OFFSET
1,1
LINKS
F. T. Adams-Watters, SeqFan Discussion, Oct 2009
FORMULA
{k: 10+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*(3 +2*x +3*x^2 -2*x^3 -4*x^4)/((1-x) * (x^2-2*x-1) * (x^2+2*x-1))
G.f.: ( 8 + (-1-6*x)/(x^2+2*x-1) + (8+17*x)/(x^2-2*x-1) + 1/(x-1) )/2. (End)
a(1..4) = (3,5,26,36); a(n) = 6*a(n-2) - a(n-4) + 2, for n > 4. - Ctibor O. Zizka, Nov 10 2009
EXAMPLE
3*(3+1)/2+10 = 4^2. 5*(5+1)/2+10 = 5^2. 26*(26+1)/2+10 = 19^2. 36*(36+1)/2+10 = 26^2.
MATHEMATICA
Join[{3, 5}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 10 &]] (* G. C. Greubel, Sep 03 2016 *)
Select[Range[0, 2 10^7], IntegerQ[Sqrt[10 + # (# + 1) / 2]] &] (* Vincenzo Librandi, Sep 03 2016 *)
PROG
(PARI) isok(n) = issquare(10 + n*(n+1)/2); \\ Michel Marcus, Sep 03 2016
(Magma) [n: n in [0..2*10^7] | IsSquare(10+n*(n+1)/2)]; /* or */ [3, 5] cat [n: n in [0..2*10^7] | (Ceiling(Sqrt(n*(n+ 1)/2)))^2-n*(n+1)/2 eq 10]; // Vincenzo Librandi, Sep 03 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
R. J. Mathar, Oct 18 2009
EXTENSIONS
a(17)-a(24) from Donovan Johnson, Nov 01 2010
a(25)-a(30) from Lars Blomberg, Jul 07 2015
STATUS
approved