A117867 Number of palindromes (in base 8) below 8^n.

7, 14, 70, 126, 574, 1022, 4606, 8190, 36862, 65534, 294910, 524286, 2359294, 4194302, 18874366, 33554430, 150994942, 268435454, 1207959550, 2147483646, 9663676414, 17179869182, 77309411326, 137438953470, 618475290622, 1099511627774, 4947802324990

Offset: 1

Author

Martin Renner, May 02 2006

Keywords

• nonn
• base
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,8,-8).

Crossrefs

Cf. A050250.

Programs

Mathematica

Table[If[OddQ[n],9*8^((n-1)/2)-2, 2*8^(n/2)-2], {n,1,25}] (* or *) LinearRecurrence[{1,8,-8}, {7,14,70}, 25] (* G. C. Greubel, Oct 27 2016 *)
Rest@ CoefficientList[Series[7 x (x + 1)/((x - 1) (8 x^2 - 1)), {x, 0, 27}], x] (* Michael De Vlieger, Oct 27 2016 *)

Formula

a(n) = 9*8^((n-1)/2)-2 (n odd), 2*8^(n/2)-2 (n even).

G.f.: 7*x*(x+1) / ((x-1)*(8*x^2-1)). - Colin Barker, Feb 15 2013

From G. C. Greubel, Oct 27 2016: (Start)

a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3).

a(n) = (1/(4*sqrt(2)))*( -8*sqrt(2) + 9*(1 - (-1)^n)*2^(3*n/2) + (1 + (-1)^n)*2^((3*n+5)/2) ).

E.g.f.: 2*cosh(2*sqrt(2)*x) + (9/(2*sqrt(2)))*sinh(2*sqrt(2)*x) -2*cosh(x) - 2*sinh(x). (End)

Extensions

More terms from Colin Barker, Feb 15 2013