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A024490 a(n) = C(n-1,1) + C(n-3,3) + ... + C(n-2*m-1,2*m+1), where m = floor((n-2)/4).

%I A024490 #76 Apr 14 2023 03:56:28
%S A024490 1,2,3,4,6,10,17,28,45,72,116,188,305,494,799,1292,2090,3382,5473,
%T A024490 8856,14329,23184,37512,60696,98209,158906,257115,416020,673134,
%U A024490 1089154,1762289,2851444,4613733,7465176,12078908,19544084,31622993,51167078,82790071
%N A024490 a(n) = C(n-1,1) + C(n-3,3) + ... + C(n-2*m-1,2*m+1), where m = floor((n-2)/4).
%C A024490 Essentially both the first difference sequence and partial sum of A005252, so its own shifted second difference and indeed virtually the same as A005252, so close to being its own shifted first difference.
%C A024490 From _Paul Curtz_, Jun 22 2011: (Start)
%C A024490 b(n) = 0,0,0,1,2,3,4,6, and differences are
%C A024490    0,  0,  0,  1,  2,  3,  4,  6,
%C A024490    0,  0,  1,  1,  1,  1,  2,  4,
%C A024490    0,  1,  0,  0,  0,  1,  2,  3,
%C A024490    1, -1,  0,  0,  1,  1,  1,  1,
%C A024490   -2,  1,  0,  1,  0,  0,  0,  1,
%C A024490    3, -1,  1  -1,  0,  0,  1,  1,
%C A024490   -4,  2, -2,  1,  0,  1,  0,  0,
%C A024490    6, -4,  3, -1,  1, -1,  0,  0;
%C A024490 b(n) is an autosequence (sequence identical to its inverse binomial transform signed) of first kind, i.e., its main diagonal is A000004.
%C A024490 Examples: A000045, A001045, A113405, A191754 (array). (End)
%C A024490 a(n) is the number of vertices of the Fibonacci cube Gamma(n-1) having an odd number of ones. The Fibonacci cube Gamma(n) can be defined as the graph whose vertices are the binary strings of length n without two consecutive 1's and in which two vertices are adjacent when their Hamming distance is exactly 1. Example: a(4) = 3; indeed, the Fibonacci cube Gamma(3) has the five vertices 000, 010, 001, 100, 101, three of which have an odd number of ones. See the E. Munarini et al. reference, p. 323. - _Emeric Deutsch_, Jun 28 2015
%C A024490 a(n) is the number of odd permutations p of 1..n such that |p(i)-i| <= 1 for i=1..n. - _Dmitry Efimov_, Jan 08 2016
%H A024490 T. D. Noe, <a href="/A024490/b024490.txt">Table of n, a(n) for n = 2..502</a>
%H A024490 Shishuo Fu and Dazhao Tang, <a href="http://arxiv.org/abs/1604.04028">Partitions with fixed largest hook length</a>, arXiv:1604.04028 [math.CO], 2016.
%H A024490 Fumio Hazama, <a href="http://dx.doi.org/10.1016/j.disc.2011.06.008">Spectra of graphs attached to the space of melodies</a>, Discr. Math., 311 (2011), 2368-2383. See Table 2.1.
%H A024490 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=886">Encyclopedia of Combinatorial Structures 886</a>
%H A024490 S. Klavzar, <a href="http://www.fmf.uni-lj.si/~klavzar/preprints/FibonacciCubesRevised.pdf">Structure of Fibonacci cubes: a survey</a>, J. Comb. Optim., 25, 2013, 505-522.
%H A024490 E. Munarini and N. Z. Salvi, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00407-1">Structural and enumerative properties of the Fibonacci cubes</a>, Discrete Math., 255, 2002, 317-324.
%H A024490 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1).
%F A024490 a(n) = A000045(n+1) - A005252(n).
%F A024490 a(n) = (A000045(n+1) - A010892(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003
%F A024490 a(n) = Sum_{k=0..n} Fibonacci(k+1)*2*sin(Pi*(n-k)/3 + Pi/3)/sqrt(3). - _Paul Barry_, May 18 2004
%F A024490 G.f.: x^2/((1-x-x^2)(1-x+x^2)). - _Jon Perry_, Jun 22 2004
%F A024490 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+1,k+1)*(1+(-1)^k)/2. - _Paul Barry_, Jul 05 2007
%F A024490 G.f.: (1 + Q(0)*x^4/2)/(1-x)^2, where Q(k) = 1 + 1/(1 - x*( 4*k + 2 - x + x^3)/( x*(4*k + 4 - x + x^3) + 1/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jan 07 2014
%F A024490 E.g.f.: exp(x/2)*(15*cosh(sqrt(5)*x/2) - 5*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/30. - _Stefano Spezia_, Aug 03 2022
%t A024490 LinearRecurrence[{2,-1,0,1},{1,2,3,4},39] (* _Ray Chandler_, Sep 23 2015 *)
%t A024490 CoefficientList[Series[1/((1-x-x^2)(1-x+x^2)), {x,0,40}], x] (* _Vincenzo Librandi_, Jan 09 2016 *)
%o A024490 (Magma) [n le 4 select n else 2*Self(n-1)-Self(n-2)+Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jan 09 2016
%o A024490 (PARI) Vec(x^2/((1-x-x^2)*(1-x+x^2)) + O(x^50)) \\ _Michel Marcus_, Feb 03 2016
%o A024490 (SageMath)
%o A024490 def A024490(n): return (fibonacci(n+1) -chebyshev_U(n,1/2))/2
%o A024490 [A024490(n) for n in range(2,60)] # _G. C. Greubel_, Apr 10 2023
%Y A024490 Cf. A000004, A000045, A001045, A005252, A010892, A113405, A191754.
%K A024490 nonn,easy
%O A024490 2,2
%A A024490 _Clark Kimberling_
%E A024490 Additional comments from _Henry Bottomley_, Apr 07 2000
%E A024490 Corrected by Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003
%E A024490 Further corrections from _Hugo van der Sanden_, Oct 05 2006