[seqfan] Fibonacci numbers in Golden Ratio Base
Dale Gerdemann
dale.gerdemann at gmail.com
Sun Oct 12 19:11:11 CEST 2014
Hello Seqfans,
In the table below, i've listed the powers of phi used in golden ratio base
to represent the Fibonacci numbers. So, for example, 5 = phi^3 + phi^-1 +
phi^-4 = f(3)+f(-1)+f(-4) [where f(n) = Fibonacci(n+1), generalized to
negative indices using "precursion"].
The pattern strikes me as remarkable. Maybe deserving a place in the OEIS?
[0] 1
[0] 1
[1, -2] 2
[2, -2] 3
[3, -1, -4] 5
[4, 0, -4] 8
[5, 1, -3, -6] 13
[6, 2, -2, -6] 21
[7, 3, -1, -5, -8] 34
[8, 4, 0, -4, -8] 55
[9, 5, 1, -3, -7, -10] 89
[10, 6, 2, -2, -6, -10] 144
[11, 7, 3, -1, -5, -9, -12] 233
[12, 8, 4, 0, -4, -8, -12] 377
[13, 9, 5, 1, -3, -7, -11, -14] 610
[14, 10, 6, 2, -2, -6, -10, -14] 987
[15, 11, 7, 3, -1, -5, -9, -13, -16] 1597
[16, 12, 8, 4, 0, -4, -8, -12, -16] 2584
[17, 13, 9, 5, 1, -3, -7, -11, -15, -18] 4181
[18, 14, 10, 6, 2, -2, -6, -10, -14, -18] 6765
[19, 15, 11, 7, 3, -1, -5, -9, -13, -17, -20] 10946
[20, 16, 12, 8, 4, 0, -4, -8, -12, -16, -20] 17711
[21, 17, 13, 9, 5, 1, -3, -7, -11, -15, -19, -22] 28657
[22, 18, 14, 10, 6, 2, -2, -6, -10, -14, -18, -22] 46368
[23, 19, 15, 11, 7, 3, -1, -5, -9, -13, -17, -21, -24] 75025
[24, 20, 16, 12, 8, 4, 0, -4, -8, -12, -16, -20, -24] 121393
[25, 21, 17, 13, 9, 5, 1, -3, -7, -11, -15, -19, -23, -26] 196418
[26, 22, 18, 14, 10, 6, 2, -2, -6, -10, -14, -18, -22, -26] 317811
[27, 23, 19, 15, 11, 7, 3, -1, -5, -9, -13, -17, -21, -25, -28] 514229
[28, 24, 20, 16, 12, 8, 4, 0, -4, -8, -12, -16, -20, -24, -28] 832040
[29, 25, 21, 17, 13, 9, 5, 1, -3, -7, -11, -15, -19, -23, -27, -30] 1346269
[30, 26, 22, 18, 14, 10, 6, 2, -2, -6, -10, -14, -18, -22, -26, -30] 2178309
[31, 27, 23, 19, 15, 11, 7, 3, -1, -5, -9, -13, -17, -21, -25, -29, -32]
3524578
[32, 28, 24, 20, 16, 12, 8, 4, 0, -4, -8, -12, -16, -20, -24, -28, -32]
5702887
[33, 29, 25, 21, 17, 13, 9, 5, 1, -3, -7, -11, -15, -19, -23, -27, -31,
-34] 9227465
[34, 30, 26, 22, 18, 14, 10, 6, 2, -2, -6, -10, -14, -18, -22, -26, -30,
-34] 14930352
[35, 31, 27, 23, 19, 15, 11, 7, 3, -1, -5, -9, -13, -17, -21, -25, -29,
-33, -36] 24157817
[36, 32, 28, 24, 20, 16, 12, 8, 4, 0, -4, -8, -12, -16, -20, -24, -28, -32,
-36] 39088169
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