A053537 Expansion of 1/((1+5*x)*(1-15*x)).

1, 10, 175, 2500, 38125, 568750, 8546875, 128125000, 1922265625, 28832031250, 432490234375, 6487304687500, 97309814453125, 1459645996093750, 21894696044921875, 328420410156250000, 4926306304931640625, 73894593811035156250, 1108418910980224609375

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..845
• Index entries for linear recurrences with constant coefficients, signature (10,75).

Crossrefs

Cf. A015518.

Programs

GAP

a:=[1,10];; for n in [3..30] do a[n]:=10*a[n-1]+75*a[n-2]; od; a; # G. C. Greubel, May 16 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/((1+5*x)*(1-15*x)) )); // G. C. Greubel, May 16 2019

Mathematica

LinearRecurrence[{10, 75}, {1, 10}, 30] (* G. C. Greubel, May 16 2019 *)
CoefficientList[Series[1/((1+5x)(1-15x)),{x,0,20}],x] (* Harvey P. Dale, Jun 15 2022 *)

PARI

Vec(1/((1+5*x)*(1-15*x)) + O(x^30)) \\ Michel Marcus, Dec 03 2014

Sage

(1/((1+5*x)*(1-15*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 16 2019

Formula

a(n) = (5^n/4)*(3^(n+1) + (-1)^n).

a(n) = 10*a(n-1) + 75*a(n-2), with a(0)=1, a(1)=10.

E.g.f.: (3*exp(15*x) + exp(-5*x))/4. - G. C. Greubel, May 16 2019

Extensions

Terms a(11) onward added by G. C. Greubel, May 16 2019