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

0,2

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

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).

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

Example

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, ...).

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 *)

Prog

(Magma) [n*2^(n-1) + 2^n + 1 - 0^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 26 2014
(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
(GAP) Concatenation([1], List([1..30], n-> n*2^(n-1) + 2^n + 1 )); # G. C. Greubel, Nov 20 2019

Crossrefs

Cf. A065190, A132749, A135224.

Sequence in context: A103040 A084861 A122498 * A297296 A048582 A069231

Adjacent sequences: A132747 A132748 A132749 * A132751 A132752 A132753

Keyword

nonn,easy

Author

Gary W. Adamson, Aug 28 2007

Status

approved