Search: id:A005578 Results 1-1 of 1 results found. %I A005578 M0788 %S A005578 1,1,2,3,6,11,22,43,86,171,342,683,1366,2731,5462,10923,21846,43691, %T A005578 87382,174763,349526,699051,1398102,2796203,5592406,11184811,22369622, %U A005578 44739243,89478486,178956971,357913942,715827883,1431655766 %N A005578 a(2n)=2a(2n-1), a(2n+1)=2a(2n)-1. %C A005578 Let u(k), v(k), w(k) be the 3 sequences defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k), v(k+1)=u(k)+w(k), w(k+1)=v(k)+w(k); let M(k)=Max(u(k), v(k),w(k)); then a(n)=M(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 25 2002 %C A005578 Unimodal analogue of Fibonacci numbers: a(n+1)=sum_{k = 0 .. n/2} A071922(n-k, n-2k). Based on the observation that F_{n+1}=sum_k binomial (n-k, k). - Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 30, 2002 %C A005578 Numbers n at which the length of the symmetric signed digit expansion of n with q=2 (i.e. the length of the representation of n in the (-1,0,1)_2 number system) increases. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 30 2003 %C A005578 Row sums of Riordan array (1/(1-x), x/(1-2x^2)). - Paul Barry (pbarry(AT)wit.ie), Apr 24 2005 %C A005578 For n>0 record-values of A107910: a(n)=A107910(A023548(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 28 2005 %C A005578 2^(n+1) = 2*a(n) + 2*A001045(n) + A000975(n-1); e.g. 2^6 = 64 = 2*a(5) + 2*A001045(5) + 2*A000975(4) = 2*11 + 2*11 + 2*10. Let a(n), A001045(n) and A000975(n-1) = the legs of a triangle (a, b, c). Then a(n-1), A001045(n-1) and A000975(n-2) = (S-c), (S-b), (S-a), where S = the triangle semiperimeter. Example: a(5), A001045(5) and A000975(4) = triangle (a, b, c) = (11, 11, 10). Then a(4), A001045(4), A000975(3) = (S-c), (S-b), (S-a) = (6, 5, 5). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 24 2007 %C A005578 a(n)=1+2^(n-1)-a(n-1)=a(n-1)+2a(n-2)-1=a(n-2)+2^(n-2). [From Paul Curtz (bpcrtz(AT)free.fr), Jan 31 2009] %D A005578 Fan Chung and Shlomo Sternberg: "Mathematics and the Buckyball", Fan Chung Graham homepage. %D A005578 R. K. Guy, Graphs and the strong law of small numbers. Graph theory, combinatorics and applications. Vol. 2 (Kalamazoo, MI, 1988), 597-614, Wiley-Intersci. Publ., Wiley, New York, 1991. %D A005578 C. Heuberger and H. Prodinger, On minimal expansions in redundant number systems: Algorithms and quantitative analysis, Computing 66(2001), 377-393. %D A005578 J. P. McSorley, Counting structures in the Moebius ladder, Discrete Math., 184 (1998), 137-164. %D A005578 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A005578 Joerg Arndt, Fxtbook %H A005578 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A005578 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A005578 Eric Weisstein's World of Mathematics, Walsh Function %H A005578 Index entries for sequences related to linear recurrences with constant coefficients %F A005578 ceiling(2^k / 3). %F A005578 The 30 listed terms are given by a(0)=1, a(1)=1 and, for n>1, by a(n)=a(n-1)+a(n-2)+Sum[F(i)*a(n-4-i), i=0, n-4] where the F(i) are Fibonacci numbers - John W. Layman (layman(AT)math.vt.edu), Jan 07 2000. %F A005578 a(n) = (2^(n+1)+3+(-1)^n)/6. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 02 2002 %F A005578 a(n)=A001045(n)+A000035(n+1). G.f.: (1-x-x^2)/((1-x^2)(1-2x)); E.g.f. : (exp(2x)-exp(-x))/3+cosh(x)=(2exp(2x)+3exp(x)+exp(-x))/6. - Paul Barry (pbarry(AT)wit.ie), Jul 20 2003 %F A005578 Binomial transform of A001045(n-1)(-1)^n+0^n/2. - Paul Barry (pbarry(AT)wit.ie), Apr 28 2004 %F A005578 a(n)=(1+A001045(n+1))/2. - Paul Barry (pbarry(AT)wit.ie), Apr 28 2004 %F A005578 a(n)=sum{k=0..n, (-1)^k*sum{j=0..n-k, if(mod(j-k, 2)=0, binomial(n-k, j), 0}} - Paul Barry (pbarry(AT)wit.ie), Jan 25 2005 %F A005578 Let M = the 6x6 adjacency matrix of a benzene ring, (ref.): [0,1,0,0, 0,1; 1,0,1,0,0,0; 0,1,0,1,0,0; 0,0,1,0,1,0; 0,0,0,1,0,1; 1,0,0,0, 1,0]. Then a(n) = leftmost nonzero term of M^n * [1,0,0,0,0,0]. E.g.: a(6) = 22 since M^6 * [1,0,0,0,0,0] = [22,0,21,0,21,0]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 14 2006 %F A005578 Starting (1, 2, 3, 6, 11, 22,...), = row sums of triangle A135229. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 23 2007 %F A005578 Let T = the 3 X 3 matrix [1,1,0; 1,0,1; 0,1,1]. Then T^n * [1,0,0] = [A005578(n), A001045(n), A000975(n-1]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 24 2007 %p A005578 A005578:=-(-1+z+z**2)/(z-1)/(2*z-1)/(z+1); [Conjectured by S. Plouffe in his 1992 dissertation.] %p A005578 with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), a):ZL1:=Prod(begin_blockP, Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, Z), card>=1), Z, end_blockRL):Q:=subs([a=Union(ZL3), b=ZL3], ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, {Q}, unlabelled], size=n), n=2..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 08 2008 %t A005578 a=0;Table[a=2^n-a;(a/2+1)/2,{n,5!}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 22 2009] %o A005578 sage: from sage.combinat.sloane_functions import recur_gen2b sage: it = recur_gen2b(1,1,1,2, lambda n: -1) sage: [it.next() for i in range(33)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008 %Y A005578 Cf. A071922, A072176. %Y A005578 Bisections: A007583 and A047849. %Y A005578 Cf. A000975, (first differences) A001045. %Y A005578 Cf. A135229. %Y A005578 Cf. A001045, A000975. %Y A005578 Sequence in context: A002083 A124973 A043327 this_sequence A058050 A026418 A063895 %Y A005578 Adjacent sequences: A005575 A005576 A005577 this_sequence A005579 A005580 A005581 %K A005578 easy,nonn,new %O A005578 0,3 %A A005578 N. J. A. Sloane (njas(AT)research.att.com). Search completed in 0.002 seconds