OFFSET
1,3
COMMENTS
The y solution to the generalized Pell equation x^2 + x = 2 + 4*y + 6*y^2. - T. D. Noe, Apr 28 2011
Also numbers n such that the sum of the pentagonal numbers P(n) and P(n+1) is equal to a hexagonal number. - Colin Barker, Dec 15 2014
Also numbers n such that the sum of the pentagonal numbers P(n) and P(n+1) is equal to a triangular number. - Colin Barker, Dec 15 2014
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,98,-98,-1,1).
FORMULA
a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5). - Colin Barker, Dec 15 2014
G.f.: x^2*(5*x^3+9*x^2-45*x-1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)). - Colin Barker, Dec 15 2014
a(n) = (-(5/8)*sqrt(6)-3/2)*(5-2*sqrt(6))^n+(-3/2+(5/8)*sqrt(6))*(5+2*sqrt(6))^n-1/3+(-(1/3)*sqrt(6)-5/6)*(-5+2*sqrt(6))^n+((1/3)*sqrt(6)-5/6)*(-5-2*sqrt(6))^n. - Robert Israel, Dec 15 2014
EXAMPLE
Corresponding values of triangular numbers tri = m(m+1)/2 and m's are
tri = 1, 6, 6441, 54946, 61843881, 527588886, 593824936321
m = 1, 3, 113, 331, 11121, 32483, 1089793.
MAPLE
ivs:=[0, 1, 46, 135, 4540]:
rec:= a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5):
f:= gfun:-rectoproc({rec, seq(a(i)=ivs[i], i=1..5)}, a(n), remember):
seq(f(n), n=1..100); # Robert Israel, Dec 15 2014
MATHEMATICA
triQ[n_] := IntegerQ[ Sqrt[8n + 1]]; lst = {}; Do[ If[ triQ[1 + 2n + 3n^2], AppendTo[lst, n]; Print@n], {n, 0, 65000000}] (* Robert G. Wilson v, Jan 08 2007 *)
LinearRecurrence[{1, 98, -98, -1, 1}, {1, 46, 135, 4540, 13261}, 30] (* T. D. Noe, Apr 28 2011 *)
PROG
(PARI) concat(0, Vec(x^2*(5*x^3+9*x^2-45*x-1)/((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^100))) \\ Colin Barker, Dec 15 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Zak Seidov, Oct 20 2006
EXTENSIONS
a(8) and a(9) from Robert G. Wilson v, Jan 08 2007
a(10) and a(11) from Donovan Johnson, Apr 28 2011
Extended by T. D. Noe, Apr 28 2011
STATUS
approved