OFFSET
1,2
LINKS
Robert Israel, Table of n, a(n) for n = 1..2351
F. T. Adams-Watters, SeqFan Discussion, Oct 2009
FORMULA
{k: 9+k*(k+1)/2 in A000290}
Conjectures: (Start)
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - a(n-4) + a(n-5).
G.f.: x^2*(10 +3*x -10*x^2 -x^3)/((1-x) * (x^2-2*x-1) * (x^2+2*x-1))
G.f.: ( 2 + (-5+4*x)/(x^2+2*x-1) + (6+17*x)/(x^2-2*x-1) + 1/(x-1) )/2. (End)
a(1..4) = (0,10,13,63); a(n) = 6*a(n-2) - a(n-4) + 2, for n > 4. - Ctibor O. Zizka, Nov 10 2009
From Robert Israel, Jul 07 2015: (Start)
These conjectures follow from the theory of Pell-like equations.
EXAMPLE
0*(0+1)/2+9 = 3^2. 10*(10+1)/2+9 = 8^2. 13*(13+1)/2+9 = 10^2. 63*(63+1)/2+9 = 45^2.
MAPLE
seq(seq((8*orthopoly[U](k+j, 3) - (8 - (-1)^j)*orthopoly[T](k+j, 3)-1)/2, j=0..1), k=0..20); # Robert Israel, Jul 07 2015
MATHEMATICA
Join[{0}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 9 &]] (* G. C. Greubel, Sep 03 2016 *)
Select[Range[0, 2 10^7], IntegerQ[Sqrt[9 + # (# + 1) / 2]] &] (* Vincenzo Librandi, Sep 03 2016 *)
(Sqrt[8#+1]-1)/2&/@Select[Accumulate[Range[0, 5*10^6]], IntegerQ[Sqrt[#+9]]&] (* The program generates the first 17 terms of the sequence. *) (* Harvey P. Dale, Oct 21 2024 *)
PROG
(Magma) [n: n in [0..2*10^7] | IsSquare(9 + n*(n+1)/2)];
/* or */ [0] cat [n: n in [0..2*10^7] | (Ceiling(Sqrt(n*(n+ 1)/2)))^2-n*(n+1)/2 eq 9]; // Vincenzo Librandi, Sep 03 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
R. J. Mathar, Oct 18 2009
EXTENSIONS
a(16)-a(24) from Donovan Johnson, Nov 01 2010
a(25)-a(30) from Lars Blomberg, Jul 07 2015
STATUS
approved