Search: id:A048909
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%I A048909
%S A048909 1,325,82621,20985481,5330229625,1353857339341,343874433963061,
%T A048909 87342752369278225,22184715227362706161,5634830324997758086741,
%U A048909 1431224717834203191326125,363525443499562612838749081
%N A048909 9-gonal (or nonagonal) triangular numbers.
%C A048909 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.
%C A048909 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.
%C A048909 Also, numbers simultaneously 9-gonal and centered 9-gonal, the intersection 
               of A001106 and A060544. - Steven Schlicker (schlicks(AT)gvsu.edu), 
               Apr 24 2007
%D A048909 S. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, 
               submitted.
%H A048909 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NonagonalTriangularNumber.html">Link to a section of The World of 
               Mathematics.</a>
%F A048909 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
%F A048909 a(n+1)=254*a(n+1)-a(n)+72. - Richard Choulet, Sep 22 2007
%F A048909 a(n+1)=127*a(n+1)+36+6*(448*a(n)^2+256*a(n)+25)^0.5. - Richard Choulet, 
               Sep 22 2007
%F A048909 G.f.: f(z)=a(1)*z+a(2)*z^2+...=((z*(1+70*z+z^2))/((1-z)*(1-254*z+z^2)). 
               - Richard Choulet, Sep 22 2007
%p A048909 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
%Y A048909 Cf. A001106, A060544, A048907, A048908.
%Y A048909 Sequence in context: A145414 A166220 A121000 this_sequence A097739 A048918 
               A031516
%Y A048909 Adjacent sequences: A048906 A048907 A048908 this_sequence A048910 A048911 
               A048912
%K A048909 nonn
%O A048909 1,2
%A A048909 Eric Weisstein (eric(AT)weisstein.com)
%E A048909 Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion 
               of Richard Choulet, Sep 22 2007

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