This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A048909 #37 Feb 16 2025 08:32:40
%S A048909 1,325,82621,20985481,5330229625,1353857339341,343874433963061,
%T A048909 87342752369278225,22184715227362706161,5634830324997758086741,
%U A048909 1431224717834203191326125,363525443499562612838749081,92334031424171069457850940521,23452480456295952079681300143325
%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_, Apr 24 2007
%H A048909 Colin Barker, <a href="/A048909/b048909.txt">Table of n, a(n) for n = 1..416</a>
%H A048909 S. C. Schlicker, <a href="http://www.jstor.org/stable/10.4169/math.mag.84.5.339">Numbers Simultaneously Polygonal and Centered Polygonal</a>, Mathematics Magazine, Vol. 84, No. 5, December 2011, pp. 339-350.
%H A048909 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NonagonalTriangularNumber.html">Nonagonal Triangular Number.</a>
%H A048909 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (255,-255,1).
%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_, 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.: z*(1+70*z+z^2)/((1-z)*(1-254*z+z^2)). - _Richard Choulet_, Sep 22 2007
%F A048909 From _Ant King_, Nov 03 2011: (Start)
%F A048909 a(n) = 255*a(n-1) - 255*a(n-2) + a(n-3).
%F A048909 a(n) = 1/112*(9*(8 + 3*sqrt(7))^(2n-1) + 9*(8-3* sqrt(7))^(2n-1) - 32).
%F A048909 a(n) = floor(9/112*(8 + 3*sqrt(7))^(2n-1)).
%F A048909 Limit_{n -> oo} a(n)/a(n-1) = (8 + 3*sqrt(7))^2. (End)
%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_, Apr 24 2007
%t A048909 LinearRecurrence[{255, -255, 1}, {1, 325, 82621}, 12]; (* _Ant King_, Nov 03 2011 *)
%o A048909 (PARI) Vec(-x*(x^2+70*x+1)/((x-1)*(x^2-254*x+1)) + O(x^20)) \\ _Colin Barker_, Jun 22 2015
%Y A048909 Cf. A001106, A060544, A048907, A048908.
%K A048909 nonn,easy
%O A048909 1,2
%A A048909 _Eric W. Weisstein_
%E A048909 Edited by _N. J. A. Sloane_ at the suggestion of _Richard Choulet_, Sep 22 2007