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%I #46 May 20 2023 08:22:05
%S 0,3,21,54,102,165,243,336,444,567,705,858,1026,1209,1407,1620,1848,
%T 2091,2349,2622,2910,3213,3531,3864,4212,4575,4953,5346,5754,6177,
%U 6615,7068,7536,8019,8517,9030,9558,10101,10659,11232,11820
%N 3 times heptagonal numbers: a(n) = 3n(5n-3)/2.
%C Also the number of 6-cycles in the (n+5)-triangular honeycomb acute knight graph. - _Eric W. Weisstein_, Jun 25 2017
%H Ivan Panchenko, <a href="/A152773/b152773.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = (15n^2 - 9n)/2 = A000566(n)*3.
%F a(n) = a(n-1)+15*n-12 with n>0, a(0)=0. - _Vincenzo Librandi_, Nov 26 2010
%F G.f.: 3*x*(1+4*x)/(1-x)^3. - _Bruno Berselli_, Jan 21 2011
%F a(0)=0, a(1)=3, a(2)=21, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - _Harvey P. Dale_, May 08 2012
%F a(n) = n + A226489(n). - _Bruno Berselli_, Jun 11 2013
%F Sum_{n>=1} 1/a(n) = tan(Pi/10)*Pi/9 - sqrt(5)*log(phi)/9 + 5*log(5)/18, where phi is the golden ratio (A001622). - _Amiram Eldar_, May 20 2023
%t Table[3 n (5 n - 3)/2, {n, 0, 50}] (* _Harvey P. Dale_, May 08 2012 *)
%t LinearRecurrence[{3, -3, 1}, {0, 3, 21}, 50] (* _Harvey P. Dale_, May 08 2012 *)
%t CoefficientList[Series[-((3 x^5 (1 + 4 x))/(-1 + x)^3), {x, 0, 20}], x] (* _Eric W. Weisstein_, Jun 25 2017 *)
%o (PARI) a(n)=3*n*(5*n-3)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000566, A001622, A135706.
%Y 3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152751, A152759, A152767, A153783, A153448, A153875.
%Y Cf. numbers of the form n*(n*k-k+6))/2, this sequence is the case k=15: see Comments lines of A226492.
%Y Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A045943 (4-cycles), A028896 (5-cycles).
%K nonn,easy
%O 0,2
%A _Omar E. Pol_, Dec 13 2008
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