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A132355 Numbers of the form 9*h^2 + 2*h, for h an integer.

%I A132355 #80 Mar 15 2022 04:33:20
%S A132355 0,7,11,32,40,75,87,136,152,215,235,312,336,427,455,560,592,711,747,
%T A132355 880,920,1067,1111,1272,1320,1495,1547,1736,1792,1995,2055,2272,2336,
%U A132355 2567,2635,2880,2952,3211,3287,3560,3640,3927,4011,4312,4400,4715,4807
%N A132355 Numbers of the form 9*h^2 + 2*h, for h an integer.
%C A132355 X values of solutions to the equation 9*X^3 + X^2 = Y^2.
%C A132355 The set of all m such that 9*m + 1 is a perfect square. - _Gary Detlefs_, Feb 22 2010
%C A132355 The concatenation of any term with 11..11 (1 repeated an even number of times, see A099814) belongs to the list. Example: 87 is a term, so also 8711, 871111, 87111111, 871111111111, ... are terms of this sequence. - _Bruno Berselli_, May 15 2017
%H A132355 Jason Kimberley, <a href="/A132355/b132355.txt">Table of n, a(n) for n = 1..2108</a>
%H A132355 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 A(q).
%H A132355 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A132355 a(2*k) = k*(9*k-2), a(2*k+1) = k*(9*k+2).
%F A132355 a(n) = n^2 - n + 5*floor(n/2)^2. - _Gary Detlefs_, Feb 23 2010
%F A132355 From _R. J. Mathar_, Mar 17 2010: (Start)
%F A132355 a(n) = +a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
%F A132355 G.f.: x^2*(7 + 4*x + 7*x^2)/((1 + x)^2*(1 - x)^3). (End)
%F A132355 a(n) = (2*n - 1 + (-1)^n)*(9*(2*n - 1) + (-1)^n)/16. - _Luce ETIENNE_, Sep 13 2014
%F A132355 Sum_{n>=2} 1/a(n) = 9/4 - cot(2*Pi/9)*Pi/2. - _Amiram Eldar_, Mar 15 2022
%p A132355 readlib(issqr); for n from 0 to 3560 do if(issqr(9*n+1)) then print(n) fi od; # _Gary Detlefs_, Feb 22 2010
%p A132355 seq(n^2+n+5*ceil(n/2)^2,n=0..39); # _Gary Detlefs_, Feb 23 2010
%t A132355 f[n_]:=IntegerQ[Sqrt[1+9*n]]; Select[Range[0,8! ],f[ # ]&] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010 *)
%t A132355 Sort[Table[9n^2+2n,{n,-30,30}]] (* _Harvey P. Dale_, Dec 06 2013 *)
%o A132355 (Magma) a:=func<n | 9*n^2+2*n>; [0] cat [a(n*m): m in [-1,1], n in [1..25]]; // _Jason Kimberley_, Nov 08 2012
%o A132355 (PARI) a(n)=n^2-n+5*(n\2)^2 \\ _Charles R Greathouse IV_, Sep 28 2015
%Y A132355 A205808 is the characteristic function.
%Y A132355 Cf. A000217, A001082, A002378, A005563, A028347, A036666, A046092, A054000, A056220, A062717, A087475, A132209, A010701, A056020.
%Y A132355 Numbers of the form 9*n^2+k*n, for integer n: A016766 (k=0), this sequence (k=2), A185039 (k=4), A057780 (k=6), A218864 (k=8). - _Jason Kimberley_, Nov 09 2012
%Y A132355 For similar sequences of numbers m such that 9*m+k is a square, see list in A266956.
%K A132355 nonn,easy
%O A132355 1,2
%A A132355 _Mohamed Bouhamida_, Nov 08 2007
%E A132355 Simpler definition and minor edits from _N. J. A. Sloane_, Feb 03 2012
%E A132355 Since this is a list, offset changed to 1 and formulas translated by _Jason Kimberley_, Nov 18 2012