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%I #36 Sep 08 2022 08:46:09
%S 1,8,29,64,113,176,253,344,449,568,701,848,1009,1184,1373,1576,1793,
%T 2024,2269,2528,2801,3088,3389,3704,4033,4376,4733,5104,5489,5888,
%U 6301,6728,7169,7624,8093,8576,9073,9584,10109,10648,11201,11768,12349,12944,13553
%N a(n) = 7*n^2 + 1.
%H Michael De Vlieger, <a href="/A247541/b247541.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.: (1 + 5*x + 8*x^2)/(1 - x)^3. - _Vincenzo Librandi_, Sep 19 2014
%F From _Amiram Eldar_, Jul 15 2020: (Start)
%F Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(7))*coth(Pi/sqrt(7)))/2.
%F Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(7))*csch(Pi/sqrt(7)))/2. (End)
%F From _Amiram Eldar_, Feb 05 2021: (Start)
%F Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(7))*sinh(sqrt(2/7)*Pi).
%F Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(7))*csch(Pi/sqrt(7)). (End)
%F E.g.f.: exp(x)*(1 + 7*x + 7*x^2). - _Stefano Spezia_, Feb 05 2021
%t a247541[n_Integer] := 7 n^2 + 1; a247541 /@ Range[0, 120] (* _Michael De Vlieger_, Sep 18 2014 *)
%t CoefficientList[Series[(1 + 5 x + 8 x^2)/(1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Sep 19 2014 *)
%t LinearRecurrence[{3,-3,1},{1,8,29},50] (* _Harvey P. Dale_, Jun 09 2015 *)
%o (Python)
%o for n in range (0,500) : print (7*n**2+1)
%o (PARI) vector(100,n,7*(n-1)^2+1) \\ _Derek Orr_, Sep 18 2014
%o (Magma) [7*n^2+1: n in [0..50]]; // _Vincenzo Librandi_, Sep 19 2014
%Y Cf. A201602 (primes of the form 7n^2 + 1).
%K nonn,easy
%O 0,2
%A _Karl V. Keller, Jr._, Sep 18 2014
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