A330722 -id:A330722 - cp's OEIS frontend

A382176 Numbers k such that the period of 1/k when expanded in golden ratio base is palindromic.

1, 2, 36, 38, 644, 646, 682, 11556, 11558, 11592, 12198, 12238

Offset: 1

Amiram Eldar, Mar 17 2025

a(13) > 80000, if it exists.

			  n | a(n) | period (row n of A382172)
  --+------+-----------------------------------------------------------------------
  1 |    1 | 0
  2 |    2 | 0, 1, 0
  3 |   36 | 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
  4 |   38 | 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Cf. A330722, A362781, A373085, A382172, A382173, A382174, A382175.

q[k_] := PalindromeQ[RealDigits[1/k, GoldenRatio, A001175[k], -1][[1]]]; Select[Range[1000], q] (* using A001175[n] from A001175 *)

A330672 Numbers whose base phi representation is symmetrical with respect to the radix point.

Original entry on oeis.org

0, 2, 14, 36, 38, 94, 96, 246, 248, 260, 644, 646, 658, 680, 682, 1686, 1688, 1700, 1722, 1724, 1780, 1782, 4414, 4416, 4428, 4450, 4452, 4508, 4510, 4660, 4662, 4674, 11556, 11558, 11570, 11592, 11594, 11650, 11652, 11802, 11804, 11816, 12200, 12202, 12214

Offset: 1

Rémy Sigrist, Apr 23 2020

There is a 13-state finite automaton that accepts the Zeckendorf expansions of the members of this sequence. - Jeffrey Shallit, May 03 2023

			The first terms, alongside their base phi representation, are:
  n   a(n)  phi(a(n))
  --  ----  -------------------------
   1     0             0.0
   2     2            10.01
   3    14        100100.001001
   4    36      10010000.00001001
   5    38      10010010.01001001
   6    94    1001000000.0000001001
   7    96    1001000010.0100001001
   8   246  100100000000.000000001001
   9   248  100100000010.010000001001
  10   260  100100100100.001001001001

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.
Rémy Sigrist, PARI program for A330672
Wikipedia, Golden ratio base

See A330722 for a weaker variant.

Cf. A057148, A130600, A178482.

PARI
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See Links section.
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A130600(a(n)) belongs to A057148 for any n >= 0.

cp's OEIS Frontend

A382176 Numbers k such that the period of 1/k when expanded in golden ratio base is palindromic.

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