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%I #91 Sep 08 2022 08:44:41
%S 0,36,144,324,576,900,1296,1764,2304,2916,3600,4356,5184,6084,7056,
%T 8100,9216,10404,11664,12996,14400,15876,17424,19044,20736,22500,
%U 24336,26244,28224,30276,32400,34596,36864,39204,41616,44100,46656,49284,51984,54756,57600,60516,63504,66564,69696,72900
%N a(n) = (6*n)^2.
%C Areas A of two classes of triangles with integer sides (a,b,c) where a = 9k, b=10k and c = 17k, or a = 3k, b = 25k and c = 26k for k=0,1,2,... These areas are given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) = (6k)^2, with the semiperimeter s = (a+b+c)/2. This sequence is a subsequence of A188158. - _Michel Lagneau_, Oct 11 2013
%C Sequence found by reading the line from 0, in the direction 0, 36, ..., in the square spiral whose vertices are the generalized 20-gonal numbers A218864. - _Omar E. Pol_, May 13 2018.
%H Vincenzo Librandi, <a href="/A016910/b016910.txt">Table of n, a(n) for n = 0..2000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F From _Ilya Gutkovskiy_, Jun 09 2016: (Start)
%F O.g.f.: 36*x*(1 + x)/(1 - x)^3.
%F E.g.f.: 36*x*(1 + x)*exp(x).
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F Sum_{n>=1} 1/a(n) = Pi^2/216 = A086726. (End)
%F Product_{n>=1} a(n)/A136017(n) = Pi/3. - _Fred Daniel Kline_, Jun 09 2016
%F a(n) = t(9*n) - 9*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(9*n) - 9*A000217(n). - _Bruno Berselli_, Aug 31 2017
%F a(n) = 36*A000290(n) = 18*A001105(n) = 12*A033428 = 9*A016742(n) = 6*A033581(n) = 4*A016766(n) = 3*A135453(n) = 2*A195321(n). - _Omar E. Pol_, Jun 07 2018
%F Sum_{n>=1) (-1)^(n+1)/a(n) = Pi^2/432. - _Amiram Eldar_, Jun 27 2020
%F From _Amiram Eldar_, Jan 25 2021: (Start)
%F Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/6)/(Pi/6).
%F Product_{n>=1} (1 - 1/a(n)) = sin(Pi/6)/(Pi/6) = 3/Pi (A089491). (End)
%t (6*Range[0,40])^2 (* or *) LinearRecurrence[{3,-3,1},{0,36,144},40] (* _Harvey P. Dale_, Dec 25 2017 *)
%t Table[36 n^2, {n, 0, 45}] (* _Omar E. Pol_, Jun 07 2018 *)
%o (Magma) [(6*n)^2: n in [0..40]]; // _Vincenzo Librandi_, May 03 2011
%o (PARI) a(n)=36*n^2 \\ _Charles R Greathouse IV_, Jun 10 2016
%Y Cf. A000217, A089491, A188158, A243941.
%Y Cf. similar sequences of the type k*n^2: A000290 (k=1), A001105 (k=2), A033428 (k=3), A016742 (k=4), A033429 (k=5), A033581 (k=6), A033582 (k=7), A139098 (k=8), A016766 (k=9), A033583 (k=10), A033584 (k=11), A135453 (k=12), A152742 (k=13), A144555 (k=14), A064761 (k=15), A016802 (k=16), A244630 (k=17), A195321 (k=18), A244631 (k=19), A195322 (k=20), A064762 (k=21), A195323 (k=22), A244632 (k=23), A195824 (k=24), A016850 (k=25), A244633 (k=26), A244634 (k=27), A064763 (k=28), A244635 (k=29), A244636 (k=30).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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