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