OFFSET
0,2
COMMENTS
Numbers k such that x=2*k+1 satisfies the Pell-type equation x^2 = 8*y^2 - 127. - Robert Israel, Jul 18 2019
LINKS
Robert Israel, Table of n, a(n) for n = 0..2608
F. T. Adams-Watters, SeqFan Discussion, Oct 2009
Robert Israel, Proof of conjectured recurrence
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-1,1).
FORMULA
{k: 16+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*(-14-3*x+14*x^2+x^3)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).
G.f.: ( 2 + (8+23*x)/(x^2-2*x-1) + 1/(x-1) + (-7+6*x)/(x^2+2*x-1) )/2. (End)
Conjectures confirmed: see link. - Robert Israel, Jul 18 2019
EXAMPLE
0, 14, 17, and 87 are terms:
0* (0+1)/2 + 16 = 4^2,
14*(14+1)/2 + 16 = 11^2,
17*(17+1)/2 + 16 = 13^2,
87*(87+1)/2 + 16 = 62^2.
MAPLE
f:= gfun:-rectoproc({a(n+4)-6*a(n+2)+a(n)=2, a(0)=0, a(1)=14, a(2)=17, a(3)=87}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Jul 18 2019
MATHEMATICA
Join[{0}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 16 &]] (* G. C. Greubel, Sep 03 2016 *)
PROG
(PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 16), print1(n, ", ") ) ); }
CROSSREFS
KEYWORD
nonn,less
AUTHOR
R. J. Mathar, Oct 18 2009
STATUS
approved