A107075 Centered square numbers that are also centered pentagonal numbers.

1, 181, 58141, 18721081, 6028129801, 1941039074701, 625008553923781, 201250813324382641, 64802136881897286481, 20866086825157601864101, 6718815155563865902953901, 2163437614004739663149291881

Offset: 1

Author

Richard Choulet, Aug 30 2007, Sep 20 2007

Keywords

• nonn
• easy

Comments

The centered square numbers are n^2 + (n+1)^2 while the centered pentagonal numbers are (5*r^2 + 5*r + 2)/2. A number has both properties iff 5*(2*r+1)^2 = (4*n+2)^2 + 1. We solve the equation 5*Y^2 - 1 = X^2 whose solutions in positive integers are given by A075796 and A007805 respectively. The r values are 0,8,..., i.e., A053606. The n values define A119032.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..399
• Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Crossrefs

Cf. A007805, A053606, A075796, A119032.

Programs

Maple

a:= n-> (Matrix([181,1,1]). Matrix([[323,1,0], [ -323,0,1], [1,0,0]])^n)[1,3]: seq(a(n), n=1..20); # Alois P. Heinz, Aug 14 2008

Mathematica

LinearRecurrence[{323,-323,1},{1,181,58141},20] (* Harvey P. Dale, Nov 15 2018 *)

Formula

G.f.: (z*(1-142*z+z^2))/((1-z)*(1-322*z+z^2)).

a(n+2) = 322*a(n+1)-a(n)-140 with a(1)=1 and a(2)=181.

a(n+1) = 161*a(n)-70+18*(80*a(n)^2-70*a(n)+15)^0.5.

a(n) = (14+(9-4*sqrt(5))^(2*n-1)+(9+4*sqrt(5))^(2*n-1))/32. - Gerry Martens, Jun 06 2015

Extensions

More terms from Alois P. Heinz, Aug 14 2008