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A048909
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9-gonal (or nonagonal) triangular numbers.
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3
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1, 325, 82621, 20985481, 5330229625, 1353857339341, 343874433963061, 87342752369278225, 22184715227362706161, 5634830324997758086741, 1431224717834203191326125, 363525443499562612838749081
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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S. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Math. Mag. 84 (5) (2011) 339.
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LINKS
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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).
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FORMULA
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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)
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MAPLE
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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
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MATHEMATICA
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LinearRecurrence[{255, -255, 1}, {1, 325, 82621}, 12]; (* Ant King, Nov 03 2011 *)
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CROSSREFS
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Cf. A001106, A060544, A048907, A048908.
Sequence in context: A145414 A166220 A121000 * A097739 A203188 A048918
Adjacent sequences: A048906 A048907 A048908 * A048910 A048911 A048912
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KEYWORD
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nonn
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AUTHOR
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Eric W. Weisstein
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EXTENSIONS
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Edited by N. J. A. Sloane at the suggestion of Richard Choulet, Sep 22 2007
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STATUS
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approved
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