A171071 -id:A171071 - cp's OEIS frontend

A178482 Phi-antipalindromic numbers.

1, 3, 4, 7, 8, 10, 11, 18, 19, 21, 22, 25, 26, 28, 29, 47, 48, 50, 51, 54, 55, 57, 58, 65, 66, 68, 69, 72, 73, 75, 76, 123, 124, 126, 127, 130, 131, 133, 134, 141, 142, 144, 145, 148, 149, 151, 152, 170, 171, 173, 174

Offset: 1

Vladimir Shevelev, May 28 2010

We call m a phi-antipalindromic number if for the vector (a,...,b) (a<...=2, either a(n)+1 or a(n)-1 is in the sequence; also either a(n)+3 or a(n)-3 is in the sequence.
Conjecture: this is the sequence of numbers k for which f(k) is an integer, where f(x) is the change-of-base function defined at A214969 using b=phi and c=b^2. - Clark Kimberling, Oct 17 2012
There is a 21-state automaton accepting the Zeckendorf representations of those n in this sequence. - Jeffrey Shallit, May 03 2023
Kimberling's conjecture has been proven by Ingrid Vukusic and myself. Along the way we prove an alternate characterization of the sequence:  they are the positive integers whose base-phi expansion consists only of even exponents of phi. - Jeffrey Shallit, Aug 28 2025
Alternatively, this sequence consists of those numbers k such that either k or k-1 can be written as the (possibly empty) sum of distinct Lucas numbers L_i where i>=2 and i is even. - Jeffrey Shallit, Aug 28 2025

			The vectors of exponents of 4 and 5 are (-2,0,2) and (-4,-1,3) correspondingly (cf.A104605). Therefore by definition 4 is a phi-antipalindromic number, while 5 is not. Let n=38. Then k=5. Thus a(38)=A005248(5)+a(6)=123+10=133. The vector of exponents of phi in the base-phi expansion of 133 is (-10,-4,-2,2,4,10).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3071 from R. J. Mathar)
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.

Cf. A005248, A055778, A104605, A104626, A104627, A104628.

For bisections see A171070, A171071.

phiAPQ[1] = True; phiAPQ[n_] := Module[{d = RealDigits[n, GoldenRatio, 2*Ceiling[Log[GoldenRatio, n]]]}, e = d[[2]] - Flatten @ Position[d[[1]], 1]; Reverse[e] == -e]; Select[Range[200], phiAPQ] (* Amiram Eldar, Apr 23 2020 *)

For k>=1, a(2^k)=A005248(k); if 2^k

A171070 A bisection of A178482.

Original entry on oeis.org

1, 4, 8, 11, 19, 22, 26, 29, 48, 51, 55, 58, 66, 69, 73, 76, 124, 127, 131, 134, 142, 145, 149, 152, 171, 174, 178, 181, 189, 192, 196, 199, 323, 326, 330, 333, 341, 344, 348, 351, 370, 373, 377, 380, 388, 391, 395, 398, 446, 449, 453, 456, 464, 467, 471, 474, 493, 496, 500

Offset: 1

N. J. A. Sloane, Sep 05 2010

nonn

An initial 1, followed by terms of A171071 plus 1. - Hans Havermann, Sep 05 2010

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A171071, A178482.

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A178482 Phi-antipalindromic numbers.

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