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A104605 Triangle read by rows: row n gives list of powers of phi in the representation of the integer n as a sum of increasing nonconsecutive powers of the golden ratio.

%I A104605 #30 Feb 16 2025 08:32:57
%S A104605 0,-2,1,-2,2,-2,0,2,-4,-1,3,-4,1,3,-4,4,-4,0,4,-4,-2,1,4,-4,-2,2,4,-4,
%T A104605 -2,0,2,4,-6,-3,-1,5,-6,-3,1,5,-6,-3,2,5,-6,-3,0,2,5,-6,-1,3,5,-6,1,3,
%U A104605 5,-6,6,-6,0,6,-6,-2,1,6,-6,-2,2,6,-6,-2,0,2,6,-6,-4,-1,3,6,-6,-4,1,3,6,-6,-4,4,6,-6,-4,0,4,6,-6,-4,-2,1,4,6,-6,-4
%N A104605 Triangle read by rows: row n gives list of powers of phi in the representation of the integer n as a sum of increasing nonconsecutive powers of the golden ratio.
%C A104605 Let f(n) = F(n+1) = A000045(n) and extend n to include negative indices. Then each row n can equally well be thought of as a sequence a_1, a_2,..., a_k such that f(a_1) + f(a_2) + ... + f(a_k) = n. For example, the fifth row is -4 -1 3, so f(-4) + f(-1) + f(3) = 2 + 0 + 3 = 5. - _Dale Gerdemann_, Apr 01 2012
%H A104605 T. D. Noe, <a href="/A104605/b104605.txt">Rows n = 1..1000, flattened</a>
%H A104605 Dale Gerdemann, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/46_47-3/Gerdemann2.pdf">Combinatorial proofs of Zeckendorf family identities</a>, Fib. Quart. 46/47 (2009) 249.
%H A104605 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PhiNumberSystem.html">Phi Number System</a>
%e A104605    0
%e A104605   -2  1
%e A104605   -2  2
%e A104605   -2  0  2
%e A104605   -4 -1  3
%e A104605   -4  1  3
%e A104605   -4  4
%e A104605   -4  0  4
%e A104605   ...
%e A104605 phi^0, phi^(-2) + phi, phi^(-2) + phi^2, phi^(-2) + phi^0 + phi^2, ...
%t A104605 nn = 100; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; Reverse[d[[2]] - Flatten[Position[d[[1]], 1]]], {n, nn}] (* _T. D. Noe_, May 20 2011 *)
%Y A104605 Cf. A055778 (length of row n), A105424, A178482 (phi-antipalindromic numbers).
%K A104605 sign,nice,tabf
%O A104605 1,2
%A A104605 _Eric W. Weisstein_, Mar 17 2005