A053311 Partial sums of A000285.

1, 5, 10, 19, 33, 56, 93, 153, 250, 407, 661, 1072, 1737, 2813, 4554, 7371, 11929, 19304, 31237, 50545, 81786, 132335, 214125, 346464, 560593, 907061, 1467658, 2374723, 3842385, 6217112, 10059501, 16276617, 26336122

Offset

0,2

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., pp. 189, 194-196.

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.

Links

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

Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Formula

a(n) = a(n-1) + a(n-2) + 4; a(0)=1, a(1)=5; n >= 1.

a(n) = 4*F(n+2) + F(n+1) - 4, where F(k) is A000045(k).

From R. J. Mathar, Apr 29 2013: (Start)

G.f.: ( 1+3*x ) / ( (x-1)*(x^2+x-1) ).

a(n) = A000071(n+3) + 3*A000071(n+2) = A000285(n+2) - 4. (End)

Mathematica

CoefficientList[Series[(1+3*x)/((x-1)*(x^2+x-1)), {x, 0, 50}], x] (* G. C. Greubel, May 24 2018 *)

Prog

(PARI) x='x+O('x^30); Vec((1+3*x)/((x-1)*(x^2+x-1))) \\ G. C. Greubel, May 24 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)/((x-1)*(x^2+x-1)))); // G. C. Greubel, May 24 2018

Crossrefs

Cf. A000285.

a(n) = A101220(4, 1, n+1).

Sequence in context: A191595 A047882 A184260 * A325056 A147390 A345715

Adjacent sequences: A053308 A053309 A053310 * A053312 A053313 A053314

Keyword

easy,nonn

Author

Barry E. Williams, Mar 06 2000

Status

approved