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%I #162 Feb 08 2024 01:39:59
%S 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,
%T 26,26,28,28,30,30,32,32,34,34,36,36,38,38,40,40,42,42,44,44,46,46,48,
%U 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
%N The even numbers repeated.
%C a(n) is also the binary rank of the complete graph K(n). - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 07 2009
%C 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
%C 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
%C The arithmetic function v_2(n,1) as defined in A289187. - _Robert Price_, Aug 22 2017
%C For n > 1, also the chromatic number of the n X n white bishop graph. - _Eric W. Weisstein_, Nov 17 2017
%C For n > 2, also the maximum vertex degree of the n-polygon diagonal intersection graph. - _Eric W. Weisstein_, Mar 23 2018
%C 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
%D C. D. Godsil and G. Royle, Algebraic Graph Theory, Springer, 2001, page 181. - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 07 2009
%D V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19.
%H Vincenzo Librandi, <a href="/A052928/b052928.txt">Table of n, a(n) for n = 0..1000</a>
%H C. Beierle, A. Biryukov and A. Udovenko, <a href="https://doi.org/10.1007/s12095-019-00415-0">On degree-d zero-sum sets of full rank</a>, Cryptography and Communications, November 2019.
%H W. Eisfeld and A. Viel, <a href="http://dx.doi.org/10.1063/1.1904594">Higher order (A+E)xe pseudo-Jahn-Teller coupling</a>, J. Chem. Phys., 122, 204317 (2005).
%H Nathan Fox, <a href="https://arxiv.org/abs/1609.06342">Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation</a>, arXiv:1609.06342 [math.NT], 2016.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=914">Encyclopedia of Combinatorial Structures 914</a>
%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/WallisFormula.html">MathWorld: Wallis Formula</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChromaticNumber.html">Chromatic Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html">Legendre-Gauss Quadrature</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximumVertexDegree.html">Maximum Vertex Degree</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygonDiagonalIntersectionGraph.html">Polygon Diagonal Intersection Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RandomMatrix.html">Random Matrix</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WhiteBishopGraph.html">White Bishop Graph</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1)
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%F a(n) = 2*floor(n/2).
%F G.f.: 2*x^2/((-1+x)^2*(1+x)).
%F a(n) + a(n+1) + 2 - 2*n = 0.
%F a(n) = n - 1/2 + (-1)^n/2.
%F a(n) = n + Sum_{k=1..n} (-1)^k. - _William A. Tedeschi_, Mar 20 2008
%F a(n) = a(n-1) + a(n-2) - a(n-3). - _R. J. Mathar_, Feb 19 2010
%F a(n) = |A123684(n) - A064455(n)| = A032766(n) - A008619(n-1). - _Jaroslav Krizek_, Mar 22 2011
%F For n > 0, a(n) = floor(sqrt(n^2+(-1)^n)). - _Francesco Daddi_, Aug 02 2011
%F 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
%F a(n) = A109613(n) - 1. - _M. F. Hasler_, Oct 22 2012
%F a(n) = n - (n mod 2). - _Wesley Ivan Hurt_, Jun 29 2013
%F a(n) = a(a(n-1)) + a(n-a(n-1)) for n>2. - _Nathan Fox_, Jul 24 2016
%F a(n) = 2*A004526(n). - _Filip Zaludek_, Oct 28 2016
%F E.g.f.: x*exp(x) - sinh(x). - _Ilya Gutkovskiy_, Oct 28 2016
%F a(-n) = -a(n+1); a(n) = A005843(A004526(n)). - _Guenther Schrack_, Sep 11 2018
%F From _Guenther Schrack_, May 29 2019: (Start)
%F a(b(n)) = b(n) + ((-1)^b(n) - 1)/2 for any sequence b(n) of offset 0.
%F a(a(n)) = a(n), idempotent.
%F a(A086970(n)) = A124356(n-1) for n > 1.
%F a(A000124(n)) = A192447(n+1).
%F a(n)*a(n+1)/2 = A007590(n), also equals partial sums of a(n).
%F A007590(a(n)) = 2*A008794(n). (End)
%p spec := [S,{S=Union(Sequence(Prod(Z,Z)),Prod(Sequence(Z),Sequence(Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t Flatten[Table[{2n, 2n}, {n, 0, 39}]] (* _Alonso del Arte_, Jun 24 2012 *)
%t With[{ev=2Range[0,40]},Riffle[ev,ev]] (* _Harvey P. Dale_, May 08 2021 *)
%o (PARI) a(n)=n\2*2 \\ _Charles R Greathouse IV_, Nov 20 2011
%o (Magma) [2*Floor(n/2) : n in [0..50]]; // _Wesley Ivan Hurt_, Sep 13 2014
%o (Haskell)
%o a052928 = (* 2) . flip div 2
%o a052928_list = 0 : 0 : map (+ 2) a052928_list
%o -- _Reinhard Zumkeller_, Jun 20 2015
%Y Cf. A000034, A000124, A004001, A004526, A005843, A007590, A008619, A008794, A032766, A064455, A099392, A109613, A118266, A123684, A124356, A192442, A289187, A342819.
%Y First differences: A010673; partial sums: A007590; partial sums of partial sums: A212964(n+1).
%Y Complement of A109613 with respect to universe A004526. - _Guenther Schrack_, Dec 07 2017
%Y Is first differences of A099392. Fixed point sequence: A005843. - _Guenther Schrack_, May 30 2019
%Y 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
%K nonn,easy
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Jun 05 2000
%E Removed duplicate of recurrence; corrected original recurrence and g.f. against offset - _R. J. Mathar_, Feb 19 2010
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