A128922 Numbers simultaneously 10-gonal and centered 10-gonal.

1, 451, 145351, 46802701, 15070324501, 4852597686751, 1562521384809451, 503127033310956601, 162005342204743216201, 52165217062894004660251, 16797037888909664757384751

Offset

0,2

Links

Table of n, a(n) for n=0..10.

S. C. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine, Vol. 84, No. 5, December 2011, pp. 339-350.

Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Formula

Let x(n) + y(n)*sqrt(80) =: (10+sqrt(80))*(9+sqrt(80))^n, s(n) = (y(n)+1)/2; then a(n) = (1/2)*(2+10*(s(n)^2-s(n))).

From Richard Choulet, Oct 01 2007: (Start)

a(n+2) = 322*a(n+1)-a(n)+130.

a(n+1) = 161*a(n)+65+9*(320*a(n)^2+260*a(n)+45)^0.5.

G.f.: z*(1+128*z+z^2)/((1-z)*(1-322*z+z^2)). (End)

Example

a(1) = 451 because 451 is the tenth centered 10-gonal number and the eleventh 10-gonal number.

Maple

CP := n -> 1+1/2*10*(n^2-n): N:=10: u:=9: v:=1: x:=10: y:=1: k_pcp:=[1]: for i from 1 to N do tempx:=x; tempy:=y; x:=tempx*u+80*tempy*v: y:=tempx*v+tempy*u: s:=(y+1)/2: k_pcp:=[op(k_pcp), CP(s)]: end do: k_pcp;

Crossrefs

Cf. A001107, A062786, A048909.

Sequence in context: A020268 A345319 A066322 * A224560 A224568 A224561

Adjacent sequences: A128919 A128920 A128921 * A128923 A128924 A128925

Keyword

easy,nonn

Author

Steven Schlicker, Apr 24 2007

Status

approved