A117864 Number of palindromes (in base 5) below 5^n.

4, 8, 28, 48, 148, 248, 748, 1248, 3748, 6248, 18748, 31248, 93748, 156248, 468748, 781248, 2343748, 3906248, 11718748, 19531248, 58593748, 97656248, 292968748, 488281248, 1464843748, 2441406248, 7324218748, 12207031248, 36621093748, 61035156248

Offset: 1

Author

Martin Renner, May 02 2006

Keywords

• nonn
• base
• easy
• changed

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

Crossrefs

Cf. A050250, A238366.

Programs

Mathematica

pal5s[n_]:=If[OddQ[n],6 5^((n-1)/2)-2,2 5^(n/2)-2]; Array[pal5s,40] (* or *) LinearRecurrence[{1,5,-5},{4,8,28},40] (* Harvey P. Dale, May 30 2012 *)

PARI

Vec(4*x*(x+1)/((x-1)*(5*x^2-1)) + O(x^100)) \\ Colin Barker, Apr 26 2015

Formula

a(n) = 6*5^((n-1)/2)-2 (n odd), 2*5^(n/2)-2 (n even).

a(1)=4, a(2)=8, a(3)=28, a(n)=a(n-1)+5*a(n-2)-5*a(n-3). - Harvey P. Dale, May 30 2012

G.f.: 4*x*(x+1) / ((x-1)*(5*x^2-1)). - Colin Barker, Apr 26 2015

a(n) = 4*A238366(n-1). - Michel Marcus, Apr 26 2015

E.g.f.: 2*cosh(sqrt(5)*x) - 2*cosh(x) - 2*sinh(x) + 6*sinh(sqrt(5)*x)/sqrt(5). - Stefano Spezia, Aug 29 2025

Extensions

More terms from Harvey P. Dale, May 30 2012