# [seqfan] Re: Fibonacci numbers in Golden Ratio Base

Neil Sloane njasloane at gmail.com
Mon Oct 13 23:37:41 CEST 2014

```Dale, definitely submit that - it will be an irregular triangle, keyword
tabf, of course

Interesting triangle!

Neil

On Sun, Oct 12, 2014 at 1:11 PM, Dale Gerdemann <dale.gerdemann at gmail.com>
wrote:

> 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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>
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>

--
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.