Two Sequences with Interesting Results

9/30/2015

Today I want to put together Sequence number for the 1, -1, 1 Tribonacci Sequence and the 1, 0, -1, 1 Tetranacci Sequence.  They produce some intriguing results.

The 1, -1, 1 Tribonacci Sequence: a(0) = a(1) = 1, a(2) = 1, and when n>2 then a(n) = a(n-1) – a(n-2) + a(n-3). The Sequence Number is: 909 (I can use a shorter Sequence Number because all I need for this sequence is single digit strings.) 1/909 = 0. 0  0  1  1  0  0  1  1  0  0  1  1  0  0  1  1  0  0  1  1  0  0  1  1  … This is a repeating decimal with a period of four. All term are accurate up to infinity. Compare with OEIS sequence A021913.

The 1, 0, -1, 1 Tetranacci Sequence: a(0) = a(1) = a(2) = 0, a(3) = 1, and when n>3 then a(n) = a(n-1) – a(n-3) + a(n-4). The Sequence Number is: 9009 1/9009 = 0. 0  0  0  1  1  1  0  0  0  1  1  1  0  0  0  1  1  1  0  0  0  1  1  1  … This is a repeating decimal with a period of 6. All terms are accurate up to infinity! This sequence is not listed in the OEIS – yet.

David

A Tale of Two Phone Numbers

9/29/2015

Music, History and the Mathematics of Sequence Numbers:

A Tale of Two Famous Phone Numbers

“Beechwood 4-5789" is a song written by Marvin Gaye, William "Mickey" Stevenson and George Gordy.  It was a 1962 hit single for the Motown girl group “The Marvelettes”.  The title of the song refers to a phone number – 234-5789.  If you want to listen to this song and see a picture of the Marvelettes you can visit this YouTube website: https://www.youtube.com/watch?v=Us18AUBM2RI

See the Wikipedia article about this song at:
https://en.wikipedia.org/wiki/Beechwood_4-5789

We can create a “Beechwood 4-5789" Heptanacci Sequence and it’s associated Sequence Number, based on this famous phone number.

The “Beechwood 4-5789" Heptanacci Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = 0, a(6) = 1, and when n>6 then a(n) = 2*a(n-1) + 3*a(n-2) + 4*a(n-3) + 5*a(n-4) + 7*a(n-5) + 8*a(n-6) + 9*a(n-7). The Sequence Number is: 9,999,999,999,997,999,999,999,999,699,999,999,999,959,999, 999,999,994,999,999,999,999,299,999,999,999,919,999,999, 999,991 1/99999999999979999999999996999999999999599999999 99994999999999999299999999999919999999999991 = 0. 0000000000000  0000000000000  0000000000000  0000000000000  0000000000000  0000000000000  0000000000001  0000000000002  0000000000007  0000000000024  0000000000082  0000000000281  0000000000961  0000000003287  0000000011233  0000000038405  0000000131301  0000000448897  0000001534708  0000005246912  0000017938323  0000061328143  0000209670775  0000716829672  0002450721977  0008378612714  0028645089715  0097932819316  0334816095709  … Terms are written in 13 digit strings so that all terms up to 12 digits long will be accurate. Terms are accurate up to the 23rd non-zero term. This sequence is not listed in the OEIS – yet.  Please compare these results with the table below.

	"The Beachwood 4 - 5789"	"867-5309/Jenny"
	Heptanaci Sequence	Heptanacci Sequence
a(0)	0	0
a(1)	0	0
a(2)	0	0
a(3)	0	0
a(4)	0	0
a(5)	0	0
a(6)	1	1
a(7)	2	8
a(8)	7	70
a(9)	24	615
a(10)	82	5,401
a(11)	281	47,431
a(12)	961	416,533
a(13)	3,287	3,657,951
a(14)	11,233	32,123,745
a(15)	38,405	282,107,385
a(16)	131,301	2,477,437,700
a(17)	448,897	21,756,600,088
a(18)	1,534,708	191,064,198,056
a(19)	5,246,912
a(20)	17,938,323
a(21)	61,328,143
a(22)	209,670,775
a(23)	716,829,672
a(24)	2,450,721,977
a(25)	8,378,612,714
a(26)	28,645,089,715
a(27)	97,932,819,316
a(28)	334,816,095,709

"867-5309/Jenny" is a song written by Alex Call and Jim Keller and performed by Tommy Tutone that was released on the album Tommy Tutone 2, on the Columbia Records label. It peaked at #4 on the Billboard Hot 100 chart and #16 on the Billboard Top Tracks chart in 1982.  If you want to see a video of Tommy Tutone singing this song you can visit the YouTube website: https://www.youtube.com/watch?v=6WTdTwcmxyo

This song actually started a nation wide disturbance because of the number of people calling this number and asking for Jenny.  See the Wikipedia article: https://en.wikipedia.org/wiki/867-5309/Jenny

We can create a "867-5309/Jenny" Heptanacci Sequence and it’s associated Sequence Number, based on this famous phone number.

The "867-5309/Jenny" Heptanacci Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = 0, a(6) = 1, and when n>6 then a(n) = 8*a(n-1) + 6*a(n-2) + 7*a(n-3) + 5*a(n-4) + 3*a(n-5) + 0*a(n-6) + 9*a(n-7). The Sequence Number is: 9,999,999,999,991,999,999,999,999,399,999,999,999,929,999, 999,999,994,999,999,999,999,699,999,999,999,999,999,999, 999,991 1/999999999999199999999999939999999999992999999 9999994999999999999699999999999999999999999991 = 0. 0000000000000  0000000000000  0000000000000  0000000000000  0000000000000  0000000000000  0000000000001  0000000000008  0000000000070  0000000000615  0000000005401  0000000047431  0000000416533  0000003657951  0000032123745  0000282107385  0002477437700  0021756600088  0191064198056  … Terms are written in 13 digit strings so that all terms up to 12 digits long will be accurate. Terms are accurate up to the 13th non-zero term. This sequence is not listed in the OEIS – yet.  Please compare these results with the table above.

Many people who were alive in the 1960 and the 1980 can still remember these two phone numbers.

David