%I A020522
%S A020522 0,2,12,56,240,992,4032,16256,65280,261632,1047552,4192256,
%T A020522 16773120,67100672,268419072,1073709056,4294901760,17179738112,
%U A020522 68719214592,274877382656,1099510579200,4398044413952,17592181850112
%N A020522 4^n - 2^n.
%C A020522 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
%C A020522 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
%C A020522 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
%C A020522 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
%C A020522 Number of monic (irreducible) polynomials of degree 1 over GF(2^n). - 
               Max Alekseyev (maxale(AT)gmail.com), Jan 13 2006
%C A020522 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
%C A020522 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
%D A020522 Putnam Exam. Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; 
               see p. 725.
%D A020522 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]
%F A020522 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
%F A020522 E.g.f.: e^(4*x)-e^(2*x). [From Mohammad K. Azarian (azarian(AT)evansville.edu), 
               Jan 14 2009]
%e A020522 n=5: a(5)=4^5-2^5=1024-32=992 -> '1111100000'.
%p A020522 [seq (((stirling2(n,2))^2-1)/4,n=2..24)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Dec 20 2006
%p A020522 with(finance):seq(add(futurevalue(2, 1, n+k),k=0..n),n=-1..21); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2008
%o A020522 (Other) sage: [4^n - 2^n for n in xrange(0,23)] # [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 05 2009]
%Y A020522 Ratio of successive terms of A028365.
%Y A020522 Cf. A000225, A060867.
%Y A020522 Sequence in context: A127216 A006659 A127221 this_sequence A037130 A078543 
               A084128
%Y A020522 Adjacent sequences: A020519 A020520 A020521 this_sequence A020523 A020524 
               A020525
%K A020522 nonn
%O A020522 0,2
%A A020522 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)