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 A048909 9-gonal (or nonagonal) triangular numbers. 4
 1, 325, 82621, 20985481, 5330229625, 1353857339341, 343874433963061, 87342752369278225, 22184715227362706161, 5634830324997758086741, 1431224717834203191326125, 363525443499562612838749081, 92334031424171069457850940521, 23452480456295952079681300143325 (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 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 (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 *) 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 Cf. A001106, A060544, A048907, A048908. Sequence in context: A266365 A166220 A121000 * A097739 A203188 A048918 Adjacent sequences:  A048906 A048907 A048908 * A048910 A048911 A048912 KEYWORD nonn,easy AUTHOR EXTENSIONS Edited by N. J. A. Sloane at the suggestion of Richard Choulet, Sep 22 2007 STATUS approved

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