0,2

Number of walks of length 2n+2 between any two diametrically opposite vertices of the cycle graph C_8. - Herbert Kociemba (kociemba(AT)t-online.de), Jul 02 2004

If we consider a(4k+2), then 2^4 == 3^4 == 3 (mod 13); 2^(4k+2) + 3^(4k+2) == 3^k(4+9) == 3*0 == 0 (mod 13). So a(4k+2) can never be prime. - Jose Brox, Dec 27 2005

If k is odd, then a(nk) is divisible by a(n), since: a(nk) = (2^n)^k + (3^n)^k = (2^n + 3^n) [(2^n)^(k-1) - (2^n)^(k-2) (3^n) + - ... + (3^n)^(k-1)]. So the only possible primes in the sequence are a(0) and a(2^n) for n>=1. I've checked that a(2^n) is composite for 3 <= n <= 15. As with Fermat primes, a probabilistic argument suggests that there are only finitely many primes in the sequence. - Dean Hickerson, Dec 27 2005

Let x,y,z be elements from some power set P(n), i.e. the power set of a set of n elements. Define a function f(x,y,z) in the following manner: f(x,y,z) = 1 if x is a subset of y and y is a subset of z and x does not equal z; f(x,y,z) = 0 if x is not a subset of y or y is not a subset of z or x equals z. Now sum f(x,y,z) for all x,y,z of P(n). This gives a(n). - Ross La Haye (rlahaye(AT)new.rr.com), Dec 26 2005

Number of monic (irreducible) polynomials of degree 1 over GF(2^n). - Max Alekseyev (maxale(AT)gmail.com), Jan 13 2006

a(n) = A099393(n)-A000225(n+1) = A083420(n)-A099393(n); in binary representation, n>0: n ones followed by n zeros (A138147(n)); A000120(a(n))=n; A023416(a(n))=n; A070939(a(n))=2*n; 2*a(n)+1=A030101(A099393(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 07 2006, Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Aug 02 2009

Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then a(n) = the number of (x,y) of B for which x does not equal y. - Ross La Haye (rlahaye(AT)new.rr.com), Jan 02 2008

Putnam Exam. Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.

Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From Ross La Haye (rlahaye(AT)new.rr.com), Mar 13 2009]

G.f.: 2x/((-1 + 2x)(-1 + 4x)), a(n)=6a(n-1)-8a(n-2). - Herbert Kociemba (kociemba(AT)t-online.de), Jul 02 2004

E.g.f.: e^(4*x)-e^(2*x). [From Mohammad K. Azarian (azarian(AT)evansville.edu), Jan 14 2009]

n=5: a(5)=4^5-2^5=1024-32=992 -> '1111100000'.

[seq (((stirling2(n, 2))^2-1)/4, n=2..24)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 20 2006

with(finance):seq(add(futurevalue(2, 1, n+k), k=0..n), n=-1..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2008

(Other) sage: [4^n - 2^n for n in xrange(0, 23)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2009]

Ratio of successive terms of A028365.

Cf. A000225, A060867.

Sequence in context: A127216 A006659 A127221 this_sequence A037130 A078543 A084128

Adjacent sequences: A020519 A020520 A020521 this_sequence A020523 A020524 A020525

nonn

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)