OFFSET
0,3
COMMENTS
Octagonal numbers that are one-eighth of another octagonal number. - Kelvin Voskuijl, Jun 19 2025
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 40.
LINKS
Colin Barker, Table of n, a(n) for n = 0..327
Eric Weisstein's World of Mathematics, Octagonal Pentagonal Number.
Index entries for linear recurrences with constant coefficients, signature (1,1331714,-1331714,-1,1).
FORMULA
From Ant King, Dec 16 2011: (Start)
a(n) = 1331714*a(n-2) - a(n-4) + 249696.
a(n) = a(n-1) + 1331714*a(n-2) - 1331714*a(n-3) - a(n-4) + a(n-5).
a(n) = (1/96)*((11-6*sqrt(2)*(-1)^n)*(1+sqrt(2))^(8*n-6)+(11+6*sqrt(2)*(-1)^n)*(1-sqrt(2))^(8*n-6)-18).
a(n) = floor(1/96*(11-6*sqrt(2)*(-1)^n)*(1+sqrt(2))^(8*n-6)).
G.f.: x*(1+175*x+243535*x^2+5945*x^3+40*x^4)/((1-x)*(1-1154*x+x^2)*(1+1154*x+x^2)).
Limit_{n->oo} a(2n+1)/a(2n) = (1/49)*(219073+154908*sqrt(2)).
Limit_{n->oo} a(2n)/a(2n-1) = (1/49)*(3649+2580*sqrt(2)). (End)
MATHEMATICA
LinearRecurrence[{1, 1331714, -1331714, -1, 1}, {1, 176, 1575425, 234631320, 2098015778145}, 11] (* Ant King, Dec 16 2011 *)
PROG
(PARI) Vec(x*(1+175*x+243535*x^2+5945*x^3+40*x^4)/((1-x)*(1-1154*x+x^2)*(1+1154*x+x^2)) + O(x^20), -20) \\ Colin Barker, Jun 23 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Initial term 0 inserted by Kelvin Voskuijl, Jun 19 2025
STATUS
approved