A159694 a(n) = 2*a(n-1) + 2^(n-1), for n > 0, with a(0)=6.

6, 13, 28, 60, 128, 272, 576, 1216, 2560, 5376, 11264, 23552, 49152, 102400, 212992, 442368, 917504, 1900544, 3932160, 8126464, 16777216, 34603008, 71303168, 146800640, 301989888, 620756992, 1275068416, 2617245696, 5368709120

Offset: 0

Author

Philippe Deléham, Apr 20 2009

Keywords

• easy
• nonn

Comments

Diagonal of triangles A062111, A152920.

Examples

			a(0) = 6,
a(1) = 2* 6 + 1 =  13,
a(2) = 2*13 + 2 =  28,
a(3) = 2*28 + 4 =  60,
a(4) = 2*60 + 8 = 128, ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Milan Janjić and Boris Petković, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
• Milan Janjić and Boris Petković, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq., Vol. 17 (2014), Article 14.3.5.
• Index entries for linear recurrences with constant coefficients, signature (4,-4).

Crossrefs

Cf. A000079, A001787, A001792, A034007, A045623.

Cf. A045891, A062111, A111297, A152920.

Seventh row of triangle A062111. - Klaus Brockhaus, Sep 27 2009

Programs

Magma

[(12+n)*2^(n-1): n in [0..30]]; // G. C. Greubel, Sep 27 2022

Mathematica

Table[(6 + n/2)*2^n, {n, 0, 30}] (* Amiram Eldar, Jan 19 2021 *)

SageMath

[(12+n)*2^(n-1) for n in range(30)] # G. C. Greubel, Sep 27 2022

Formula

a(n) = Sum_{k=0..n} (k+6)*binomial(n,k).

From Klaus Brockhaus, Sep 27 2009: (Start)

a(n) = (6 + n/2)*2^n.

G.f.: (6 - 11*x)/(1-2*x)^2. (End)

From Amiram Eldar, Jan 19 2021: (Start)

Sum_{n>=0} 1/a(n) = 8192*log(2) - 3934820/693.

Sum_{n>=0} (-1)^n/a(n) = 11509636/3465 - 8192*log(3/2). (End)

E.g.f.: (6 + x)*exp(2*x). - G. C. Greubel, Sep 27 2022