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 A007582 a(n) = 2^(n-1)*(1+2^n). (Formerly M2849) 54
 1, 3, 10, 36, 136, 528, 2080, 8256, 32896, 131328, 524800, 2098176, 8390656, 33558528, 134225920, 536887296, 2147516416, 8590000128, 34359869440, 137439215616, 549756338176, 2199024304128, 8796095119360, 35184376283136 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 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, Apr 01 2004 Smallest number containing in its binary representation two equal non-overlapping subwords of length n: A097295(a(n))=n and A097295(m)0: A000120(a(n))=2, A023414(a(n))=2*(n-1), A087117(a(n))=n-1. - Reinhard Zumkeller, Jun 23 2009 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)). - Jaroslav Krizek, Jul 27 2009 a(n) for n >= 1 is a bisection of A001445(n+1). - Jaroslav Krizek, Aug 14 2009 Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times sum_{k=0..n} binomial(n,k)*k^q, then A007582(x)= sum_{k=0..x-1} T(x,k)*2^k. - John M. Campbell, Nov 16 2011 a(n) gives the number of pairs (r, s) such that 0 <= r <= s <= (2^n)-1 that satisfy AND(r, s, XOR(r, s)) = 0. - Ramasamy Chandramouli, Aug 30 2012 a(n) = A000217(2^n) = 2^(2n-1) + 2^(n-1) is the nearest triangular number above 2^(2n-1); cf. A006516, A233327. - Antti Karttunen, Feb 26 2014 Consider the quantum spin-1/2 chain with even number of sites L (physics, condensed matter theory). The spectrum of the Hamiltonian can be classified according to symmetries. If the only symmetry of the spin Hamiltonian is Parity, i.e., reflection with respect to the middle of the chain (see e.g. the transverse-field Ising model with open boundary conditions), then the dimension of the p=+1 parity sector is given by a(n) with n=L/2. - Marin Bukov, Mar 11 2016 a(n) is also the total number of words of length n, over an alphabet of four letters, of which one of them appears an even number of times. See the Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter odd case), and the Balakrishnan reference there. For the 1- to 11-letter cases, see the crossrefs. - Wolfdieter Lang, Jul 17 2017 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n=0..200 S. Hong and J. H. Kwak, Regular fourfold coverings with respect to the identity automorphism, J. Graph Theory, 17 (1993), 621-627. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 168 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. Mircea Merca, A Note on Cosine Power Sums, J. Integer Sequences, Vol. 15 (2012), Article 12.5.3. Index entries for linear recurrences with constant coefficients, signature (6,-8) FORMULA 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. Binomial transform of A007051. Inverse binomial transform of A081186. - Paul Barry, Apr 07 2003 E.g.f.: exp(3*x)*cosh(x). - Paul Barry, Apr 07 2003 a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)3^(n-2k). - Paul Barry, May 08 2003 a(n+1) = 4*a(n) - 2^n; see also A049775. a(n) = 2^(n-1)*A000051(n). - Philippe Deléham, Feb 20 2004 a(n) = 6*a(n-1) - 8*a(n-2). - Emeric Deutsch, Apr 01 2004 Row sums of triangle A134308. - Gary W. Adamson, Oct 19 2007 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, Mar 01 2008 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, Apr 02 2008 a(n) = A000079(n) + A006516(n). - Yosu Yurramendi, Aug 06 2008 a(n) = A028403(n+1) / 4. - Jaroslav Krizek, Jul 27 2009 a(n) = sum_{k=-floor(n/4)..floor(n/4)} binomial(2*n,n+4*k)/2. - Mircea Merca, Jan 28 2012 G.f.: Q(0)/2 where Q(k)=  1 + 2^k/(1 - 2*x/(2*x + 2^k/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 10 2013 a(n) = Sum_{k=1..2^n} k. - Joerg Arndt, Sep 01 2013 a(n) = (Sum_{k=2^n..2^(n+1)}k)/3. - J. M. Bergot, Jan 26 2015 a(n+1) = 2*a(n) + 4^n. - Yuchun Ji, Mar 10 2017 MAPLE seq(binomial(-2^n, 2), n=0..23); # Zerinvary Lajos, Feb 22 2008 MATHEMATICA Table[ Binomial[2^n + 1, 2], {n, 0, 23}] (* Robert G. Wilson v, Jul 30 2004 *) LinearRecurrence[{6, -8}, {1, 3}, 30] (* Harvey P. Dale, Apr 08 2013 *) PROG (PARI) a(n)=if(n<0, 0, 2^(n-1)*(1+2^n)) (PARI) a(n)=sum(k=-n\4, n\4, binomial(2*n+1, n+1+4*k)) (Maxima) A007582(n):=2^(n-1)*(1+2^n)\$ makelist(A007582(n), n, 0, 30); /* Martin Ettl, Nov 15 2012 */ CROSSREFS Cf. A000217, A049773, A049775. Cf. A006516. Cf. A134308. Cf. A000225, A000392, A032263, A028243, A000079. Cf. A102573. The number of words of length n with m letters, one of them appearing an even number of times is for m = 1..11: A000035, A011782,  A007051, A007582, A081186, A081187, A081188, A081189, A081190, A060531, A081192. - Wolfdieter Lang, Jul 17 2017 Sequence in context: A081909 A126189 A122448 * A026854 A136576 A129156 Adjacent sequences:  A007579 A007580 A007581 * A007583 A007584 A007585 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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