A052996 Expansion of (1+x^2-x^3)/((1-x)*(1-2*x)).

1, 3, 8, 17, 35, 71, 143, 287, 575, 1151, 2303, 4607, 9215, 18431, 36863, 73727, 147455, 294911, 589823, 1179647, 2359295, 4718591, 9437183, 18874367, 37748735, 75497471, 150994943, 301989887, 603979775, 1207959551, 2415919103

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 3-4, 14.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1074
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Cf. A050524 (primes of this sequence).

Programs

Magma

[Floor(9*2^(n-2) - 1): n in [0..40]]; // Vincenzo Librandi, Sep 24 2011

Maple

spec := [S, {S=Prod(Union(Prod(Z,Z), Sequence(Z)), Sequence(Union(Z,Z)))}, unlabeled ]: seq(combstruct[count](spec, size=n), n=0..20);

Mathematica

a[0] := 1; a[1] := 3; a[2] := 8; a[n_] := 2*a[n - 1] + 1; Table[a[n], {n, 0, 12}] (* L. Edson Jeffery, Dec 18 2014 *)
CoefficientList[ Series[(1 + x^2 - x^3)/((1 - x) (1 - 2 x)), {x, 0, 30}], x] (* Robert G. Wilson v, Jul 29 2015 *)
LinearRecurrence[{3,-2},{1,3,8,17},40] (* Harvey P. Dale, Feb 11 2018 *)

PARI

Vec((1+x^2-x^3)/((1-x)*(1-2*x)) + O(x^50)) \\ Michel Marcus, Jul 30 2015

Formula

Recurrence: {-2*a(n)+a(n+1)-1=0, a(0)=1, a(1)=3, a(2)=8}.

a(n) = 9*2^(n-2) - 1 for n > 1. - Brad Clardy, Sep 23 2011

Extensions

More terms from James Sellers, Jun 06 2000