A176691 a(n) = 2^n + 2*n + 1.

2, 5, 9, 15, 25, 43, 77, 143, 273, 531, 1045, 2071, 4121, 8219, 16413, 32799, 65569, 131107, 262181, 524327, 1048617, 2097195, 4194349, 8388655, 16777265, 33554483, 67108917, 134217783, 268435513, 536870971, 1073741885, 2147483711, 4294967361, 8589934659, 17179869253

Offset: 0

Author

Jonathan Vos Post, Apr 23 2010

Keywords

• nonn
• easy

Comments

The subsequence of primes in this sequence is A163115.

Also the number of connected dominating sets in the (n+1)-wheel graph. - Eric W. Weisstein, Aug 30 2017

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000
• Eric Weisstein's World of Mathematics, Connected Dominating Set
• Eric Weisstein's World of Mathematics, Wheel Graph
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A000051, A000079, A000290, A005126, A005843, A163115, A194455.

Programs

GAP

List([0..35],n->2^n+2*n+1); # Muniru A Asiru, Mar 25 2018

Magma

[2^n + 2*n + 1: n in [0..40]]; // Vincenzo Librandi, Aug 12 2015

Maple

seq(2^n+2*n+1,n=0..35); # Muniru A Asiru, Mar 25 2018

Mathematica

Table[2^n + 2 n + 1, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)
LinearRecurrence[{4, -5, 2}, {2, 5, 9}, 40] (* Vincenzo Librandi, Aug 12 2015 *)
CoefficientList[Series[(-2 + 3 x + x^2)/((-1 + x)^2 (-1 + 2 x)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 30 2017 *)

PARI

vector(40, n, n--; 2^n + 2*n + 1) \\ Michel Marcus, Aug 12 2015

Formula

a(n) = 2^n + 2*n + 1 = A000079(n) + A005843(n) + 1 = A000051(n) + A005843(n).

From R. J. Mathar, Apr 28 2010: (Start)

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).

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

E.g.f.: exp(x)*(1 + exp(x) + 2*x). - Stefano Spezia, May 06 2023

Extensions

Corrected (one 1048617 replaced by 2097195) by R. J. Mathar, Apr 28 2010