A001445 a(n) = (2^n + 2^[ n/2 ] )/2.

3, 5, 10, 18, 36, 68, 136, 264, 528, 1040, 2080, 4128, 8256, 16448, 32896, 65664, 131328, 262400, 524800, 1049088, 2098176, 4195328, 8390656, 16779264, 33558528, 67112960, 134225920, 268443648

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is union of A007582(n-1) and A164051(n). - Jaroslav Krizek, Aug 14 2009

Number of binary strings of length n+1, not counting strings which are the reversal, the complement, or the reversal of the complement of each other as different. - Christian Barrientos, Jun 06 2025

Examples

			G.f. = 3*x^2 + 5*x^3 + 10*x^4 + 18*x^5 + 36*x^6 + 68*x^7 + 136*x^8 + ...
		

Links

• James Spahlinger, Table of n, a(n) for n = 2..1001
• Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Crossrefs

Cf. A056309, A052957.

Programs

Maple

f := n->(2^n+2^floor(n/2))/2;

Mathematica

Table[(2^n + 2^(Floor[n/2]))/2, {n, 2, 50}] (* G. C. Greubel, Sep 08 2017 *)
LinearRecurrence[{2,2,-4},{3,5,10},30] (* Harvey P. Dale, Sep 12 2021 *)

PARI

for(n=2,50, print1((2^n + 2^(n\2))/2, ", ")) \\ G. C. Greubel, Sep 08 2017

Formula

a(n) = (1/2)*A005418(n+2).

G.f.: x^2*(3-x-6*x^2)/((1-2*x)*(1-2*x^2)).

G.f.: 3*G(0) where G(k) = 1 + x*(4*2^k + 1)*(1 + 2*x*G(k+1))/(1 + 2*2^k). - Sergei N. Gladkovskii, Dec 12 2011 [Edited by Michael Somos, Sep 09 2013]

a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3) for n > 4. - Chai Wah Wu, Sep 10 2020

E.g.f.: (2*cosh(2*x) + 2*cosh(sqrt(2)*x) + 2*sinh(2*x) + sqrt(2)*sinh(sqrt(2)*x) - 4 - 6*x)/4. - Stefano Spezia, Jun 14 2025