This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A193828 #49 Jan 21 2025 09:02:26
%S A193828 0,2,12,22,26,40,70,92,100,126,176,210,222,260,330,376,392,442,532,
%T A193828 590,610,672,782,852,876,950,1080,1162,1190,1276,1426,1520,1552,1650,
%U A193828 1820,1926,1962,2072,2262,2380,2420,2542,2752,2882,2926,3060,3290,3432,3480
%N A193828 Even generalized pentagonal numbers.
%C A193828 Even numbers in A001318.
%C A193828 Exponents in the expansion of Sum_{n >= 0} q^(2*n)/(Product_{k = 1..2*n} 1 + q^(2*k)) = 1 + q^2 - q^12 - q^22 + q^26 + q^40 - - + + ... (follows from Berndt et al., Theorem 3.3). Cf. A067589. - _Peter Bala_, Jan 21 2025
%H A193828 G. C. Greubel, <a href="/A193828/b193828.txt">Table of n, a(n) for n = 0..1000</a>
%H A193828 B. C. Berndt, B. Kim, and A. J. Yee, <a href="http://dx.doi.org/10.1016/j.jcta.2009.07.005">Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions</a>, J. Comb. Thy. Ser. A, 117 (2010), 957-973.
%H A193828 Mircea Merca, <a href="https://doi.org/10.1016/j.jnt.2015.05.008">The bisectional pentagonal number theorem</a>, Journal of Number Theory, Volume 157 (December 2015), Pages 223-232.
%H A193828 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-5,7,-7,5,-3,1).
%F A193828 a(n) = A000217(A108752(n+1))/3 = 2*A154293(n+1).
%F A193828 G.f.: -2*x*(x^2-x+1)*(x^2+4*x+1)/((x-1)^3*(x^2+1)^2). - _Colin Barker_, Sep 12 2012
%F A193828 Sum_{n>=1} 1/a(n) = 6 - (1+4/sqrt(3))*Pi/2. - _Amiram Eldar_, Mar 18 2022
%t A193828 CoefficientList[Series[-2*x*(x^2 - x + 1)*(x^2 + 4*x + 1)/((x - 1)^3*(x^2 + 1)^2), {x, 0, 50}], x] (* _G. C. Greubel_, Jun 06 2017 *)
%t A193828 LinearRecurrence[{3,-5,7,-7,5,-3,1},{0,2,12,22,26,40,70},50] (* _Harvey P. Dale_, Apr 09 2019 *)
%o A193828 (PARI) my(x='x+O('x^50)); concat([0], Vec(-2*x*(x^2-x+1)*(x^2+4*x+1)/((x-1)^3*(x^2+1)^2))) \\ _G. C. Greubel_, Jun 06 2017
%Y A193828 Cf. A001318, A067589, A154293.
%K A193828 nonn,easy
%O A193828 0,2
%A A193828 _Omar E. Pol_, Aug 19 2011