A097175 a(n) = Sum_{k=0..n} binomial(floor((n+1)/2), floor((k+1)/2)) * 4^k.

1, 5, 21, 105, 361, 2045, 6141, 38865, 104401, 726245, 1774821, 13394745, 30171961, 244487885, 512923341, 4424729505, 8719696801, 79515368885, 148234845621, 1420480747785, 2519992375561, 25247684340125, 42839870384541

Offset

0,2

Comments

a(n) = (5/4)*{1, 17, 17, 289, 289, 4913, ...} - 16*{0, 1, 0, 16, 0, 256, ...} - (1/4)*{1, 1, 1, 1, 1, 1, ...}.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,33,-33,-272,272).

Formula

G.f.: (1+4*x-17*x^2-48*x^3)/((1-x)*(1-16*x^2)*(1-17*x^2)).

a(n) = (5/8)*((1-sqrt(17))*(-sqrt(17))^n + (1+sqrt(17))*(sqrt(17))^n) - 2*(4^n - (-4)^n) - 1/4.

a(n) = a(n-1) + 33*a(n-2) - 33*a(n-3) - 272*a(n-4) + 272*a(n-5).

Maple

seq(coeff(series((1+4*x-17*x^2-48*x^3)/((1-x)*(1-16*x^2)*(1-17*x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 17 2019

Mathematica

CoefficientList[Series[(1+4*x-17*x^2-48*x^3)/((1-x)*(1-16*x^2)*(1-17*x^2)), {x, 0, 30}], x] (* G. C. Greubel, Sep 17 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1+4*x-17*x^2-48*x^3)/((1-x)*(1-16*x^2)*(1-17*x^2))) \\ G. C. Greubel, Sep 17 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+4*x-17*x^2-48*x^3)/((1-x)*(1-16*x^2)*(1-17*x^2)) )); // G. C. Greubel, Sep 17 2019
(Sage)
def A097175_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P((1+4*x-17*x^2-48*x^3)/((1-x)*(1-16*x^2)*(1-17*x^2))).list()
A097175_list(30) # G. C. Greubel, Sep 17 2019
(GAP) a:=[1, 5, 21, 105, 361];; for n in [6..30] do a[n]:=a[n-1] + 33*a[n-2] - 33*a[n-3] - 272*a[n-4] + 272*a[n-5]; od; a; # G. C. Greubel, Sep 17 2019

Crossrefs

Cf. A097176, A097177.

Sequence in context: A046633 A280623 A203154 * A100284 A337168 A341853

Adjacent sequences: A097172 A097173 A097174 * A097176 A097177 A097178

Keyword

easy,nonn

Author

Paul Barry, Jul 30 2004

Status

approved