A159694 - OEIS

%I #29 Sep 28 2022 04:09:05

%S 6,13,28,60,128,272,576,1216,2560,5376,11264,23552,49152,102400,

%T 212992,442368,917504,1900544,3932160,8126464,16777216,34603008,

%U 71303168,146800640,301989888,620756992,1275068416,2617245696,5368709120

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

%C Diagonal of triangles A062111, A152920.

%H G. C. Greubel, <a href="/A159694/b159694.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjić and Boris Petković, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013.

%H Milan Janjić and Boris Petković, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq., Vol. 17 (2014), Article 14.3.5.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).

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

%F From _Klaus Brockhaus_, Sep 27 2009: (Start)

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

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

%F From _Amiram Eldar_, Jan 19 2021: (Start)

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

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

%F E.g.f.: (6 + x)*exp(2*x). - _G. C. Greubel_, Sep 27 2022

%e a(0) = 6,

%e a(1) = 2* 6 + 1 =  13,

%e a(2) = 2*13 + 2 =  28,

%e a(3) = 2*28 + 4 =  60,

%e a(4) = 2*60 + 8 = 128, ...

%t Table[(6 + n/2)*2^n, {n, 0, 30}] (* _Amiram Eldar_, Jan 19 2021 *)

%o (Magma) [(12+n)*2^(n-1): n in [0..30]]; // _G. C. Greubel_, Sep 27 2022

%o (SageMath) [(12+n)*2^(n-1) for n in range(30)] # _G. C. Greubel_, Sep 27 2022

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

%Y Cf. A045891, A062111, A111297, A152920.

%Y Seventh row of triangle A062111. - _Klaus Brockhaus_, Sep 27 2009

%K easy,nonn

%O 0,1

%A _Philippe Deléham_, Apr 20 2009