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%I #81 May 01 2022 07:55:35
%S 0,1,9,12,28,33,57,64,96,105,145,156,204,217,273,288,352,369,441,460,
%T 540,561,649,672,768,793,897,924,1036,1065,1185,1216,1344,1377,1513,
%U 1548,1692,1729,1881,1920,2080,2121,2289,2332,2508,2553,2737,2784,2976,3025
%N Generalized 12-gonal numbers: k*(5*k-4) for k = 0, +-1, +-2, ...
%C Also generalized dodecagonal numbers.
%C Second 12-gonal numbers (A135705) and positive terms of A051624 interleaved. - _Omar E. Pol_, Aug 04 2012
%C The characteristic function of this sequence is A205988. - _Jason Kimberley_, Nov 15 2012
%C Also, integer values of m*(m+4)/5. - _Bruno Berselli_, Dec 05 2012
%C Also, numbers h such that 5*h + 4 is a square. - _Bruno Berselli_, Oct 10 2013
%C Exponents in expansion of Product_{n >= 1} (1 + x^(10*n-9))*(1 + x^(10*n-1))*(1 - x^(10*n)) = 1 + x + x^9 + x^12 + x^28 + .... - _Peter Bala_, Dec 10 2020
%H Vincenzo Librandi, <a href="/A195162/b195162.txt">Table of n, a(n) for n = 0..10000</a>
%H S. Cooper and M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00079-7">Results of Hurwitz type for three squares</a>, Discrete Math. , Vol. 274, No. 1-3 (2004), pp. 9-24. See E(q).
%H John Elias, <a href="/A195162/a195162.png">Generalized 12-gonal & 20-gonal Cross Configurations</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F From _R. J. Mathar_, Sep 24 2011: (Start)
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F a(n) = A008805(n-1) + A008805(n-3) + 8*A008805(n-2). (End)
%F From _Bruno Berselli_, Sep 26 2011: (Start)
%F G.f.: x*(1+8*x+x^2)/((1+x)^2*(1-x)^3).
%F a(n) = (10*n*(n+1) + 3*(2*n+1)*(-1)^n - 3)/8.
%F a(n) = a(-n-1). (End)
%F Sum_{n>=1} 1/a(n) = (5 + 4*sqrt(1 + 2/sqrt(5))*Pi)/16. - _Vaclav Kotesovec_, Oct 05 2016
%F E.g.f.: (3*(1 - 2*x)*exp(-x) + (-3 +20*x +10*x^2)*exp(x))/8. - _G. C. Greubel_, Jul 04 2019
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(5)/8 + sqrt(5)*log(phi)/4 - 5/16, where phi is the golden ratio (A001622). - _Amiram Eldar_, Feb 28 2022
%t nn = 25; Sort[Table[n*(5*n - 4), {n, -nn, nn}]] (* _T. D. Noe_, Sep 23 2011 *)
%o (Magma) [0] cat &cat[[5*n^2-4*n, 5*n^2+4*n]: n in [1..25]]; // _Vincenzo Librandi_, Sep 26 2011
%o (PARI) vector(50, n, n--; (10*n^2 +10*n -3 +3*(-1)^n*(2*n+1))/8) \\ _G. C. Greubel_, Jul 04 2019
%o (Sage) [(10*n^2 +10*n -3 +3*(-1)^n*(2*n+1))/8 for n in (0..50)] # _G. C. Greubel_, Jul 04 2019
%o (GAP) List([0..50], n-> (10*n^2 +10*n -3 +3*(-1)^n*(2*n+1))/8) # _G. C. Greubel_, Jul 04 2019
%Y Partial sums of A195161.
%Y Column 8 of A195152.
%Y Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), this sequence (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
%Y Cf. A001622, A051624, A135705.
%Y Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [_Bruno Berselli_, Jul 25 2016]
%K nonn,easy
%O 0,3
%A _Omar E. Pol_, Sep 10 2011
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