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%I #61 Apr 02 2023 12:51:14
%S 0,10,36,78,136,210,300,406,528,666,820,990,1176,1378,1596,1830,2080,
%T 2346,2628,2926,3240,3570,3916,4278,4656,5050,5460,5886,6328,6786,
%U 7260,7750,8256,8778,9316,9870,10440,11026,11628,12246,12880,13530,14196,14878,15576
%N a(n) = 2*n*(4*n + 1).
%C If Y is a fixed 3-subset of a (4n+1)-set X then a(n) is the number of (4n-1)-subsets of X intersecting Y. - _Milan Janjic_, Oct 28 2007
%C Sequence found by reading the line from 0, in the direction 0, 10, ..., in the square spiral whose vertices are the triangular numbers A000217. - _Omar E. Pol_, Sep 03 2011
%H G. C. Greubel, <a href="/A033585/b033585.txt">Table of n, a(n) for n = 0..5000</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>.
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3470205">The groupoid of the Triangular Numbers and the generation of related integer sequences</a>, Politecnico di Torino, Italy (2019).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 2*A007742(n).
%F a(n) = A000217(4*n) = A014105(2*n). - _Reinhard Zumkeller_, Sep 17 2008
%F a(n) = 16*n + a(n-1) - 6 with a(0) = 0. - _Vincenzo Librandi_, Aug 05 2010
%F a(n) = A005843(n)*A016813(n). - _Omar E. Pol_, Oct 31 2013
%F G.f.: -2*x*(5+3*x)/(x-1)^3 . - _R. J. Mathar_, Feb 06 2017
%F E.g.f.: (8*x^2 + 10*x)*exp(x). - _G. C. Greubel_, Jul 18 2017
%F From _Amiram Eldar_, Jul 22 2020: (Start)
%F Sum_{n>=1} 1/a(n) = 2 - Pi/4 - 3*log(2)/2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/4 + sqrt(2)*arcsinh(1)/2 + log(2)/2 - 2. (End)
%p seq(binomial(4*n+1,2), n=0..36); # _Zerinvary Lajos_, Jan 21 2007
%t f[n_]:=2*n*(4*n+1);f[Range[0,60]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 05 2011 *)
%o (PARI) a(n)=2*n*(4*n+1) \\ _Charles R Greathouse IV_, Jun 16 2017
%Y Cf. A081266, A144312, A144314. - _Reinhard Zumkeller_, Sep 17 2008
%Y Cf. A000217.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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