%I A007582 M2849
%S A007582 1,3,10,36,136,528,2080,8256,32896,131328,524800,2098176,8390656,
%T A007582 33558528,134225920,536887296,2147516416,8590000128,34359869440,
%U A007582 137439215616,549756338176,2199024304128,8796095119360,35184376283136
%N A007582 2^(n-1)*(1+2^n).
%C A007582 Let G_n be the elementary Abelian group G_n = (C_2)^n for n >= 1: A006516 
               is the number of times the number -1 appears in the character table 
               of G_n and A007582 is the number of times the number 1. Together 
               the two sequences cover all the values in the table i.e. A006516(n) 
               + A007582(n) = 2^(2n). - Ahmed Fares (ahmedfares(AT)my-deja.com), 
               Jun 01 2001
%C A007582 Number of walks of length 2n+1 between two adjacent vertices in the cycle 
               graph C_8. Example: a(1)=3 because in the cycle ABCDEFGH we have 
               three walks of length 3 between A and B: ABAB, ABCB and AHAB. - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004
%C A007582 Smallest number containing in its binary representation two equal non-overlapping 
               subwords of length n: A097295(a(n))=n and A097295(m)<n for m<a(n). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 04 2004
%C A007582 a(n)^2 + (A006516(n))^2 = a(2n). E.g. a(3) = 36, A006516(3) = 28, a(6) 
               = 2080. 36^2 + 28^2 = 2080. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jun 17 2006
%C A007582 Let P(A) be the power set of an n-element set A. Then a(n) = the number 
               of pairs of elements {x,y} of P(A) for which either x equals y or 
               x does not equal y. - Ross La Haye (rlahaye(AT)new.rr.com), Jan 02 
               2008
%C A007582 Let P(A) be the power set of an n-element set A. Then a(n) = the number 
               of pairs of elements {x,y} of P(A). This is just a simpler statement 
               of my previous comment for this sequence. - Ross La Haye (rlahaye(AT)new.rr.com), 
               Jan 10 2008
%C A007582 For n>0: A000120(a(n))=2, A023414(a(n))=2*(n-1), A087117(a(n))=n-1. [From 
               Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 23 2009]
%C A007582 a(n+1) written in base 2: 11, 1010, 100100, 10001000, 1000010000, ..., 
               i.e. number 1, n times 0, number 1, n times 0 (A163449(n)). [From 
               Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jul 27 2009]
%C A007582 a(n) for n >= 1 is a bisection of A001445(n+1). [From Jaroslav Krizek 
               (jaroslav.krizek(AT)atlas.cz), Aug 14 2009]
%D A007582 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007582 S. Hong and J. H. Kwak, Regular fourfold covering with respect to the 
               identity automorphism, J. Graph Theory, 17 (1993), 621-627.
%D A007582 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), Feb 22 2009]
%H A007582 T. D. Noe, <a href="b007582.txt">Table of n, a(n) for n=0..200</a>
%H A007582 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A007582 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=168">
               Encyclopedia of Combinatorial Structures 168</a>
%F A007582 G.f.: (1-3*x)/((1-2*x)*(1-4*x)). C(1+2^n, 2) where C(n, 2) is n-th triangular 
               number A000217.
%F A007582 Binomial transform of A007051. Inverse binomial transform of A081186. 
               - Paul Barry (pbarry(AT)wit.ie), Apr 07 2003
%F A007582 E.g.f. exp(3x)cosh(x) - Paul Barry (pbarry(AT)wit.ie), Apr 07 2003
%F A007582 a(n)=sum{k=0..floor(n/2); C(n, 2k)3^(n-2k) } - Paul Barry (pbarry(AT)wit.ie), 
               May 08 2003
%F A007582 a(n+1) = 4*a(n) - 2^n; see also A049775. a(n) = 2^(n-1)*A000051(n). - 
               DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 20 2004
%F A007582 a(n)=6a(n-1)-8a(n-2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 
               01 2004
%F A007582 Row sums of triangle A134308 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 19 2007
%F A007582 a(n) = StirlingS2(2^n + 1,2^n) = 1 + 2*StirlingS2(n+1,2) + 3*StirlingS2(n+1,
               3) + 3*StirlingS2(n+1,4) = StirlingS2(n+2,2) + 3(StirlingS2(n+1,3) 
               + StirlingS2(n+1,4)). - Ross La Haye (rlahaye(AT)new.rr.com), Mar 
               01 2008
%F A007582 a(n) = StirlingS2(2^n + 1,2^n) = 1 + 2*StirlingS2(n+1,2) + 3*StirlingS2(n+1,
               3) + 3*StirlingS2(n+1,4) = StirlingS2(n+2,2) + 3(StirlingS2(n+1,3) 
               + StirlingS2(n+1,4)). - Ross La Haye (rlahaye(AT)new.rr.com), Apr 
               02 2008 Ross
%F A007582 a(n)= A000079(n)+A006516(n). - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), 
               Aug 06 2008
%F A007582 a(n) = A028403(n+1) / 4. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               Jul 27 2009]
%p A007582 seq(binomial(-2^n, 2), n=0..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 22 2008
%t A007582 Table[ Binomial[2^n + 1, 2], {n, 0, 23}] (from Robert G. Wilson v Jul 
               30 2004)
%o A007582 (PARI) a(n)=if(n<0,0,2^(n-1)*(1+2^n))
%o A007582 (PARI) a(n)=sum(k=-n\4,n\4,binomial(2*n+1,n+1+4*k))
%Y A007582 Cf. A000217, A049773, A049775.
%Y A007582 Cf. A006516.
%Y A007582 Cf. A134308.
%Y A007582 Cf. A000225, A000392, A032263, a028243, A000079.
%Y A007582 Sequence in context: A081909 A126189 A122448 this_sequence A026854 A136576 
               A129156
%Y A007582 Adjacent sequences: A007579 A007580 A007581 this_sequence A007583 A007584 
               A007585
%K A007582 nonn,easy,nice
%O A007582 0,2
%A A007582 Simon Plouffe (simon.plouffe(AT)gmail.com)