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

0, 7, 11, 32, 40, 75, 87, 136, 152, 215, 235, 312, 336, 427, 455, 560, 592, 711, 747, 880, 920, 1067, 1111, 1272, 1320, 1495, 1547, 1736, 1792, 1995, 2055, 2272, 2336, 2567, 2635, 2880, 2952, 3211, 3287, 3560, 3640, 3927, 4011, 4312, 4400, 4715, 4807

Offset: 1

Author

Mohamed Bouhamida, Nov 08 2007

Keywords

• nonn
• easy

Comments

X values of solutions to the equation 9*X^3 + X^2 = Y^2.

The set of all m such that 9*m + 1 is a perfect square. - Gary Detlefs, Feb 22 2010

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

Links

• Jason Kimberley, Table of n, a(n) for n = 1..2108
• S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See A(q).
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

A205808 is the characteristic function.

Cf. A000217, A001082, A002378, A005563, A028347, A036666, A046092, A054000, A056220, A062717, A087475, A132209, A010701, A056020.

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

For similar sequences of numbers m such that 9*m+k is a square, see list in A266956.

Programs

Magma

a:=func; [0] cat [a(n*m): m in [-1,1], n in [1..25]]; // Jason Kimberley, Nov 08 2012

Maple

readlib(issqr); for n from 0 to 3560 do if(issqr(9*n+1)) then print(n) fi od; # Gary Detlefs, Feb 22 2010
seq(n^2+n+5*ceil(n/2)^2,n=0..39); # Gary Detlefs, Feb 23 2010

Mathematica

f[n_]:=IntegerQ[Sqrt[1+9*n]]; Select[Range[0,8! ],f[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
Sort[Table[9n^2+2n,{n,-30,30}]] (* Harvey P. Dale, Dec 06 2013 *)

PARI

a(n)=n^2-n+5*(n\2)^2 \\ Charles R Greathouse IV, Sep 28 2015

Formula

a(2*k) = k*(9*k-2), a(2*k+1) = k*(9*k+2).

a(n) = n^2 - n + 5*floor(n/2)^2. - Gary Detlefs, Feb 23 2010

From R. J. Mathar, Mar 17 2010: (Start)

a(n) = +a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).

G.f.: x^2*(7 + 4*x + 7*x^2)/((1 + x)^2*(1 - x)^3). (End)

a(n) = (2*n - 1 + (-1)^n)*(9*(2*n - 1) + (-1)^n)/16. - Luce ETIENNE, Sep 13 2014

Sum_{n>=2} 1/a(n) = 9/4 - cot(2*Pi/9)*Pi/2. - Amiram Eldar, Mar 15 2022

Extensions

Simpler definition and minor edits from N. J. A. Sloane, Feb 03 2012

Since this is a list, offset changed to 1 and formulas translated by Jason Kimberley, Nov 18 2012