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%I #83 Feb 28 2022 08:17:45
%S 0,1,17,20,52,57,105,112,176,185,265,276,372,385,497,512,640,657,801,
%T 820,980,1001,1177,1200,1392,1417,1625,1652,1876,1905,2145,2176,2432,
%U 2465,2737,2772,3060,3097,3401,3440,3760,3801,4137,4180,4532,4577,4945,4992
%N Numbers of the form 9*k^2 + 8*k, k an integer.
%C Numbers m such that 9*m + 16 is a square. - _Vincenzo Librandi_, Apr 07 2013
%C Equivalently, integers of the form h*(h + 8)/9 (nonnegative values of h are listed in A090570). - _Bruno Berselli_, Jul 15 2016
%C Generalized 20-gonal (or icosagonal) numbers: r*(9*r - 8) with r = 0, +1, -1, +2, -2, +3, -3, ... - _Omar E. Pol_, Jun 06 2018
%C Partial sums of A317316. - _Omar E. Pol_, Jul 28 2018
%C Exponents in expansion of Product_{n >= 1} (1 + x^(18*n-17))*(1 + x^(18*n-1))*(1 - x^(18*n)) = 1 + x + x^17 + x^20 + x^52 + .... - _Peter Bala_, Dec 10 2020
%H Jason Kimberley, <a href="/A218864/b218864.txt">Table of n, a(n) for n = 1..2000</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 C(q).
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F a(n) = (18*n*(n - 1) - 7*(-1)^n*(2*n - 1) - 7)/8. - _Bruno Berselli_, Nov 13 2012
%F G.f.: x*(1 + 16*x + x^2)/((1 + x)^2*(1 - x)^3). - _Bruno Berselli_, Nov 14 2012
%F Sum_{n>=2} 1/a(n) = (9 + 8*Pi*cot(Pi/9))/64. - _Amiram Eldar_, Feb 28 2022
%t Array[(18 # (# - 1) - 7 (-1)^#*(2 # - 1) - 7)/8 &, 48] (* or *)
%t CoefficientList[Series[x (1 + 16 x + x^2)/((1 + x)^2*(1 - x)^3), {x, 0, 47}], x] (* _Michael De Vlieger_, Jun 06 2018 *)
%o (Magma) a:=func<n | 9*n^2+8*n>; [0]cat[a(n*m): m in [-1,1], n in [1..20]];
%Y Characteristic function is A205987.
%Y Numbers of the form 9*m^2+k*m, for integer n: A016766 (k=0), A132355 (k=2), A185039 (k=4), A057780 (k=6), this sequence (k=8).
%Y Cf. A074377 (numbers m such that 16*m+9 is a square).
%Y Cf. A317316.
%Y For similar sequences of numbers m such that 9*m+i is a square, see list in A266956.
%Y Cf. sequences of the form m*(m+i)/(i+1) listed in A274978. [_Bruno Berselli_, Jul 25 2016]
%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), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), this sequence (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).
%K nonn,easy
%O 1,3
%A _Jason Kimberley_, Nov 08 2012
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