# [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

```