login
This site is supported by donations to The OEIS Foundation.

 

Logo

110 people attended OEIS-50 (videos, suggestions); annual fundraising drive to start soon (donate); editors, please edit! (stack is over 300), your editing is more valuable than any donation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A048909 9-gonal (or nonagonal) triangular numbers. 3
1, 325, 82621, 20985481, 5330229625, 1353857339341, 343874433963061, 87342752369278225, 22184715227362706161, 5634830324997758086741, 1431224717834203191326125, 363525443499562612838749081 (list; graph; refs; listen; history; text; internal format)
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 (schlicks(AT)gvsu.edu), Apr 24 2007

lim(n -> Infinity, a(n)/a(n-1) = (8 + 3*sqrt(7))^2. - Ant King, Nov 03 2011

REFERENCES

S. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Math. Mag. 84 (5) (2011) 339.

LINKS

Table of n, a(n) for n=1..12.

Eric Weisstein's World of Mathematics, Nonagonal Triangular Number.

Index to sequences with 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 (schlicks(AT)gvsu.edu), 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))

(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 (schlicks(AT)gvsu.edu), Apr 24 2007

MATHEMATICA

LinearRecurrence[{255, -255, 1}, {1, 325, 82621}, 12]; (* Ant King, Nov 03 2011 *)

CROSSREFS

Cf. A001106, A060544, A048907, A048908.

Sequence in context: A145414 A166220 A121000 * A097739 A203188 A048918

Adjacent sequences:  A048906 A048907 A048908 * A048910 A048911 A048912

KEYWORD

nonn

AUTHOR

Eric W. Weisstein

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified October 31 06:33 EDT 2014. Contains 248845 sequences.