1,2

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

S. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, submitted.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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.: 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

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

Cf. A001106, A060544, A048907, A048908.

Adjacent sequences: A048906 A048907 A048908 this_sequence A048910 A048911 A048912

Sequence in context: A031606 A145414 A121000 this_sequence A097739 A048918 A031516

nonn

Eric Weisstein (eric(AT)weisstein.com)

Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Richard Choulet, Sep 22 2007