OFFSET
1,2
COMMENTS
We want solutions to m(7m-5)/2 = n(n+1)/2, or equivalently (14m-5)^2 = 7(2n+1)^2 + 18. This is the Pell-type equation x^2 - 7y^2 = 18.
This equation has unit solutions (x,y) = (5,1), (9, 3) and (19, 7), which lead to the family of solutions (5, 1), (9, 3), (19, 7), (61, 23), (135, 51), (299, 113), (971, 367), .... The corresponding integer solutions are (m,n) = (1,1), (10, 25), (154, 406), (2449, 6478), ... (A048907 and A048908), giving the nonagonal triangular numbers 1, 325, 82621, 20985481, ... shown here.
Also, numbers simultaneously 9-gonal and centered 9-gonal, the intersection of A001106 and A060544. - Steven Schlicker, Apr 24 2007
LINKS
Colin Barker, Table of n, a(n) for n = 1..416
S. C. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine, Vol. 84, No. 5, December 2011, pp. 339-350.
Eric Weisstein's World of Mathematics, Nonagonal Triangular Number.
Index entries for linear recurrences with constant coefficients, signature (255,-255,1).
FORMULA
Define x(n) + y(n)*sqrt(63) = (9+sqrt(63))*(8+sqrt(63))^n, s(n) = (y(n)+1)/2; then a(n) = (2+9*(s(n)^2-s(n)))/2. - Steven Schlicker, Apr 24 2007
a(n+1) = 254*a(n+1)-a(n)+72. - Richard Choulet, Sep 22 2007
a(n+1) = 127*a(n+1)+36+6*(448*a(n)^2+256*a(n)+25)^0.5. - Richard Choulet, Sep 22 2007
G.f.: z*(1+70*z+z^2)/((1-z)*(1-254*z+z^2)). - Richard Choulet, Sep 22 2007
From Ant King, Nov 03 2011: (Start)
a(n) = 255*a(n-1) - 255*a(n-2) + a(n-3).
a(n) = 1/112*(9*(8 + 3*sqrt(7))^(2n-1) + 9*(8-3* sqrt(7))^(2n-1) - 32).
a(n) = floor(9/112*(8 + 3*sqrt(7))^(2n-1)).
Limit_{n -> oo} a(n)/a(n-1) = (8 + 3*sqrt(7))^2. (End)
MAPLE
CP := n -> 1+1/2*9*(n^2-n): N:=10: u:=8: v:=1: x:=9: y:=1: k_pcp:=[1]: for i from 1 to N do tempx:=x; tempy:=y; x:=tempx*u+63*tempy*v: y:=tempx*v+tempy*u: s:=(y+1)/2: k_pcp:=[op(k_pcp), CP(s)]: end do: k_pcp; # Steven Schlicker, Apr 24 2007
MATHEMATICA
LinearRecurrence[{255, -255, 1}, {1, 325, 82621}, 12]; (* Ant King, Nov 03 2011 *)
PROG
(PARI) Vec(-x*(x^2+70*x+1)/((x-1)*(x^2-254*x+1)) + O(x^20)) \\ Colin Barker, Jun 22 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane at the suggestion of Richard Choulet, Sep 22 2007
STATUS
approved