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A004526
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Nonnegative integers repeated, floor(n/2).
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205
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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, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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]
Let A be the Hessenberg n by n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n+1)=(-1)^n det(A). [From Milan R. Janjic (agnus(AT)blic.net), Jan 24 2010]
Let RT abbreviate rank transform (A187224). Then
RT(A004526)=A187484;
RT(A004526 without 1st term)=A026371;
RT(A004526 without 1st 2 terms)=A026367;
RT(A004526 without 1st 3 terms)=A026363. [From Clark Kimberling, Mar 10 2011]
The diameter (longest path) of the n-cycle. -Cade Herron (herrona (AT) goldmail.etsu.edu) April 14, 2011
For n>=3, a(n-1) is the number of two-color bracelets of n beads, three of them are black, having a diameter of symmetry. [Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 3 2011]
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REFERENCES
| 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).
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LINKS
| 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, Prime Partition
Index entries for "core" sequences
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FORMULA
| 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+1) = A002378(n) - A035608(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 27 2010]
a(n+1) = A002620(n+1) - A002620(n) = floor((n+1)/2)*ceiling((n+1)/2) - floor(n^2/4). [From Jonathan Vos Post (jvospost3(AT)gmail.com), May 20 2010]
For n>=2, a(n)=floor(log_2(2^a(n-1)+2^a(n-2))) [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jun 22 2010]
a(n)=A180969(2,n) - Adriano Caroli (adriano_caroli(AT)virgilio.it), Nov 24 2010
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MAPLE
| A004526 := n->floor(n/2); [ seq(floor(i/2), i=0..50) ];
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MATHEMATICA
| Table[(2n - 1)/4 + (-1)^n/4, {n, 0, 70}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006
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PROG
| (PARI) a(n)=n\2 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 25 2009]
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CROSSREFS
| 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. Second row of A180969.
Cf. A002264, A002265, A002266, A010761, A010762, A110532, A110533.
Partial sums: A002620. Other related sequences: A002264, A002265, A002266, A010872, A010873, A010874.
Sequence in context: A001057 A130472 A076938 * A140106 A123108 A008619
Adjacent sequences: A004523 A004524 A004525 * A004527 A004528 A004529
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KEYWORD
| nonn,easy,core,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Partially edited by Joerg Arndt (arndt(AT)jjj.de), Mar 11 2010
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