A250493 -id:A250493 - cp's OEIS frontend

A054783 (n^2)-th Fibonacci number.

0, 1, 3, 34, 987, 75025, 14930352, 7778742049, 10610209857723, 37889062373143906, 354224848179261915075, 8670007398507948658051921, 555565404224292694404015791808, 93202207781383214849429075266681969, 40934782466626840596168752972961528246147

Offset: 0

Jeff Burch, May 22 2000

nonn

The sequence (5*a(n+1)){n>=1} = (5, 15, 170, 4935, ...) is realizable in the sense that there is a self-map on a set T:X->X with the property that a(n) = #{x in X:T^nx=x} for all n >= 1. This is the simplest illustrative example of two different phenomena. The Fibonacci sequence sampled along an odd power cannot be made realizable after multiplication by a constant; the Fibonacci sequence sampled along an even power becomes realizable after multiplication by 5 (the discriminant of the sequence). This is now known to be an instance of a more general phenomenon in the following sense. If (a(n)) is a linear recurrence sequence whose characteristic polynomial F has simple zeros then the sequence (Ma(n^s)) satisfies the Dold congruence, where M=|discriminant(F)| and s is an integer multiple of the exponent of the Galois group of the splitting field of F over the rationals. Under an additional hypothesis on the signs of the coefficients of F, the sequence (Ma(n^s)) is realizable. - _Thomas Ward, May 06 2022

Seiichi Manyama, Table of n, a(n) for n = 0..69
Jakub Byszewski, Grzegorz Graff and Thomas Ward, Dold sequences, periodic points, and dynamics, arXiv:2007.04031 [math.DS], 2020-2021; Bull. Lond. Math. Soc. 53 (2021), no. 5, 1263-1298.
T. Kotek and J. A. Makowsky, Recurrence Relations for Graph Polynomials on Bi-iterative Families of Graphs, arXiv preprint arXiv:1309.4020 [math.CO], 2013.
Florian Luca and Tom Ward, On (almost) realizable subsequences of linearly recurrent sequences, arXiv:2204.02711 [math.NT], 2022.
Piotr Miska and Tom Ward, Stirling numbers and periodic points, arXiv:2102.07561 [math.NT], 2021; Acta Arith. 201 (2021), no. 4, 421-435.
Patrick Moss and Tom Ward, Fibonacci along even powers is (almost) realizable, arXiv:2011.13068 [math.NT], 2020; Fibonacci Quart. 60 (2022), no. 1, 40-47.

Cf. (n^k)-th Fibonacci number: A000045 (k=1), this sequence (k=2), A182149 (k=3), A250490 (k=4), A250491 (k=5), A250492 (k=6), A250493 (k=7), A250494 (k=8).
Cf. A081667.
Cf. A341617 shows a similar property for the Stirling numbers of the second kind.

[Fibonacci(n^2): n in [0..50]]; // Vincenzo Librandi, Apr 09 2011

a:= n-> (<<0|1>, <1|1>>^(n^2))[1, 2]:
seq(a(n), n=0..15);  # Alois P. Heinz, Jun 10 2018

Table[Fibonacci[n^2], {n, 15}] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

a(n)=fibonacci(n^2) \\ Charles R Greathouse IV, Oct 07 2016

a(n) = Sum_{k=1..T(n-1)+1} binomial(T(n-1), k-1)*F(n-1+k), where F(n) is A000045 and T(n) is A000217. - Tony Foster III, Sep 03 2018

A250486 A(n,k) is the n^k-th Fibonacci number; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 2, 1, 0, 1, 21, 34, 3, 1, 0, 1, 987, 196418, 987, 5, 1, 0, 1, 2178309, 37889062373143906, 10610209857723, 75025, 8, 1

Offset: 0

Alois P. Heinz, Nov 24 2014

			Square array A(n,k) begins:
  1,  0,  0,      0,       0,    0,  0,  0,   ...
  1,  1,  1,      1,       1,    1,  1,  1,   ...
  1,  1,  3,      21,      987,  2178309,     ...
  1,  2,  34,     196418,  37889062373143906, ...
  1,  3,  987,    10610209857723,             ...
  1,  5,  75025,  59425114757512643212875125, ...
  1,  8,  14930352,                           ...
  1,  13, 7778742049,                         ...

Alois P. Heinz, Antidiagonals n = 0..10, flattened
Wikipedia, Fibonacci number

Columns k=0-8 give: A000012, A000045, A054783, A182149, A250490, A250491, A250492, A250493, A250494.
Rows n=0-10 give: A000007, A000012, A058635, A045529, A145231, A145232, A145233, A145234, A250487, A250488, A250489.
Main diagonal gives A250495.
Cf. A000045.

A:= (n, k)-> (<<0|1>, <1|1>>^(n^k))[1, 2]:
seq(seq(A(n, d-n), n=0..d), d=0..8);

A[n_, k_] := MatrixPower[{{0, 1}, {1, 1}}, n^k][[1, 2]]; A[0, 0] = 1;
Table[A[n, d-n], {d, 0, 8}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 28 2019, from Maple *)

A(n,k) = [0, 1; 1, 1]^(n^k)[1,2].

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A054783 (n^2)-th Fibonacci number.

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