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%I #67 May 21 2022 08:29:44
%S 1,21,65,133,225,341,481,645,833,1045,1281,1541,1825,2133,2465,2821,
%T 3201,3605,4033,4485,4961,5461,5985,6533,7105,7701,8321,8965,9633,
%U 10325,11041,11781,12545,13333,14145,14981,15841,16725,17633,18565,19521,20501,21505
%N Odd octagonal numbers: (2n+1)*(6n+1).
%C Sequence found by reading the line from 1, in the direction 1, 21, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - _Omar E. Pol_, Jul 18 2012
%H Muniru A Asiru, <a href="/A014641/b014641.txt">Table of n, a(n) for n = 0..5000</a>
%H Richard P. Brent, <a href="https://arxiv.org/abs/1407.3533">Generalising Tuenter's binomial sums</a>, arXiv:1407.3533 [math.CO], 2014.
%H Richard P. Brent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Brent/brent5.html">Generalising Tuenter's binomial sums</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.2.
%H Leo Tavares, <a href="/A014641/a014641.jpg">Illustration: Square Block Stars</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = a(n-1) + 24*n - 4, with n > 0, a(0)=1. - _Vincenzo Librandi_, Dec 28 2010
%F G.f.: (1 + 18*x + 5*x^2)/(1 - 3*x + 3*x^2 - x^3). - _Colin Barker_, Jan 06 2012
%F a(n) = A289873(6*n+2). - _Hugo Pfoertner_, Jul 15 2017
%F From _Peter Bala_, Jan 22 2018: (Start)
%F This is the polynomial Qbar(2,n) in Brent. See A160485 for the triangle of coefficients (with signs) of the Qbar polynomials.
%F a(n) = (1/4^n) * Sum_{k = 0..n} (2*k + 1)^4*binomial(2*n + 1, n - k).
%F a(n-1) = (2/4^n) * binomial(2*n,n) * ( 1 + 3^4*(n - 1)/(n + 1) + 5^4*(n - 1)*(n - 2)/((n + 1)*(n + 2)) + 7^4*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)) + ... ). (End)
%F From _Amiram Eldar_, Feb 27 2022: (Start)
%F Sum_{n>=0} 1/a(n) = (sqrt(3)*Pi + 3*log(3))/8.
%F Sum_{n>=0} (-1)^n/a(n) = Pi/8 + sqrt(3)*log(2+sqrt(3))/4. (End)
%F E.g.f.: exp(x)*(1 + 20*x + 12*x^2). - _Stefano Spezia_, Apr 16 2022
%F a(n) = A016754(n) + 4*A014105(n). - _Leo Tavares_, May 20 2022
%p A014641:=n->(2*n+1)*(6*n+1); seq(A014641(n), n=0..50); # _Wesley Ivan Hurt_, Jun 08 2014
%t Table[(2n + 1)(6n + 1), {n, 0, 49}] (* _Harvey P. Dale_, Mar 24 2011 *)
%o (Magma) [ (2*n+1)*(6*n+1) : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 08 2014
%o (PARI) a(n)=(2*n+1)*(6*n+1) \\ _Charles R Greathouse IV_, Jun 17 2017
%o (GAP) List([0..50],n->(2*n+1)*(6*n+1)); # _Muniru A Asiru_, Feb 05 2019
%Y Cf. A000567, A001082, A014642, A014793, A014794, A243201, A289873.
%Y Cf. A160485, A245244.
%Y Cf. A016754, A014105.
%K nonn,easy
%O 0,2
%A _Mohammad K. Azarian_, Dec 11 1999
%E More terms from _Patrick De Geest_
%E Better description from _N. J. A. Sloane_