A048909 9-gonal (or nonagonal) triangular numbers.

1, 325, 82621, 20985481, 5330229625, 1353857339341, 343874433963061, 87342752369278225, 22184715227362706161, 5634830324997758086741, 1431224717834203191326125, 363525443499562612838749081, 92334031424171069457850940521, 23452480456295952079681300143325

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A001106, A060544, A048907, A048908.

Programs

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 *)

PARI

Vec(-x*(x^2+70*x+1)/((x-1)*(x^2-254*x+1)) + O(x^20)) \\ Colin Barker, Jun 22 2015

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)

Extensions

Edited by N. J. A. Sloane at the suggestion of Richard Choulet, Sep 22 2007