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A185039 Numbers of the form 9*m^2 + 4*m, m an integer.

%I A185039 #34 Sep 08 2022 08:45:55
%S A185039 0,5,13,28,44,69,93,128,160,205,245,300,348,413,469,544,608,693,765,
%T A185039 860,940,1045,1133,1248,1344,1469,1573,1708,1820,1965,2085,2240,2368,
%U A185039 2533,2669,2844,2988,3173,3325,3520,3680,3885,4053,4268,4444,4669,4853,5088
%N A185039 Numbers of the form 9*m^2 + 4*m, m an integer.
%C A185039 Also, numbers m such that 9*m+4 is a square. After 0, therefore, there are no squares in this sequence. - _Bruno Berselli_, Jan 07 2016
%H A185039 Bruno Berselli, <a href="/A185039/b185039.txt">Table of n, a(n) for n = 1..1000</a>
%H A185039 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. 274 (2004), no. 1-3, 9-24. See B(q).
%H A185039 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A185039 From _Bruno Berselli_, Feb 04 2012: (Start)
%F A185039 G.f.: x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3).
%F A185039 a(n) = a(-n+1) = (18*n*(n-1)+(2*n-1)*(-1)^n+1)/8 = A004526(n)*A156638(n). (End).
%t A185039 CoefficientList[Series[x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3), {x,0,50}], x] (* _G. C. Greubel_, Jun 20 2017 *)
%t A185039 LinearRecurrence[{1,2,-2,-1,1},{0,5,13,28,44},50] (* _Harvey P. Dale_, Jan 23 2018 *)
%o A185039 (Magma) [0] cat &cat[[9*n^2-4*n,9*n^2+4*n]: n in [1..32]]; // _Bruno Berselli_, Feb 04 2011
%o A185039 (PARI) x='x+O('x^50); Vec(x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3)) \\ _G. C. Greubel_, Jun 20 2017
%Y A185039 Characteristic function is A205809.
%Y A185039 Numbers of the form 9*n^2+k*n, for integer n: A016766 (k=0), A132355 (k=2), this sequence (k=4), A057780 (k=6), A218864 (k=8). [_Jason Kimberley_, Nov 08 2012]
%Y A185039 For similar sequences of numbers m such that 9*m+k is a square, see list in A266956.
%K A185039 nonn,easy
%O A185039 1,2
%A A185039 _N. J. A. Sloane_, Feb 04 2012