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%I #51 Sep 17 2023 10:00:15
%S 0,7,24,51,88,135,192,259,336,423,520,627,744,871,1008,1155,1312,1479,
%T 1656,1843,2040,2247,2464,2691,2928,3175,3432,3699,3976,4263,4560,
%U 4867,5184,5511,5848,6195,6552,6919,7296,7683,8080,8487,8904,9331,9768,10215,10672
%N a(n) = n*(2 + 5*n).
%C Appears on the main diagonal of the following table of terms of the Hydrogen series, A169603:
%C 0, 3, 8, 15, 24, ... A005563
%C 0, 7, 16, 1, 40, 55, ... A061039
%C 0, 11, 24, 39, 56, 3, 96, ... A061043
%C 0, 15, 32, 51, 72, 95, 120, ... A061047
%C 0, 19, 40, 63, 88, 115, 144, 175, 208, 1, ...
%H Vincenzo Librandi, <a href="/A168668/b168668.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature(3,-3,1).
%F G.f.: x*(7 + 3*x)/(1-x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F First differences: a(n) - a(n-1) = 10*n-3.
%F Second differences: a(n) - 2*a(n-1) + a(n-2) = 10 = A010692(n).
%F a(n) = A131242(10n+6). - _Philippe Deléham_, Mar 27 2013
%F a(n) = A000384(n) + 6*A000217(n). - _Luciano Ancora_, Mar 28 2015
%F a(n) = A000217(n) + A000217(3*n). - _Bruno Berselli_, Jul 01 2016
%F E.g.f.: x*(7 + 5*x)*exp(x). - _G. C. Greubel_, Jul 29 2016
%F Sum_{n>=1} 1/a(n) = 5/4 - sqrt(1-2/sqrt(5))*Pi/4 + sqrt(5)*log(phi)/4 - 5*log(5)/8, where phi is the golden ratio (A001622). - _Amiram Eldar_, Sep 17 2023
%p A168668:=n->n*(2+5*n): seq(A168668(n), n=0..50); # _Wesley Ivan Hurt_, Mar 28 2015
%t f[n_]:=n*(2+5*n);f[Range[0,60]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 05 2011*)
%t LinearRecurrence[{3,-3,1},{0,7,24},50] (* _Harvey P. Dale_, Sep 09 2021 *)
%o (Magma) [n*(2+5*n): n in [0..50] ]; // _Vincenzo Librandi_, Aug 06 2011
%o (PARI) vector(50,n,n--;n*(2+5*n)) \\ _Derek Orr_, Jun 26 2015
%Y Cf. A000217, A000384, A001622, A010692, A131242, A254963 (comment).
%Y Cf. A005563, A061039, A061043, A061047, A169603.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, Dec 02 2009
%E Edited and extended by _R. J. Mathar_, Dec 05 2009
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