Search: id:A004526
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%I A004526
%S A004526 0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,
%T A004526 14,15,15,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,
%U A004526 25,26,26,27,27,28,28,29,29,30,30,31,31,32,32,33,33,34,34,35,35,36,36
%N A004526 Integers repeated.
%C A004526 Number of elements in the set {k: 1 <= 2k <= n}.
%C A004526 Apart from initial term(s), dimension of the space of weight 2n cusp 
               forms for Gamma_0( 2 ).
%C A004526 Number of ways 2^n is expressible as r^2-s^2 with s > 0. Proof: (r+s) 
               and (r-s) both should be powers of 2, even and distinct hence a(2k)=a(2k-1)=(k-1) 
               etc. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 20 2002
%C A004526 Lengths of sides of Ulam square spiral; i.e. lengths of runs of equal 
               terms in A063826. - Donald S. McDonald (don.mcdonald(AT)paradise.net.nz), 
               Jan 09 2003
%C A004526 Number of partitions of n into two parts. A008619 gives partitions of 
               n into at most two parts, so A008619(n) = A004526(n) + 1 for all 
               n >= 0. Partial sums are A002620 (Quarter-squares). - Rick L. Shepherd 
               (rshepherd2(AT)hotmail.com), Feb 27 2004
%C A004526 a(n+1) is the number of 1's in the binary expansion of the Jacobsthal 
               number A001045(n). - Paul Barry (pbarry(AT)wit.ie), Jan 13 2005
%C A004526 Partitions of n+1 into two distinct parts. Example: a(8)=4 because we 
               have [8,1],[7,2],[6,3] and [5,4]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Apr 14 2006
%C A004526 Complement of A000035, since A000035(n)+2*a(n)=n. - Also equal to the 
               partial sums of A000035. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Jun 01 2007
%C A004526 Number of binary bracelets of n beads, two of them 0. For n>=2 a(n-2) 
               is the number of binary bracelets of n beads, two of them 0, with 
               00 prohibited. [From Washington Bomfim (webonfim(AT)bol.com.br), 
               Aug 27 2008]
%D A004526 G. L. Alexanderson et al., The William Powell Putnam Mathematical Competition 
               - Problems and Solutions: 1965-1984, M.A.A., 1985; see Problem A-1 
               of 27-th Competition.
%D A004526 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, P(n,2).
%D A004526 Graham, Knuth and Patashnik, "Concrete Mathematics, Addison-Wesley, NY, 
               1989, page 77 (partitions of n into at most 2 parts).
%H A004526 David Wasserman, <a href="b004526.txt">Table of n, a(n) for n = 0..1000</
               a>
%H A004526 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A004526 John A. Pelesko, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Generalizing 
               the Conway-Hofstadter $10,000 Sequence</a>, Journal of Integer Sequences, 
               Vol. 7 (2004), Article 04.3.5.
%H A004526 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/dimskg0n.gp">
               Dimensions of the spaces S_k(Gamma_0(N))</a>
%H A004526 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/">The 
               modular forms database</a>
%H A004526 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimePartition.html">Link to a section of The World of Mathematics.</
               a>
%H A004526 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A004526 G.f.: x^2*(1+x)/(1-x^2)^2. a(n)=floor(n/2). a(n)=1+a(n-2). a(n)=a(n-1)+a(n-2)-a(n-3). 
               a(2n)=a(2n+1)=n.
%F A004526 For n>0, a(n)=sum(i=1, n, (1/2)/cos(Pi*(2*i-(1-(-1)^n)/2)/(2*n+1))). 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 11 2002
%F A004526 a(n)=(2n-1)/4+(-1)^n/4; a(n+1)=sum{k=0..n, k*(-1)^(n+k)}; - Paul Barry 
               (pbarry(AT)wit.ie), May 20 2003
%F A004526 E.g.f.: ((2x-1)exp(x)+exp(-x))/4; - Paul Barry (pbarry(AT)wit.ie), Sep 
               03 2003
%F A004526 G.f.: 1/(1-x) * sum(k>=0, t^2/(1-t^4), t=x^2^k). - Ralf Stephan, Feb 
               24 2004
%F A004526 a(n+1)=A000120(A001045(n)); - Paul Barry (pbarry(AT)wit.ie), Jan 13 2005
%F A004526 a(n+1)=n-a(n) - Jeremy Bem (jeremy1(AT)gmail.com), Feb 22 2007
%F A004526 a(n)=(n-(1-(-1)^n)/2)/2=1/2*(n-|sin(n*Pi/2)|). Likewise: a(n)=(n-A000035(n))/
               2. Also: a(n)=sum{0<=k<=n, A000035(k)}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Jun 01 2007
%F A004526 The expression floor((x^2-1)/(2*x)) (x >= 1) produces this sequence. 
               - Mohammad K. Azarian (azarian(AT)evansville.edu), Nov 08 2007; corrected 
               by Maximilian Hasler, Nov 17 2008
%F A004526 a(n)=n-a(n-1)-2 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 21 2009]
%e A004526 a(7) = 3, as 128 = 33^2 -31^2 = 18^2-14^2 = 12^2-4^2. a(8) = 3 as 256 
               = 20^2-12^2 = 34^2-30^2 = 65^2-63^2.
%e A004526 For n=2, a(2)=2-0-2=0; n=3, a(3)=3-0-2=1; n=4, a(4)=4-1-2=1; n=5, a(5)=5-1-2=2 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 21 2009]
%p A004526 A004526 := n->floor(n/2); [ seq(floor(i/2),i=0..50) ];
%p A004526 seq(seq(k,j=2..3),k=0..36); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 28 2007
%p A004526 with(combstruct):ZL3:=[S,{S=Set(Cycle(Z,card<3))}, unlabeled]:seq(count(ZL3,
               size=n),n=0..71); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Sep 24 2007
%p A004526 a:=n->add(chrem( [n,j], [1,2] ) ,j=1..n):seq(a(n), n=-1..72);# [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 08 2009]
%t A004526 Table[(2n - 1)/4 + (-1)^n/4, {n, 0, 70}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 02 2006
%o A004526 (PARI) a(n)=n\2 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Mar 25 2009]
%Y A004526 See A008619 for references. Cf. A008619, A001057.
%Y A004526 A001477(n)=A004526(n+1)+A004526(n). A000035(n)=A004526(n+1)-A002456(n).
%Y A004526 a(n)=A008284(n, 2), n >= 1.
%Y A004526 Zero followed by the partial sums of A000035.
%Y A004526 Cf. A002620.
%Y A004526 Column 2 of triangle A094953.
%Y A004526 Cf. A002264, A002265, A002266, A010761, A010762, A110532, A110533.
%Y A004526 Partial sums: A002620. Other related sequences: A002264, A002265, A002266, 
               A010872, A010873, A010874.
%Y A004526 Sequence in context: A065033 A001057 A130472 this_sequence A140106 A123108 
               A008619
%Y A004526 Adjacent sequences: A004523 A004524 A004525 this_sequence A004527 A004528 
               A004529
%K A004526 nonn,easy,core,nice,new
%O A004526 0,5
%A A004526 N. J. A. Sloane (njas(AT)research.att.com).

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