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
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
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Aug 28 2007
STATUS
approved