cp's OEIS Frontend

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

%I A159694 #29 Sep 28 2022 04:09:05
%S A159694 6,13,28,60,128,272,576,1216,2560,5376,11264,23552,49152,102400,
%T A159694 212992,442368,917504,1900544,3932160,8126464,16777216,34603008,
%U A159694 71303168,146800640,301989888,620756992,1275068416,2617245696,5368709120
%N A159694 a(n) = 2*a(n-1) + 2^(n-1), for n > 0, with a(0)=6.
%C A159694 Diagonal of triangles A062111, A152920.
%H A159694 G. C. Greubel, <a href="/A159694/b159694.txt">Table of n, a(n) for n = 0..1000</a>
%H A159694 Milan Janjić and Boris Petković, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013.
%H A159694 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 A159694 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).
%F A159694 a(n) = Sum_{k=0..n} (k+6)*binomial(n,k).
%F A159694 From _Klaus Brockhaus_, Sep 27 2009: (Start)
%F A159694 a(n) = (6 + n/2)*2^n.
%F A159694 G.f.: (6 - 11*x)/(1-2*x)^2. (End)
%F A159694 From _Amiram Eldar_, Jan 19 2021: (Start)
%F A159694 Sum_{n>=0} 1/a(n) = 8192*log(2) - 3934820/693.
%F A159694 Sum_{n>=0} (-1)^n/a(n) = 11509636/3465 - 8192*log(3/2). (End)
%F A159694 E.g.f.: (6 + x)*exp(2*x). - _G. C. Greubel_, Sep 27 2022
%e A159694 a(0) = 6,
%e A159694 a(1) = 2* 6 + 1 =  13,
%e A159694 a(2) = 2*13 + 2 =  28,
%e A159694 a(3) = 2*28 + 4 =  60,
%e A159694 a(4) = 2*60 + 8 = 128, ...
%t A159694 Table[(6 + n/2)*2^n, {n, 0, 30}] (* _Amiram Eldar_, Jan 19 2021 *)
%o A159694 (Magma) [(12+n)*2^(n-1): n in [0..30]]; // _G. C. Greubel_, Sep 27 2022
%o A159694 (SageMath) [(12+n)*2^(n-1) for n in range(30)] # _G. C. Greubel_, Sep 27 2022
%Y A159694 Cf. A000079, A001787, A001792, A034007, A045623.
%Y A159694 Cf. A045891, A062111, A111297, A152920.
%Y A159694 Seventh row of triangle A062111. - _Klaus Brockhaus_, Sep 27 2009
%K A159694 easy,nonn
%O A159694 0,1
%A A159694 _Philippe Deléham_, Apr 20 2009