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A097176 a(n) = Sum_{k=0..n} binomial(floor((n+1)/2), floor((k+1)/2)) * 5^k. 3
1, 6, 31, 186, 811, 5466, 21091, 157746, 548371, 4492026, 14257651, 126558306, 370698931, 3534656586, 9638172211, 98004586866, 250592477491, 2700707149146, 6515404414771, 74033083143426, 169400514784051, 2020227593369706 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = (6/5)*{1, 26, 26, 676, 676, 17576, ...} - 25*{0, 1, 0, 25, 0, 625, ...} - (1/5)*{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 (1,51,-51,-650,650).

FORMULA

G.f.: (1 +5*x -26*x^2 -100*x^3)/((1-x)*(1-25*x^2)*(1-26*x^2)).

a(n) = (3/5)*((1-sqrt(26))*(-sqrt(26))^n + (1+sqrt(26))*(sqrt(26))^n) - (5/2)*(5^n - (-5)^n) - 1/5.

a(n) = a(n-1) + 51*a(n-2) - 51*a(n-3) - 650*a(n-4) + 650*a(n-5).

MAPLE

seq(coeff(series((1+5*x-26*x^2-100*x^3)/((1-x)*(1-25*x^2)*(1-26*x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 17 2019

MATHEMATICA

CoefficientList[Series[(1+5*x-26*x^2-100*x^3)/((1-x)*(1-25*x^2)*(1- 26*x^2)), {x, 0, 30}], x] (* G. C. Greubel, Sep 17 2019 *)

LinearRecurrence[{1, 51, -51, -650, 650}, {1, 6, 31, 186, 811}, 30] (* Harvey P. Dale, Oct 27 2020 *)

PROG

(PARI) my(x='x+O('x^30)); Vec((1+5*x-26*x^2-100*x^3)/((1-x)*(1-25*x^2)*(1-26*x^2))) \\ G. C. Greubel, Sep 17 2019

(MAGMA) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (1+5*x-26*x^2-100*x^3)/((1-x)*(1-25*x^2)*(1-26*x^2)) )); // G. C. Greubel, Sep 17 2019

(Sage)

def A097194_list(prec):

    P.<x> = PowerSeriesRing(QQ, prec)

    return P((1+5*x-26*x^2-100*x^3)/((1-x)*(1-25*x^2)*(1-26*x^2))).list()

A097194_list(20) # G. C. Greubel, Sep 17 2019

(GAP) a:=[1, 6, 31, 186, 811];; for n in [6..30] do a[n]:=a[n-1]+51*a[n-2] -51*a[n-3]-650*a[n-4]+650*a[n-5]; od; a; # G. C. Greubel, Sep 17 2019

CROSSREFS

Cf. A097175, A097177.

Sequence in context: A275403 A338674 A199320 * A121754 A200775 A244255

Adjacent sequences:  A097173 A097174 A097175 * A097177 A097178 A097179

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jul 30 2004

STATUS

approved

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Last modified September 28 10:48 EDT 2021. Contains 347714 sequences. (Running on oeis4.)