OFFSET
1,2
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).
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
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 *)
PROG
(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
(PARI) a(n)=n^2-n+5*(n\2)^2 \\ Charles R Greathouse IV, Sep 28 2015
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.
KEYWORD
nonn,easy
AUTHOR
Mohamed Bouhamida, Nov 08 2007
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
STATUS
approved