0,5

Number of elements in the set {k: 1 <= 2k <= n}.

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 2 ).

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

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

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

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

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

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

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]

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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, P(n,2).

Graham, Knuth and Patashnik, "Concrete Mathematics, Addison-Wesley, NY, 1989, page 77 (partitions of n into at most 2 parts).

David Wasserman, Table of n, a(n) for n = 0..1000

Index entries for sequences related to linear recurrences with constant coefficients

John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.

William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))

William A. Stein, The modular forms database

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for "core" sequences

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.

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

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

E.g.f.: ((2x-1)exp(x)+exp(-x))/4; - Paul Barry (pbarry(AT)wit.ie), Sep 03 2003

G.f.: 1/(1-x) * sum(k>=0, t^2/(1-t^4), t=x^2^k). - Ralf Stephan, Feb 24 2004

a(n+1)=A000120(A001045(n)); - Paul Barry (pbarry(AT)wit.ie), Jan 13 2005

a(n+1)=n-a(n) - Jeremy Bem (jeremy1(AT)gmail.com), Feb 22 2007

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

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

a(n)=n-a(n-1)-2 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 21 2009]

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.

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]

A004526 := n->floor(n/2); [ seq(floor(i/2), i=0..50) ];

seq(seq(k, j=2..3), k=0..36); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 28 2007

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

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]

Table[(2n - 1)/4 + (-1)^n/4, {n, 0, 70}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006

(PARI) a(n)=n\2 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 25 2009]

See A008619 for references. Cf. A008619, A001057.

A001477(n)=A004526(n+1)+A004526(n). A000035(n)=A004526(n+1)-A002456(n).

a(n)=A008284(n, 2), n >= 1.

Zero followed by the partial sums of A000035.

Cf. A002620.

Column 2 of triangle A094953.

Cf. A002264, A002265, A002266, A010761, A010762, A110532, A110533.

Partial sums: A002620. Other related sequences: A002264, A002265, A002266, A010872, A010873, A010874.

Sequence in context: A065033 A001057 A130472 this_sequence A140106 A123108 A008619

Adjacent sequences: A004523 A004524 A004525 this_sequence A004527 A004528 A004529

nonn,easy,core,nice,new

N. J. A. Sloane (njas(AT)research.att.com).