A132750 A132749 * [1, 2, 3, ...] = A007318 * A065190.

1, 4, 9, 21, 49, 113, 257, 577, 1281, 2817, 6145, 13313, 28673, 61441, 131073, 278529, 589825, 1245185, 2621441, 5505025, 11534337, 24117249, 50331649, 104857601, 218103809, 452984833, 939524097, 1946157057, 4026531841

Offset: 0

Author

Gary W. Adamson, Aug 28 2007

Keywords

• nonn
• easy

Comments

Equals double binomial transform of [1, 2, -3, 7, -15, 31, -63, 127, -255, ...]. - Gary W. Adamson, Jul 23 2008

For n >= 1, also the number of cliques in the n-hypercube graph Q_n. - Eric W. Weisstein, Mar 31 2017

Examples

			a(3) = 21 = (1, 3, 3, 1) dot (1, 3, 2, 5) = (1 + 9 + 6 + 5) = 21; where A065190 = (1, 3, 2, 5, 4, 7, 6, 9, ...).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• V. Jelinek, T. Mansour, M. Shattuck, On multiple pattern avoiding set partitions, Adv. Appl. Math. 50 (2) (2013) 292-326, Lemma 4.21 (sequence starting 1, 1, 2, 4, 9, 21, .... with offset 0).
• Eric Weisstein's World of Mathematics, Clique
• Eric Weisstein's World of Mathematics, Hypercube Graph
• Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Crossrefs

Cf. A065190, A132749, A135224.

Programs

GAP

Concatenation([1], List([1..30], n-> n*2^(n-1) + 2^n + 1 )); # G. C. Greubel, Nov 20 2019

Magma

[n*2^(n-1) + 2^n + 1 - 0^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 26 2014

Maple

A132750:=n->n*2^(n-1)+2^n+1-0^n: seq(A132750(n), n=0..30); # Wesley Ivan Hurt, Sep 26 2014

Mathematica

Join[{1}, Table[n*2^(n-1) +2^n +1, {n, 30}]] (* Wesley Ivan Hurt, Sep 26 2014 *)
Join[{1}, LinearRecurrence[{5,-8,4}, {4,9,21}, 30]] (* Vincenzo Librandi, Apr 01 2017 *)

PARI

vector(31, n, if(n==1, 1, (n-1)*2^(n-2) + 2^(n-1) + 1)) \\ G. C. Greubel, Nov 20 2019

Sage

[1]+[n*2^(n-1) + 2^n + 1 for n in (1..30)] # G. C. Greubel, Nov 20 2019

Formula

A132749 as an infinite lower triangular matrix * vector [1, 2, 3, ...]. Binomial transform of A065190 (with an incorrect offset)

Row sums of triangle A135224. - Gary W. Adamson, Nov 23 2007

G.f.: (1-x-3*x^2+4*x^3)/((1-x)*(1-2*x)^2). - Colin Barker, Aug 09 2012

a(n) = n*2^(n-1) + 2^n + 1 - 0^n. - Tim Smith, Sep 25 2014

a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3). - Wesley Ivan Hurt, Sep 26 2014

E.g.f.: -1 + exp(x) + (1+x)*exp(2*x). - G. C. Greubel, Nov 20 2019