A099365 Squares of A052918(n-1) (generalized Fibonacci).

0, 1, 25, 676, 18225, 491401, 13249600, 357247801, 9632441025, 259718659876, 7002771375625, 188815108482001, 5091005157638400, 137268324147754801, 3701153746831741225, 99793882840309258276, 2690733682941518232225

Offset

0,3

Comments

See the comment in A099279. This is example a=5.

a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal) and (1/2,1/2)-fences if there are 5 kinds of half-squares available. A (w,g)-fence is a tile composed of two w X 1 pieces separated horizontally by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/4,1/4)-fences and (1/4,3/4)-fences if there are 5 kinds of (1/4,1/4)-fences available. - Michael A. Allen, Mar 30 2023

Links

G. C. Greubel, Table of n, a(n) for n = 0..650

Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.

Index entries for linear recurrences with constant coefficients, signature (26,26,-1).

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n) = A052918(n-1)^2, n >= 1, a(0) = 0.

a(n) = 26*a(n-1) + 26*a(n-2) - a(n-3), n >= 3; a(0)=0, a(1)=1, a(2)=25.

a(n) = 27*a(n-1) - a(n-2) - 2*(-1)^n, n >= 2; a(0)=0, a(1)=1.

a(n) = 2*(T(n, 27/2) - (-1)^n)/29 with twice the Chebyshev's T(n, x) polynomials of the first kind. 2*T(n, 27/2) = A090248(n).

G.f.: x*(1-x)/((1-27*x+x^2)*(1+x)) = x*(1-x)/(1-26*x-26*x^2+x^3).

a(n) = (1 - (-1)^n)/2 + 25*Sum_{r=1..n-1} r*a(n-r). - Michael A. Allen, Mar 30 2023

Maple

with (combinat):seq(fibonacci(n, 5)^2, n=0..16); # Zerinvary Lajos, Apr 09 2008

Mathematica

LinearRecurrence[{26, 26, -1}, {0, 1, 25}, 30] (* Harvey P. Dale, Sep 25 2019 *)

Prog

(Magma) [(2/29)*(Evaluate(ChebyshevFirst(n), 27/2) -(-1)^n): n in [0..30]]; // G. C. Greubel, Aug 21 2022
(SageMath)
def A099365(n): return (2/29)*(chebyshev_T(n, 27/2) - (-1)^n)
[A099365(n) for n in (0..30)] # G. C. Greubel, Aug 21 2022

Crossrefs

Cf. other squares of k-metallonacci numbers (for k=1 to 10): A007598, A079291, A092936, A099279, this sequence, A099366, A099367, A099369, A099372, A099374.

Sequence in context: A180811 A318183 A015697 * A215017 A266100 A369141

Adjacent sequences: A099362 A099363 A099364 * A099366 A099367 A099368

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 18 2004

Status

approved