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

Author

Barry E. Williams, Mar 06 2000

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A000285.

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

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)/((x-1)*(x^2+x-1)))); // G. C. Greubel, May 24 2018

Mathematica

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

PARI

x='x+O('x^30); Vec((1+3*x)/((x-1)*(x^2+x-1))) \\ G. C. Greubel, May 24 2018

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)