms in the fourth diagonal of Pascal’s triangle (written in nine digit strings) is 999,999,996,000,000,005,999,999,996,000,000,001.
1/999999996000000005999999996000000001 =
0.
000000000  000000000  000000000  000000001  000000004  000000010  000000020  000000035  000000056  000000084  000000120  000000165  000000220  000000286  000000364  000000455  000000560  000000680  000000816  000000969  000001140  000001330  000001540  000001771  000002024  000002300  000002600  000002925  000003276  000003654  000004060  000004495  000004960  000005456  000005984  000006545  ...
(This Sequence of numbers are also known as the “Tetrahedral Numbers”, and they are also the numbers described in C (n, 3) or “the number of ways to choose 3 items from a group of n items.”
You can compare this sequence with the Online Encyclopedia of Number Sequences: http://oeis.org/A000292 and http://oeis.org/A000292/b000292.txt.
If you desire to calculate more terms in this sequence I recommend using the free website http://www.wolframalpha.com/ and use the inverse of the Sequence Number shown above (1/99999999600000000599999999600000000) as the input.  When the output is calculate you can click on the button that say’s more digits (up to six times) to display more digits (up to about 3900 digits).  The output will be displayed in scientific notation which you will want to convert to a regular notation.  In this case you will need to add 36 zeros to the front of the number and place a decimal point after the first zero.

The Numbers in the Fifth Diagonal of Pascal’s Triangle:
A Sequence Number that will produce a list of the terms in the fifth diagonal of Pascal’s triangle (written in nine digit strings) is 999999995000000009999999990000000004999999999.
1/999,999,995,000,000,009,999,999,990,000,000,004,999,999, 999 =
0.
000000000  000000000  000000000  000000000  000000001  000000005  000000015  000000035  000000070  000000126  000000210  000000330  000000495  000000715  000001001  000001365  000001820  000002380  000003060  000003876  000004845  000005985  000007315  000008855  000010626  000012650  000014950  000017550  000020475  ...
(This Sequence of numbers are also the numbers described in C (n, 4) or “the number of ways to choose 4 items from a group of n items.”)

The Numbers in the Sixth Diagonal of Pascal’s Triangle:
A Sequence Number that will produce a list of the terms in the sixth diagonal of Pascal’s triangle (written in nine digit strings) is 999,999,994,000,000,014,999,999,980,000,000,014,999,999, 994,000,000,001.
1/999999994000000014999999980000000014999999994000 000001 =
0.
000000000  000000000  000000000  000000000  000000000  000000001  000000006  000000021  000000056  000000126  000000252  000000462  000000792  000001287  000002002  000003003  000004368  000006188  000008568  000011628  000015504  000020349  000026334  000033649  000042504  000053130  000065780  000080730  000098280  000118755  000142506  000169911  000201376  000237336  000278256  000324632  000376992  000435897  000501942  000575757  000658008  000749398  000850668  000962598  001086008  001221759  ...
(This Sequence of numbers are also the numbers described in C (n, 5) or “the number of ways to choose 5 items from a group of n items.”)

The Numbers in the Seventh Diagonal of Pascal’s Triangle:
A Sequence Number that will produce a list of the terms in the seventh diagonal of Pascal’s triangle (written in 12 digit strings) is 999,999,999,993,000,000,000,020,999,999,999,965,000,000, 000,034,999,999,999,979,000,000,000,006,999,999,999,999.
1/999999999993000000000020999999999965000000000034999999999979000000000006999999999999 =
0.
000000000000  000000000000  000000000000  000000000000  000000000000  000000000000  000000000001  000000000007  000000000028  000000000084  000000000210  000000000462  000000000924  000000001716  000000003003  000000005005  000000008008  000000012376  000000018564  000000027132  000000038760  000000054264  000000074613  000000100947  000000134596  000000177100  000000230230  000000296010  000000376740  000000475020  000000593775  000000736281  000000906192  000001107568  000001344904  000001623160  000001947792  000002324784  000002760681  000003262623  000003838380  ...
(This Sequence of numbers are also the numbers described in C (n, 6) or “the number of ways to choose 6 items from a group of n items.”)

The Numbers in the Eighth Diagonal of Pascal’s Triangle:
A Sequence Number that will produce a list of the terms in the eighth diagonal of Pascal’s triangle (written in 12 digit strings) is 999,999, 999,992,000,000,000,027,999,999,999,944,000,000,000,069,999, 999,999,944,000,000,000,027,999,999,999,992,000,000,000,001.
1/999999999992000000000027999999999944000000000069999999999944000000000027999999999992000000000001 =
0.
000000000000  000000000000  000000000000  000000000000  000000000000  000000000000  000000000000  000000000001  000000000008  000000000036  000000000120  000000000330  000000000792  000000001716  000000003432  000000006435  000000011440  000000019448  000000031824  000000050388  000000077520  000000116280  000000170544  000000245157  000000346104  000000480700  000000657800  000000888030  000001184040  000001560780  000002035800  000002629575  000003365856  000004272048  000005379616  000006724520  000008347680  000010295472  000012620256  000015380937  000018643560  000022481940  000026978328  000032224114  000038320568  000045379620  000053524680  ...
(This Sequence of numbers are also the numbers described in C (n, 7) or “the number of ways to choose 7 items from a group of n items.”)

The Numbers in the Ninth Diagonal of Pascal’s Triangle:
A Sequence Number that will produce a list of the terms in the ninth diagonal of Pascal’s triangle (written in 15 digit strings) is 999,999, 999,999,991,000,000,000,000,035,999,999,999,999,916,000,000, 000,000,125,999,999,999,999,874,000,000,000,000,083,999,999, 999,999,964,000,000,000,000,008,999,999,999,999,999.
1/999999999999991000000000000035999999999999916000000 0000001259999999999998740000000000000839999999999999 64000000000000008999999999999999 =
0.
000000000000000  000000000000000  000000000000000  000000000000000  000000000000000  000000000000000  000000000000000  000000000000000  000000000000001  000000000000009  000000000000045  000000000000165  000000000000495  000000000001287  000000000003003  000000000006435  000000000012870  000000000024310  000000000043758  000000000075582  000000000125970  000000000203490  000000000319770  000000000490314  000000000735471  000000001081575  000000001562275  000000002220075  000000003108105  000000004292145  000000005852925  000000007888725  000000010518300  000000013884156  000000018156204  000000023535820  000000030260340  000000038608020  000000048903492  000000061523748  000000076904685  000000095548245  000000118030185  000000145008513  000000177232627  000000215553195  000000260932815  000000314457495  000000377348994  000000450978066  000000536878650  000000636763050  000000752538150  000000886322710  000001040465790  000001217566350  000001420494075  000001652411475  000001916797311  000002217471399  000002558620845  000002944827765  000003381098545  000003872894697  000004426165368  000005047381560  000005743572120  000006522361560  000007392009768  000008361453672  000009440350920  000010639125640  000011969016345  ...
(This Sequence of numbers are also the numbers described in C (n, 8) or “the number of ways to choose 8 items from a group of n items.”)

The Numbers in the Tenth Diagonal of Pascal’s Triangle:
A Sequence Number that will produce a list of the terms in the tenth diagonal of Pascal’s triangle (written in 15 digit strings) is 999,999, 999,999,990,000,000,000,000,044,999,999,999,999,880,000,000, 000,000,209,999,999,999,999,748,000,000,000,000,209,999,999, 999,999,880,000,000,000,000,044,999,999,999,999,990,000,000, 000,000,001.
1/999999999999990000000000000044999999999999880000000 00000020999999999999974800000000000020999999999999988 0000000000000044999999999999990000000000000001 =
0.
000000000000000  000000000000000  000000000000000  000000000000000  000000000000000  000000000000000  000000000000000  000000000000000  000000000000000  000000000000001  000000000000010  000000000000055  000000000000220  000000000000715  000000000002002  000000000005005  000000000011440  000000000024310  000000000048620  000000000092378  000000000167960  000000000293930  000000000497420  000000000817190  000000001307504  000000002042975  000000003124550  000000004686825  000000006906900  000000010015005  000000014307150  000000020160075  000000028048800  000000038567100  000000052451256  000000070607460  000000094143280  000000124403620  000000163011640  000000211915132  000000273438880  000000350343565  000000445891810  000000563921995  000000708930508  000000886163135  000001101716330  000001362649145  000001677106640  000002054455634  000002505433700  000003042312350  000003679075400  000004431613550  000005317936260  000006358402050  000007575968400  000008996462475  000010648873950  000012565671261  000014783142660  000017341763505  000020286591270  000023667689815  000027540584512  000031966749880  000037014131440  000042757703560  000049280065120  000056672074888  000065033528560  000074473879480  000085113005120  000097082021465  000110524147514  000125595622175  000142466675900  000161322559475  000182364632450  000205811513765  000231900297200  000260887834350  000293052087900  000328693558050  000368136785016  000411731930610  000459856441980  000512916800670  000571350360240  000635627275767  000706252528630  000783768050065  000868754947060  000961835834245  001063677275518  001174992339235  001296543270880  001429144287220  001573664496040  001731030945644  001902231808400  002088319702700  002290415157800  002509710226100  002747472247520  003005047770725  003283866636050  003585446225075  003911395881900  004263421511271  004643330358810  005053035978705  005494563394320  005970054457290  006481773410772  007032112662630  007623598774440  008258898672310  008940826085620  009672348219898  010456592670160  011296854581155  012196604061070  013159493855365  014189367287524  015290266473625  016466440817750  017722355795375  019062702032000  020492404684400  022016633132000  023640810986000  025370626424000  027212042858000  029171309943776  031254974939760  033469894423680  035823246375345  038322542634090  040975641739527  043790762164380  046776495948315  049941822741810  053296124269245  056849199220528  060611278580710  064593041407180  068805631064170  073260671924440  077970286548154  082947113349100  088204324758550 
(This Sequence of numbers are also the numbers described in C (n, 9) or “the number of ways to choose 9 items from a group of n items.”)

The Numbers in the Eleventh Diagonal of Pascal’s Triangle:
A Sequence Number that will produce a list of the terms in the eleventh diagonal of Pascal’s triangle (written in 18 digit strings) is 999,999,999,999,999,989,000,000,000,000,000,054,999,999,999, 999,999,835,000,000,000,000,000,329,999,999,999,999,999,538, 000,000,000,000,000,461,999,999,999,999,999,670,000,000,000, 000,000,164,999,999,999,999,999,945,000,000,000,000,000,010, 999,999,999,999,999,999.
1/999999999999999989000000000000000054999999999999999 83500000000000000032999999999999999953800000000000000 04619999999999999996700000000000000001649999999999999 99945000000000000000010999999999999999999 =
0.
000000000000000000  000000000000000000  000000000000000000  000000000000000000  000000000000000000  000000000000000000  000000000000000000  000000000000000000  000000000000000000  000000000000000000  000000000000000000  000000000000000011  000000000000000066  000000000000000286  000000000000001001  000000000000003003  000000000000008008  000000000000019448  000000000000043758  000000000000092378  000000000000184756  000000000000352716  000000000000646646  000000000001144066  000000000001961256  000000000003268760  000000000005311735  000000000008436285  000000000013123110  000000000020030010  000000000030045015  000000000044352165  000000000064512240  000000000092561040  000000000131128140  000000000183579396  000000000254186856  000000000348330136  000000000472733756  000000000635745396  000000000847660528  000000001121099408  000000001471442973  000000001917334783  000000002481256778  000000003190187286  000000004076350421  000000005178066751  000000006540715896  000000008217822536  000000010272278170  000000012777711870  000000015820024220  000000019499099620  000000023930713170  000000029248649430  000000035607051480  000000043183019880  000000052179482355  000000062828356305  000000075394027566  000000090177170226  000000107518933731  000000127805525001  000000151473214816  000000179013799328  000000210980549208  000000247994680648  000000290752384208  000000340032449328  000000396704524216  000000461738052776  000000536211932256  000000621324937376  000000718406958841  000000828931106355  000000954526728530  000001096993404430  000001258315963905  000001440680596355  000001646492110120  000001878392407320  000002139280241670  000002432332329570  000002761025887620  000003129162672636  000003540894603246  000004000751045226  000004513667845896  000005085018206136  000005720645481903  000006426898010533  000007210666060598  000008079421007658  000009041256841903  000010104934117421  000011279926456656  000012576469727536  000014005614014756  000015579278510796  000017310309456440  000019212541264840  000021300860967540  000023591276125340  000026100986351440  000028848458598960  000031853506369685  000035137373005735  000038722819230810  000042634215112710  000046897636623981  000051540966982791  000056594002961496  000062088566355816  000068058620813106  000074540394223878  000081572506886508  000089196105660948  000097455004333258  000106395830418878  000116068178638776  000126524771308936  000137821625890091  000150018229951161  000163177723806526  000177367091094050  000192657357567675  000209123798385425  000226846154180800  000245908856212800  000266401260897200  000288417894029200  000312058705015200  000337429331439200  000364641374297200  000393812684240976  000425067659180736  000458537553604416  000494360799979761  000532683342613851  000573658984353378  000617449746517758  000664226242466073  000714168065207883  000767464189477128  000824313388697656  000884924667278366  000949517708685546  001018323339749716  001091584011674156  001169554298222310  001252501411571410  001340705736329960  001434461382227160  001534076755992935  001639875152957965  001752195368913990  001871392332785690  001997837760676615  002131920831862965  002274048887320496  002424648151381456  002584164477130236  002753064116158356  002931834513311496  003120985127073528  003321048276244908  003532580013585348  003756161027103408  003992397569688528  004241922417794061  004505395859893071  004783506715442026  005076973385101046  005386544932973061  005713002201638095  006057158960772920  006419863090160520  006801997797908170  007204482874707470  007628275984984380  008074373995802180  008543814344395330  009037676445227430  009557083137481880  010103202173909416  010677247751972451  011280482088242081  011914217037019726  012579815754171666  013278694407181203  014012323932439833  014782231840815648  015590004072554208  016437286902584328  017325788897318616 018257282924056176  019233608214112656  020256672480820776  021328454093562616  022451004309013280  023626449560794080  024856993808752105  026144920949101955  027492597286684530  028902474070617070  030377090094628145  031919074363390995  033531148826188520  035216131179263320  036976937738226486  038816586381919346  040738199569143076  ...
(This Sequence of numbers are also the numbers described in C (n, 10) or “the number of ways to choose 10 items from a group of n items.”)

The Numbers in the Twelfth Diagonal of Pascal’s Triangle:
A Sequence Number that will produce a list of the terms in the twelfth diagonal of Pascal’s triangle (written in 21 digit strings) is 999,999,999,999,999,999,988,000,000,000,000,000,000,065,999, 999,999,999,999,999,780,000,000,000,000,000,000,494,999,999, 999,999,999,999,208,000,000,000,000,000,000,923,999,999,999, 999,999,999,208,000,000,000,000,000,000,494,999,999,999,999, 999,999,780,000,000,000,000,000,000,065,999,999,999,999,999, 999,988,000,000,000,000,000,000,001.
1/999999999999999999988000000000000000000065999999999 99999999978000000000000000000049499999999999999999920 80000000000000000009239999999999999999992080000000000 00000000494999999999999999999780000000000000000000065 999999999999999999988000000000000000000001 =
0.
000000000000000000000  000000000000000000000  000000000000000000000  000000000000000000000  000000000000000000000  000000000000000000000  000000000000000000000  000000000000000000000  000000000000000000000  000000000000000000000  000000000000000000000  000000000000000000001  000000000000000000012  000000000000000000078  000000000000000000364  000000000000000001365  000000000000000004368  000000000000000012376  000000000000000031824  000000000000000075582  000000000000000167960  000000000000000352716  000000000000000705432  000000000000001352078  000000000000002496144  000000000000004457400  000000000000007726160  000000000000013037895  000000000000021474180  000000000000034597290  000000000000054627300  000000000000084672315  000000000000129024480  000000000000193536720  000000000000286097760  000000000000417225900  000000000000600805296  000000000000854992152  000000000001203322288 000000000001676056044  000000000002311801440  000000000003159461968  000000000004280561376  000000000005752004349  000000000007669339132  000000000010150595910  000000000013340783196  000000000017417133617  000000000022595200368  000000000029135916264  000000000037353738800  000000000047626016970  000000000060403728840  000000000076223753060  000000000095722852680  000000000119653565850  000000000148902215280  000000000184509266760  000000000227692286640  000000000279871768995  000000000342700125300  000000000418094152866  000000000508271323092  000000000615790256823  000000000743595781824  000000000895068996640  000000001074082795968  000000001285063345176  000000001533058025824  000000001823810410032  000000002163842859360  000000002560547383576  000000003022285436352  000000003558497368608  000000004179822305984  000000004898229264825  000000005727160371180  000000006681687099710  000000007778680504140  000000009036996468045  000000010477677064400  000000012124169174520  000000014002561581840  000000016141841823510  000000018574174153080  000000021335200040700  000000024464362713336  000000028005257316582  000000032006008361808  000000036519676207704  000000041604694413840  000000047325339895743  000000053752237906276  000000060962903966874  000000069042324974532  000000078083581816435  000000088188515933856  000000099468442390512  000000112044912118048  000000126050526132804  000000141629804643600  000000158940114100040  000000178152655364880  000000199453516332420  000000223044792457760  000000249145778809200  000000277994237408160  000000309847743777845  000000344985116783580  000000383707936014390  000000426342151127100  000000473239787751081  000000524780754733872  000000581374757695368  000000643463324051184  000000711521944864290  000000786062339088168  000000867634845974676  000000956830951635624  000001054285955968882  000001160681786387760  000001276749965026536  000001403274736335472  000001541096362225563  000001691114592176724  000001854292315983250  000002031659407077300  000002224316764644975  000002433440563030400  000002660286717211200  000002906195573424000  000003172596834321200  000003461014728350400  000003773073433365600  000004110502764804800  000004475144139102000  000004868956823342976  000005294024482523712  000005752562036128128  000006246922836107889  000006779606178721740  000007353265163075118  000007970714909592876  000008634941152058949  000009349109217266832  000010116573406743960  000010940886795441616  000011825811462719982  000012775329171405528  000013793652511155244  000014885236522829400  000016054790821051710  000017307292232623120  000018647997968953080  000020082459351180240  000021616536107173175  000023256411260131140  000025008606629045130 000026879998961830820  000028877836722507435  000031009757554370400  000033283806441690896  000035708454593072352  000038292619070202588  000041045683186360944  000043977517699672440  000047098502826745968  000050419551102990876  000053952131116576224  000057708292143679632  000061700689713368160  000065942612131162221  000070448007991055292  000075231514706497318  000080308488091598364  000085695033024571425  000091408035226209520  000097465194186982440  000103885057277142960  000110687055075051130  000117891537949758600  000125519813934742980  000133594187930545160  000142138002274940490  000151175678720167920  ...
(This Sequence of numbers are also the numbers described in C (n, 11) or “the number of ways to choose 11 items from a group of n items.”)

To see more examples of Sequence Numbers see my blog about Sequence Numbers:


Sequences of numbers seen is each diagonal of Pascal’s Triangle can be described by Sequence Numbers.  If you know how Sequence Numbers work, you simply need to know which diagonal you want to describe and how many terms you want to obtain (or more specifically, how many digits are contained in the largest term you want to obtain).
If the largest term that you want to obtain contains 12 digits then start with a string of 13 nines (9,999,999,999,999).  This will ensure that all of the 12 digit terms (and smaller terms) are accurate.  Additionally, you will get some 13 digit terms that are accurate.  However the last 13 digit term will not be accurate – it will be larger than the actual term.  If you use 13 nines, the terms will be written in 13 digit strings.  If you use 6 nines, the terms will be written in 6 digit strings.  Etc.
Now, suppose you want to get a sequence that shows you the terms in the nth diagonal.  In this case you want add an exponent at the end of your string of nines – and this exponent will be whatever number “n” is.
999,9993 will produce a sequence of terms in the third diagonal, and will write these terms in six digit strings.  All of the terms up to 5 digits long will be accurate, and some six digit terms may be accurate.
Now take the inverse and calculate the decimal expansion of this fraction:
1/999,999^3 =
(I will separate each term by spaces to make it easier to read.)
0.
000000  000000  000001  000003  000006  000010  000015  000021  000028  000036  000045  000055  000066  000078  000091  000105  000120  000136  000153  000171 ...
You can see the first 18 non-zero terms accurately listed above.
(This number sequences is also called the Triangular Numbers and it is also terms of C (n, 3) which is the number of ways to select 3 items from a group of n items.)
9,999,999,9995 will produce a sequence of terms in the fifth diagonal, and will write these terms in 10 digit strings.  All of the terms up to 9 digits long will be accurate, and some 10 digit terms may be accurate.
Now take the inverse and calculate the decimal expansion of this fraction:
1/9,999999,999^5 =
(This time I will not separate each term by spaces to make it easier to read.)
0.
00000000000000000000000000000000000000000000000001  00000000050000000015000000003500000000700000000126 00000002100000000330000000049500000007150000001001 00000013650000001820000000238000000030600000003876 00000048450000005985000000731500000088550000010626 0000012650000001495  ...
The first 23 non-zero terms are accurately listed above.

David

No comments:

Post a Comment