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A033580 Four times second pentagonal numbers: a(n) = 2*n*(3*n+1).

%I A033580 #92 Jul 26 2022 21:59:52
%S A033580 0,8,28,60,104,160,228,308,400,504,620,748,888,1040,1204,1380,1568,
%T A033580 1768,1980,2204,2440,2688,2948,3220,3504,3800,4108,4428,4760,5104,
%U A033580 5460,5828,6208,6600,7004,7420,7848,8288,8740,9204,9680,10168,10668,11180,11704,12240
%N A033580 Four times second pentagonal numbers: a(n) = 2*n*(3*n+1).
%C A033580 Subsequence of A062717: A010052(6*a(n)+1) = 1. - _Reinhard Zumkeller_, Feb 21 2011
%C A033580 Sequence found by reading the line from 0, in the direction 0, 8,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A139267 in the same spiral - _Omar E. Pol_, Sep 09 2011
%C A033580 a(n) is the number of edges of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference. - _Emeric Deutsch_ May 13 2018
%C A033580 The partial sums of this sequence give A035006. - _Leo Tavares_, Oct 03 2021
%H A033580 Ivan Panchenko, <a href="/A033580/b033580.txt">Table of n, a(n) for n = 0..1000</a>
%H A033580 M. K. Siddiqui, M. Naeem, N. A. Rahman, and M. Imran, <a href="https://joam.inoe.ro/articles/computing-topological-indices-of-certain-networks/">Computing topological indices of certain networks</a>, J. of Optoelectronics and Advanced Materials, 18, No. 9-10 (2016), pp. 884-892.
%H A033580 Leo Tavares, <a href="/A033580/a033580_9.jpg">Illustration: Crossed Stars</a>
%H A033580 Leo Tavares, <a href="/A033580/a033580_10.jpg">Illustration: Four Quarter Star Crosses</a>
%H A033580 Leo Tavares, <a href="/A033580/a033580_11.jpg">Illustration: Triangulated Star Crosses</a>
%H A033580 Leo Tavares, <a href="/A033580/a033580_12.jpg">Illustration: Oblong Star Crosses</a>
%H A033580 Leo Tavares, <a href="/A033580/a033580_13.jpg">Illustration: Crossed Diamond Stars</a>
%H A033580 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A033580 a(n) = a(n-1) +12*n -4 (with a(0)=0). - _Vincenzo Librandi_, Aug 05 2010
%F A033580 G.f.: 4*x*(2+x)/(1-x)^3. - _Colin Barker_, Feb 13 2012
%F A033580 a(-n) = A033579(n). - _Michael Somos_, Jun 09 2014
%F A033580 E.g.f.: 2*x*(4 + 3*x)*exp(x). - _G. C. Greubel_, Oct 09 2019
%F A033580 From _Amiram Eldar_, Jan 14 2021: (Start)
%F A033580 Sum_{n>=1} 1/a(n) = 3/2 - Pi/(4*sqrt(3)) - 3*log(3)/4.
%F A033580 Sum_{n>=1} (-1)^(n+1)/a(n) =  -3/2 + Pi/(2*sqrt(3)) + log(2). (End)
%F A033580 From _Leo Tavares_, Oct 12 2021: (Start)
%F A033580 a(n) = A003154(n+1) - A016813(n). See Crossed Stars illustration.
%F A033580 a(n) = 4*A005449(n). See Four Quarter Star Crosses illustration.
%F A033580 a(n) = 2*A049451(n).
%F A033580 a(n) = A046092(n-1) + A033996(n). See Triangulated Star Crosses illustration.
%F A033580 a(n) = 4*A000217(n-1) + 8*A000217(n).
%F A033580 a(n) = 4*A000217(n-1) + 4*A002378. See Oblong Star Crosses illustration.
%F A033580 a(n) = A016754(n) + 4*A000217(n). See Crossed Diamond Stars illustration.
%F A033580 a(n) = 2*A001105(n) + 4*A000217(n).
%F A033580 a(n) = A016742(n) + A046092(n).
%F A033580 a(n) = 4*A000290(n) + 4*A000217(n). (End)
%p A033580 seq(2*n*(3*n+1), n=0..50); # _G. C. Greubel_, Oct 09 2019
%t A033580 4*Binomial[3*Range[50]-2, 2]/3 (* _G. C. Greubel_, Oct 09 2019 *)
%o A033580 (PARI) a(n)=2*n*(3*n+1) \\ _Charles R Greathouse IV_, Sep 28 2015
%o A033580 (Magma) [2*n*(3*n+1): n in [0..50]]; // _G. C. Greubel_, Oct 09 2019
%o A033580 (Sage) [2*n*(3*n+1) for n in (0..50)] # _G. C. Greubel_, Oct 09 2019
%o A033580 (GAP) List([0..50], n-> 2*n*(3*n+1)); # _G. C. Greubel_, Oct 09 2019
%Y A033580 Cf. A033579, A049451, A045945, A186423.
%Y A033580 Cf. sequences listed in A254963.
%Y A033580 Cf. A003154, A016813.
%Y A033580 Cf. A035006.
%Y A033580 Cf. A005449, A046092, A033996, A002378, A016742, A000290, A016754, A001105.
%K A033580 nonn,easy
%O A033580 0,2
%A A033580 _N. J. A. Sloane_