A099754 a(n) = (3^n +1)/2 + 2^n.

2, 4, 9, 22, 57, 154, 429, 1222, 3537, 10354, 30549, 90622, 269817, 805354, 2407869, 7207222, 21588897, 64701154, 193972389, 581655022, 1744440777, 5232273754, 15694724109, 47079978022, 141231545457, 423677859154, 1271000023029

Offset: 0

Author

Miklos Kristof, Nov 11 2004

Keywords

• easy
• nonn

Comments

Let b(0)=1, b(n) = A005578(n-1) = {1,1,2,3,6,11,22,43,86,171,342, ...} then a(n) = Sum_{k=0..n+1} C(n+1,k)*b(k).

Binomial transform of A135351. - R. J. Mathar, Aug 05 2009

Examples

			a(6) = (3^6+1)/2 + 2^6 = 365+64 = 429.
a(6) = 1 + 7*1 + 21*1 + 35*2 + 35*3 + 21*6 + 7*11 + 1*22 = 429.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Yilmaz Simsek, New families of special numbers for computing negative order Euler numbers, arXiv:1604.05601 [math.NT], 2016.
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Crossrefs

Cf. A005578.

Programs

GAP

List([0..30], n-> (3^n +2^(n+1) +1)/2); # G. C. Greubel, Sep 03 2019

Magma

[(3^n +2^(n+1) +1)/2: n in [0..30]]; // G. C. Greubel, Sep 03 2019

Maple

seq((3^n +2^(n+1) +1)/2, n=0..30); # G. C. Greubel, Sep 03 2019

Mathematica

Table[(3^n +2^(n+1) +1)/2, {n,0,30}] (* G. C. Greubel, Sep 03 2019 *)
LinearRecurrence[{6,-11,6},{2,4,9},30] (* Harvey P. Dale, May 23 2021 *)

PARI

a(n) = (3^n+1)/2 + 2^n; \\ Michel Marcus, Aug 15 2013

Sage

[(3^n +2^(n+1) +1)/2 for n in (0..30)] # G. C. Greubel, Sep 03 2019

Formula

a(n) = (3^n + 2^(n+1) + 1)/2.

G.f.: (2-8*x+7*x^2)/((1-x)*(1-2*x)*(1-3*x)). - Jaume Oliver Lafont, Mar 06 2009

a(n) = A007051(n) + A000079(n). - Michel Marcus, Aug 15 2013

E.g.f.: (exp(x) + 2*exp(2*x) + exp(3*x))/2. - G. C. Greubel, Sep 03 2019

Extensions

Corrected and extended by T. D. Noe, Nov 07 2006