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%I #21 Jun 30 2023 00:47:01
%S 7,15,32,68,144,304,640,1344,2816,5888,12288,25600,53248,110592,
%T 229376,475136,983040,2031616,4194304,8650752,17825792,36700160,
%U 75497472,155189248,318767104,654311424,1342177280,2751463424,5637144576
%N a(0)=7, a(n) = 2*a(n-1) + 2^(n-1) for n > 0.
%C Diagonal of triangles A062111, A152920.
%H G. C. Greubel, <a href="/A159695/b159695.txt">Table of n, a(n) for n = 0..3300</a>
%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+7)*binomial(n,k).
%F From _R. J. Mathar_, Apr 20 2009: (Start)
%F a(n) = (14+n)*2^(n-1).
%F a(n) = 4*a(n-1) - 4*a(n-2).
%F G.f.: (7-13*x)/(1-2x)^2. (End)
%F E.g.f.: (x+7)*exp(2*x). - _G. C. Greubel_, Jun 02 2018
%F From _Amiram Eldar_, Jan 19 2021: (Start)
%F Sum_{n>=0} 1/a(n) = 32768*log(2) - 204619418/9009.
%F Sum_{n>=0} (-1)^n/a(n) = 598484902/45045 - 32768*log(3/2). (End)
%e a(0)=7, a(1) = 2*7 + 1 = 15, a(2) = 2*15 + 2 = 32, a(3) = 2*32 + 4 = 68, a(4) = 2*68 + 8 = 144, ...
%t LinearRecurrence[{4,-4}, {7,15}, 30] (* or *) Table[(14+n)*2^(n-1), {n, 0, 30}] (* _G. C. Greubel_, Jun 02 2018 *)
%t nxt[{n_,a_}]:={n+1,2a+2^n}; NestList[nxt,{0,7},30][[All,2]] (* _Harvey P. Dale_, Jan 01 2023 *)
%o (PARI) for(n=0, 30, print1((14+n)*2^(n-1), ", ")) \\ _G. C. Greubel_, Jun 02 2018
%o (Magma) [(14+n)*2^(n-1): n in [0..30]]; // _G. C. Greubel_, Jun 02 2018
%Y Cf. A000079, A001787, A001792, A045623, A045891, A034007, A111297, A159694.
%K easy,nonn
%O 0,1
%A _Philippe Deléham_, Apr 20 2009
%E More terms from _R. J. Mathar_, Apr 20 2009
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