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

Author

Gary W. Adamson, Aug 08 2007

Keywords

• nonn
• easy

Comments

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

Examples

			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).
		

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).

Crossrefs

Cf. A000325, A095768, A132044.

Programs

Magma

[1] cat [2^n -n +1: n in [1..35]]; // G. C. Greubel, Feb 12 2021

Mathematica

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

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

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)