A290263 Positive integers n such that the Zeckendorf (Fibonacci) representation of n^2 consists of two consecutive identical blocks.

4, 49, 306, 728, 2021, 3556, 3740, 5236, 21360, 35244, 98210, 243252, 1096099, 1625040, 1662860, 4976785, 5080514, 11408968, 31622994, 31831002, 33587514, 33599070, 56568930, 78167976, 209645093, 218297737, 220158358, 223289647, 225150268, 238764568, 535850484

Offset

1,1

References

Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, Experimental Math, 28 (2019), 428-439.

Links

Lars Blomberg, Table of n, a(n) for n = 1..47

Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, preprint arXiv:1707.03894 [math.NT], July 14 2017.

Example

For example, for n = 4, we have n^2 in Fibonacci representation is 100100, which consists of two consecutive blocks of 100.

Prog

(PARI) Z(n)=my(k=0, v, m); while(fibonacci(k)<=n, k=k+1); m=k-1; v=vector(m-1); v[1]=1; n=n-fibonacci(k-1); while(n>0, k=0; while(fibonacci(k)<=n, k=k+1); v[m-k+2]=1; n=n-fibonacci(k-1)); v; \\ after A014417
isok(n) = {my(vz = Z(n^2)); if (!(#vz % 2), vector(#vz/2, k, vz[k]) == vector(#vz/2, k, vz[k+#vz/2]); ); } \\ Michel Marcus, Aug 02 2017

Crossrefs

Cf. A014417.

Sequence in context: A163944 A000596 A113525 * A224538 A064751 A045787

Adjacent sequences: A290260 A290261 A290262 * A290264 A290265 A290266

Keyword

nonn

Author

Jeffrey Shallit, Jul 25 2017

Extensions

a(23)-a(31) from Lars Blomberg, Aug 02 2017

Status

approved