A048908 -id:A048908 - cp's OEIS frontend

A048907 Indices of 9-gonal numbers which are also triangular.

1, 10, 154, 2449, 39025, 621946, 9912106, 157971745, 2517635809, 40124201194, 639469583290, 10191389131441, 162422756519761, 2588572715184730, 41254740686435914, 657487278267789889, 10478541711598202305, 166999180107303446986, 2661508340005256949466

Offset: 1

Eric W. Weisstein

Entries are == 1 (mod 3). - N. J. A. Sloane, Sep 22 2007

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

Colin Barker, Table of n, a(n) for n = 1..832
Eric Weisstein's World of Mathematics, Nonagonal Triangular Number.
Index entries for linear recurrences with constant coefficients, signature (17,-17,1).

Cf. A001080, A073352, A048908, A048909.

LinearRecurrence[{17, -17, 1}, {1, 10, 154}, 17]; (* Ant King, Nov 03 2011 *)

Vec(-x*(x^2-7*x+1)/((x-1)*(x^2-16*x+1)) + O(x^20)) \\ Colin Barker, Jun 22 2015

G.f.: x*(1-7*x+x^2)/((1-x)*(1-16*x+x^2)).
a(n+2) = 16*a(n+1)-a(n)-5, a(n+1) = 8*a(n)-2.5+1.5*(28*a(n)^2-20*a(n)+1)^0.5. - Richard Choulet, Sep 22 2007
From Ant King, Nov 03 2011: (Start)
a(n) = 17*a(n-1) - 17*a(n-2) + a(n-3).
a(n) = ceiling(3/28*(3-sqrt(7))*(8 + 3*sqrt(7))^n).
(End)
a(n) = A097830(n-1)-7*A097830(n-2)+A097830(n-3). - R. J. Mathar, Jul 04 2024

A048909 9-gonal (or nonagonal) triangular numbers.

Original entry on oeis.org

1, 325, 82621, 20985481, 5330229625, 1353857339341, 343874433963061, 87342752369278225, 22184715227362706161, 5634830324997758086741, 1431224717834203191326125, 363525443499562612838749081, 92334031424171069457850940521, 23452480456295952079681300143325

Offset: 1

Eric W. Weisstein

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, Apr 24 2007

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).

Cf. A001106, A060544, A048907, A048908.

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

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

Vec(-x*(x^2+70*x+1)/((x-1)*(x^2-254*x+1)) + O(x^20)) \\ Colin Barker, Jun 22 2015

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
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)).
Limit_{n -> oo} a(n)/a(n-1) = (8 + 3*sqrt(7))^2. (End)

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

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A048907 Indices of 9-gonal numbers which are also triangular.

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