A053536 Expansion of 1/((1+4*x)*(1-12*x)).

1, 8, 112, 1280, 15616, 186368, 2240512, 26869760, 322502656, 3869769728, 46438285312, 557255229440, 6687079530496, 80244887257088, 962938915520512, 11555265912504320, 138663195245019136, 1663958325760360448, 19967499977843802112, 239609999459247718400

Offset: 0

Author

Barry E. Williams, Jan 15 2000

Keywords

• easy
• nonn

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..920
• Index entries for linear recurrences with constant coefficients, signature (8,48).

Crossrefs

Cf. A015518.

Programs

GAP

a:=[1,8];; for n in [3..30] do a[n]:=8*a[n-1]+48*a[n-2]; od; a; # G. C. Greubel, May 16 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/((1+4*x)*(1-12*x)) )); // G. C. Greubel, May 16 2019

Mathematica

LinearRecurrence[{8,48}, {1,8}, 30] (* G. C. Greubel, May 16 2019 *)

PARI

Vec(1/((1+4*x)*(1-12*x)) + O(x^30)) \\ Michel Marcus, Dec 03 2014

Sage

(1/((1+4*x)*(1-12*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 16 2019

Formula

a(n) = (4^n/4)*(3^(n+1) + (-1)^n).

a(n) = 8*a(n-1) + 48*a(n-2), with a(0)=1, a(1)=8.

E.g.f.: (3*exp(12*x) + exp(-4*x))/4. - G. C. Greubel, May 16 2019

a(n) = 2^n*A053524(n+1). - R. J. Mathar, Mar 08 2021

Extensions

Terms a(12) onward added by G. C. Greubel, May 16 2019