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A052928 The even numbers repeated. 65
0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 62, 62, 64, 64, 66, 66, 68, 68, 70, 70, 72, 72 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
a(n) is also the binary rank of the complete graph K(n). - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 07 2009
Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 6, a(n) is the number of (0,1) n X n matrices A <= P^(-1)+I+P having exactly two 1's in every row and column with perA=2. - Vladimir Shevelev, Apr 12 2010
a(n+2) is the number of symmetry allowed, linearly independent terms at n-th order in the series expansion of the (E+A)xe vibronic perturbation matrix, H(Q) (cf. Eisfeld & Viel). - Bradley Klee, Jul 21 2015
The arithmetic function v_2(n,1) as defined in A289187. - Robert Price, Aug 22 2017
For n > 1, also the chromatic number of the n X n white bishop graph. - Eric W. Weisstein, Nov 17 2017
For n > 2, also the maximum vertex degree of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 23 2018
For n >= 2, a(n+2) gives the minimum weight of a Boolean function of algebraic degree at most n-2 whose support contains n linearly independent elements. - Christof Beierle, Nov 25 2019
REFERENCES
C. D. Godsil and G. Royle, Algebraic Graph Theory, Springer, 2001, page 181. - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 07 2009
V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19.
LINKS
C. Beierle, A. Biryukov and A. Udovenko, On degree-d zero-sum sets of full rank, Cryptography and Communications, November 2019.
W. Eisfeld and A. Viel, Higher order (A+E)xe pseudo-Jahn-Teller coupling, J. Chem. Phys., 122, 204317 (2005).
J. Sondow and E. W. Weisstein, MathWorld: Wallis Formula
Eric Weisstein's World of Mathematics, Chromatic Number
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature
Eric Weisstein's World of Mathematics, Maximum Vertex Degree
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph
Eric Weisstein's World of Mathematics, Random Matrix
Eric Weisstein's World of Mathematics, White Bishop Graph
FORMULA
a(n) = 2*floor(n/2).
G.f.: 2*x^2/((-1+x)^2*(1+x)).
a(n) + a(n+1) + 2 - 2*n = 0.
a(n) = n - 1/2 + (-1)^n/2.
a(n) = n + Sum_{k=1..n} (-1)^k. - William A. Tedeschi, Mar 20 2008
a(n) = a(n-1) + a(n-2) - a(n-3). - R. J. Mathar, Feb 19 2010
a(n) = |A123684(n) - A064455(n)| = A032766(n) - A008619(n-1). - Jaroslav Krizek, Mar 22 2011
For n > 0, a(n) = floor(sqrt(n^2+(-1)^n)). - Francesco Daddi, Aug 02 2011
a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=0 and b(k)=2^k for k>0. - Philippe Deléham, Oct 19 2011
a(n) = A109613(n) - 1. - M. F. Hasler, Oct 22 2012
a(n) = n - (n mod 2). - Wesley Ivan Hurt, Jun 29 2013
a(n) = a(a(n-1)) + a(n-a(n-1)) for n>2. - Nathan Fox, Jul 24 2016
a(n) = 2*A004526(n). - Filip Zaludek, Oct 28 2016
E.g.f.: x*exp(x) - sinh(x). - Ilya Gutkovskiy, Oct 28 2016
a(-n) = -a(n+1); a(n) = A005843(A004526(n)). - Guenther Schrack, Sep 11 2018
From Guenther Schrack, May 29 2019: (Start)
a(b(n)) = b(n) + ((-1)^b(n) - 1)/2 for any sequence b(n) of offset 0.
a(a(n)) = a(n), idempotent.
a(A086970(n)) = A124356(n-1) for n > 1.
a(A000124(n)) = A192447(n+1).
a(n)*a(n+1)/2 = A007590(n), also equals partial sums of a(n).
A007590(a(n)) = 2*A008794(n). (End)
MAPLE
spec := [S, {S=Union(Sequence(Prod(Z, Z)), Prod(Sequence(Z), Sequence(Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
Flatten[Table[{2n, 2n}, {n, 0, 39}]] (* Alonso del Arte, Jun 24 2012 *)
With[{ev=2Range[0, 40]}, Riffle[ev, ev]] (* Harvey P. Dale, May 08 2021 *)
PROG
(PARI) a(n)=n\2*2 \\ Charles R Greathouse IV, Nov 20 2011
(Magma) [2*Floor(n/2) : n in [0..50]]; // Wesley Ivan Hurt, Sep 13 2014
(Haskell)
a052928 = (* 2) . flip div 2
a052928_list = 0 : 0 : map (+ 2) a052928_list
-- Reinhard Zumkeller, Jun 20 2015
CROSSREFS
First differences: A010673; partial sums: A007590; partial sums of partial sums: A212964(n+1).
Complement of A109613 with respect to universe A004526. - Guenther Schrack, Dec 07 2017
Is first differences of A099392. Fixed point sequence: A005843. - Guenther Schrack, May 30 2019
For n >= 3, A329822(n) gives the minimum weight of a Boolean function of algebraic degree at most n-3 whose support contains n linearly independent elements. - Christof Beierle, Nov 25 2019
Sequence in context: A161764 A293706 A131055 * A346663 A137501 A308767
KEYWORD
nonn,easy
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from James A. Sellers, Jun 05 2000
Removed duplicate of recurrence; corrected original recurrence and g.f. against offset - R. J. Mathar, Feb 19 2010
STATUS
approved

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Last modified June 19 06:35 EDT 2024. Contains 373492 sequences. (Running on oeis4.)