A132045 Row sums of triangle A132044.

1, 2, 3, 6, 13, 28, 59, 122, 249, 504, 1015, 2038, 4085, 8180, 16371, 32754, 65521, 131056, 262127, 524270, 1048557, 2097132, 4194283, 8388586, 16777193, 33554408, 67108839, 134217702, 268435429, 536870884, 1073741795, 2147483618, 4294967265, 8589934560

Offset

0,2

Comments

Apart from initial terms, and with a change of offset, same as A095768. - Jon E. Schoenfield, Jun 15 2017

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Formula

Binomial transform of (1, 1, 0, 2, 0, 2, 0, 2, 0, 2, ...).

For n>=1, a(n) = 2^n - n + 1 = A000325(n) + 1. - Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 17 2009. (Corrected by Franklin T. Adams-Watters, Jan 17 2009)

E.g.f.: U(0)- 1, where U(k) = 1 - x/(2^k + 2^k/(x - 1 - x^2*2^(k+1)/(x*2^(k+1) + (k+1)/U(k+1) ))). - Sergei N. Gladkovskii, Dec 01 2012

From Colin Barker, Mar 14 2014: (Start)

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

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

Example

a(4) = 13 = sum of row 4 terms of triangle A132044: (1 + 3 + 5 + 3 + 1).
a(4) = 13 = (1, 4, 6, 4, 1) dot (1, 1, 0, 2, 0) = (1 + 4 + 0 + 8 + 0).

Mathematica

Table[2^n -(n-1) -Boole[n==0], {n, 0, 35}] (* G. C. Greubel, Feb 12 2021 *)

Prog

(PARI) Vec((1-2*x+2*x^3)/((1-x)^2*(1-2*x)) + O(x^100)) \\ Colin Barker, Mar 14 2014
(Sage) [1]+[2^n -n +1 for n in (1..35)] # G. C. Greubel, Feb 12 2021
(Magma) [1] cat [2^n -n +1: n in [1..35]]; // G. C. Greubel, Feb 12 2021

Crossrefs

Cf. A000325, A095768, A132044.

Sequence in context: A274493 A324770 A075853 * A032143 A032160 A089735

Adjacent sequences: A132042 A132043 A132044 * A132046 A132047 A132048

Keyword

nonn,easy

Author

Gary W. Adamson, Aug 08 2007

Status

approved