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.

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, -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, 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

Offset: 1

Author

Eric W. Weisstein, Mar 17 2005

Keywords

• sign
• nice
• tabf

Comments

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

Examples

			   0
  -2  1
  -2  2
  -2  0  2
  -4 -1  3
  -4  1  3
  -4  4
  -4  0  4
  ...
phi^0, phi^(-2) + phi, phi^(-2) + phi^2, phi^(-2) + phi^0 + phi^2, ...
		

Links

• T. D. Noe, Rows n = 1..1000, flattened
• Dale Gerdemann, Combinatorial proofs of Zeckendorf family identities, Fib. Quart. 46/47 (2009) 249.
• Eric Weisstein's World of Mathematics, Phi Number System

Crossrefs

Cf. A055778 (length of row n), A105424, A178482 (phi-antipalindromic numbers).

Programs

Mathematica

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 *)