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%I #67 Sep 08 2022 08:46:09
%S 1,10,37,82,145,226,325,442,577,730,901,1090,1297,1522,1765,2026,2305,
%T 2602,2917,3250,3601,3970,4357,4762,5185,5626,6085,6562,7057,7570,
%U 8101,8650,9217,9802,10405,11026,11665,12322,12997,13690,14401,15130,15877,16642,17425,18226,19045,19882
%N a(n) = 9*n^2 + 1.
%C The odd numbers of the form 9n^2 + 1 are listed in A158591 (36n^2 + 1).
%C The even numbers of the form 9n^2 + 1 are given by 36x^2 - 36x + 10, x > 0.
%C Every integer n>0 give three perfect squares and consecutives from 2^2. The formulas for each value of n are: a(n)-6n, a(n)-1 and a(n)+6n. - _Miquel Cerda_, Sep 19 2016
%C These squares are, for n>0, A000290(3*n-1), 3*n and (3n+1) and the sum of them is 3*a(n) - 1. - _Miquel Cerda_, Sep 26 2016
%H Karl V. Keller, Jr., <a href="/A247792/b247792.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 a(n) = (3n)^2 + 1 = 9n^2 + 1 = A016766(n) + 1.
%F G.f.: (1+7*x+10*x^2)/(1-x)^3. - _Vincenzo Librandi_, Sep 27 2014
%F a(n) = ((3n-1)^2 + (3n+1)^2)/2 = (A016790(n-1) + A016778(n))/2. - _Miquel Cerda_, Jun 25 2016
%F From _Ilya Gutkovskiy_, Jun 25 2016: (Start)
%F E.g.f.: (1 + 9*x + 9*x^2)*exp(x).
%F Dirichlet g.f.: 9*zeta(s-2) + zeta(s).
%F Sum_{n>=0} 1/a(n) = (3 + Pi*coth(Pi/3))/6. (End)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - _Wesley Ivan Hurt_, Jun 25 2016
%F Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/3)*csch(Pi/3))/2. - _Amiram Eldar_, Jul 15 2020
%F From _Amiram Eldar_, Feb 05 2021: (Start)
%F Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/3)*sinh(sqrt(2)*Pi/3).
%F Product_{n>=1} (1 - 1/a(n)) = (Pi/3)*csch(Pi/3). (End)
%e a(1) = (2^2 + 4^2)/2 = 3^2 + 1 = 10, a(2) = (5^2 + 7^2)/2 = 6^2 + 1 = 37, a(3) = (8^2 + 10^2)/2 = 9^2 + 1 = 82. - _Miquel Cerda_, Jun 25 2016
%p A247792:=n->9*n^2 + 1: seq(A247792(n), n=0..80); # _Wesley Ivan Hurt_, Jun 25 2016
%t (3Range[0, 49])^2 + 1 (* _Alonso del Arte_, Sep 24 2014 *)
%t CoefficientList[Series[(1 + 7 x + 10 x^2)/(1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Sep 27 2014 *)
%o (Python) for n in range (0,100): print (9*n**2+1)
%o (PARI) a(n)=9*n^2+1 \\ _Charles R Greathouse IV_, Sep 26 2014
%o (Magma) [9*n^2+1: n in [0..60]]; // _Vincenzo Librandi_, Sep 27 2014
%Y Cf. A016766, A158591 (36n^2 + 1), A156226 (primes of the form 9n^2 + 1).
%Y Cf. also A000290.
%K nonn,easy
%O 0,2
%A _Karl V. Keller, Jr._, Sep 23 2014
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