A387455 Partial sums of squares of A000975.

0, 1, 5, 30, 130, 571, 2335, 9560, 38460, 154741, 619865, 2483090, 9935990, 39758511, 159048595, 636252620, 2545068720, 10180507881, 40722264525, 162889990150, 651560892650, 2606247298851, 10424992923655, 41699986607680, 166799961343780, 667199905027421

Offset

0,3

References

P. K. Stockmeyer, An exploration of sequence A000975, Fibonacci Quart., 55(5), (2017), 174-185.

Links

Paolo Xausa, Table of n, a(n) for n = 0..1500

A. M. Hinz, The Lichtenberg sequence, Fibonacci Quart., 55(2), (2017), 2-12.

Index entries for linear recurrences with constant coefficients, signature (5,1,-25,16,20,-16).

Formula

a(n) = (1/108)*(4^(n+3) - 9*2^(n+4) - (-2)^(n+4) + 9*(-1)^(n) + 30*n + 87).

a(n) = 5*a(n-1) + a(n-2) - 25*a(n-3) + 16*a(n-4) + 20*a(n-5) - 16*a(n-6).

G.f.: x*(1 + 4*x^2)/((1 - x)^2*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 - 4*x)).

E.g.f.: (32*cosh(4*x) - 80*cosh(2*x) + 3*(13 + 5*x)*sinh(x) + cosh(x)*(48 + 15*x + 256*sinh(x)^3))/54. - Stefano Spezia, Sep 07 2025

Example

a(0) = 0^2 = 0.
a(1) = 0^2 + 1^2 = 1.
a(2) = 0^2 + 1^2 + 2^2 = 5.
a(3) = 0^2 + 1^2 + 2^2 + 5^2 = 30.
a(4) = 0^2 + 1^2 + 2^2 + 5^2 + 10^2 = 130.

Maple

f:= n -> (1/108)*(4^(n+3) - 9*2^(n+4) - (-2)^(n+4) + 9*(-1)^(n) + 30*n + 87):
map(f, [$0..30]); # Robert Israel, Sep 07 2025

Mathematica

Accumulate[Ceiling[2*(2^Range[0, 25] - 1)/3]^2] (* or *)
LinearRecurrence[{5, 1, -25, 16, 20, -16}, {0, 1, 5, 30, 130, 571}, 26] (* Paolo Xausa, Sep 12 2025 *)

Crossrefs

Cf. A000975.

Sequence in context: A000649 A027173 A286973 * A196452 A227267 A213260

Adjacent sequences: A387452 A387453 A387454 * A387456 A387457 A387458

Keyword

nonn,easy

Author

Gamaliel Morales, Aug 29 2025

Status

approved