A132925 a(n) = 2^n - 1 + n*(n-1)/2.

1, 4, 10, 21, 41, 78, 148, 283, 547, 1068, 2102, 4161, 8269, 16474, 32872, 65655, 131207, 262296, 524458, 1048765, 2097361, 4194534, 8388860, 16777491, 33554731, 67109188, 134218078, 268435833, 536871317, 1073742258, 2147484112

Offset: 1

Author

Gary W. Adamson, Sep 05 2007

Keywords

• nonn
• easy

Comments

Row sums of triangle A132924. n-th Mersenne number + (n-1)-th triangular number.

Partial sums of A006127. - Jaroslav Krizek, Oct 16 2009

Examples

			a(4) = 21 = sum of row 4 terms of triangle A132924: (4 + 4 + 5 + 8).
a(4) = 21 = (1, 3, 3, 1) dot (1, 3, 3, 2) = (1 + 9 + 9 + 2).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Crossrefs

Cf. A006127, A132924.

Cf. A000225, A000217.

Programs

Magma

[2^n - 1 + n*(n-1)/2: n in [1..40]]; // Vincenzo Librandi, Jun 21 2011

Maple

A132925 := proc(n) 2^n-1+n*(n-1)/2 ; end proc; # R. J. Mathar, Oct 23 2009

Mathematica

Table[2^n-1+n (n-1)/2,{n,40}] (* or *) LinearRecurrence[{5,-9,7,-2},{1,4,10,21},40] (* Harvey P. Dale, Jun 19 2011 *)

PARI

a(n)=2^n+binomial(n,2)-1 \\ Charles R Greathouse IV, Jun 20 2011

Formula

Binomial transform of [1, 3, 3, 2, 2, 2, 2, ...].

a(n) = A000225(n) + A000217(n-1). - Jaroslav Krizek, Oct 16 2009

From Harvey P. Dale, Jun 19 2011: (Start)

a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4); a(1)=1, a(2)=4, a(3)=10, a(4)=21.

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

E.g.f.: exp(x)*(2*exp(x) + x^2 - 2)/2. - Stefano Spezia, Feb 01 2025