A027266 a(n) = Sum_{k=0..2n} (k+1) * A026519(n, k).

1, 6, 18, 72, 180, 648, 1512, 5184, 11664, 38880, 85536, 279936, 606528, 1959552, 4199040, 13436928, 28553472, 90699264, 191476224, 604661760, 1269789696, 3990767616, 8344332288, 26121388032, 54419558400, 169789022208

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,12,0,-36).

Formula

a(n) = Sum_{k=0..2n} (k+1) * A026519(n, k).

G.f.: (1+6*x+6*x^2)/(1-6*x^2)^2.

a(n) = 12*a(n-2) - 36*a(n-4), with a(0)=1, a(1)=6, a(2)=18, a(3)=72. - Harvey P. Dale, Jun 19 2015

a(n) = ((n+1)/2)*6^((n-1)/2)*( 3*(1-(-1)^n) + sqrt(6)*(1+(-1)^n) ). - G. C. Greubel, Dec 21 2021

Mathematica

CoefficientList[Series[(1+6x+6x^2)/(1-6x^2)^2, {x, 0, 30}], x] (* or *) LinearRecurrence[{0, 12, 0, -36}, {1, 6, 18, 72}, 30] (* Harvey P. Dale, Jun 19 2015 *)

Prog

(Magma) I:=[1, 6, 18, 72]; [n le 4 select I[n] else 12*(Self(n-2) - 3*Self(n-4)): n in [1..41]]; // G. C. Greubel, Dec 21 2021
(Sage) [((n+1)/2)*6^((n-1)/2)*( 3*(1-(-1)^n) + sqrt(6)*(1+(-1)^n) ) for n in (0..40)] # G. C. Greubel, Dec 21 2021
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -36, 0, 12, 0]^n*[1; 6; 18; 72])[1, 1] \\ Charles R Greathouse IV, Oct 18 2022

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026532, A026533, A026534, A027262, A027263, A027264, A027265.

Sequence in context: A304941 A129369 A095853 * A242278 A129796 A129790

Adjacent sequences: A027263 A027264 A027265 * A027267 A027268 A027269

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved