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%I #125 Apr 11 2023 08:42:23
%S 0,0,1,3,6,11,21,42,85,171,342,683,1365,2730,5461,10923,21846,43691,
%T 87381,174762,349525,699051,1398102,2796203,5592405,11184810,22369621,
%U 44739243,89478486,178956971,357913941,715827882,1431655765,2863311531,5726623062
%N a(n) = C(n,2) + C(n,5) + ... + C(n, 3*floor(n/3)+2).
%C Trisections give A082365, A132804, A132805. - _Paul Curtz_, Nov 18 2007
%C If the offset is changed to 1, this is the maximal number of closed regions bounded by straight lines after n straight line cuts in a plane: a(n) = a(n-1) + n - 3, a(1)=0; a(2)=0; a(3)=1; and so on. - _Srikanth K S_, Jan 23 2008
%C M^n * [1,0,0] = [A024493(n), a(n), A024494(n)]; where M = a 3x3 matrix [1,1,0; 0,1,1; 1,0,1]. Sum of terms = 2^n. Example: M^5 * [1,0,0] = [11, 11, 10], sum = 2^5 = 32. - _Gary W. Adamson_, Mar 13 2009
%C For n>=1, a(n-1) is the number of generalized compositions of n when there are i^2/2 - 3*i/2 + 1 different types of i, (i=1,2,...). - _Milan Janjic_, Sep 24 2010
%C Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1+M)^n = A024493(n) + A024494(n)*M + a(n)*M^2. - _Stanislav Sykora_, Jun 10 2012
%C {A024493, A131708, A024495} is the difference analog of the hyperbolic functions {h_1(x), h_2(x), h_3(x)} of order 3. For the definitions of {h_i(x)} and the difference analog {H_i(n)} see [Erdelyi] and the Shevelev link respectively. - _Vladimir Shevelev_, Aug 01 2017
%C This is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S^3; see A291000. - _Clark Kimberling_, Aug 24 2017
%D A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
%D D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd. ed., Problem 38, p. 70.
%H Seiichi Manyama, <a href="/A024495/b024495.txt">Table of n, a(n) for n = 0..3000</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Barry/barry594.html">A note on Krawtchouk Polynomials and Riordan Arrays</a>, JIS 11 (2008) 08.2.2, example 9.
%H Antoine-Augustin Cournot, <a href="https://archive.org/details/bulletindesscie22unkngoog/page/n101/mode/2up?view=theater">Solution d'un problème d'analyse combinatoire</a>, Bulletin des Sciences Mathématiques, Physiques et Chimiques, item 34, volume 11, 1829, pages 93-97. Also at <a href="http://books.google.com.au/books?id=B-v-eXuvoG4C">Google Books</a>. Page 97 case p=3 formula y^(2) = a(n).
%H Christian Ramus, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0011?tify=%7B%22pages%22%3A%5B365%5D%2C%22view%22%3A%22info%22%7D">Solution générale d'un problème d'analyse combinatoire</a>, Journal für die Reine und Angewandte Mathematik (Crelle's journal), volume 11, 1834, pages 353-355. Page 353 case p=3 formula y^(2) = a(n).
%H Vladimir Shevelev, <a href="https://arxiv.org/abs/1706.01454">Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n</a>, arXiv:1706.01454 [math.CO], 2017.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlaneDivisionbyLines.html">Plane division by lines</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2).
%F a(n) = ( 2^n + 2*cos((n-4)*Pi/3) )/3 = (2^n - A057079(n))/3.
%F a(n) = 2*a(n-1) + A010892(n-2) = a(n-1) + A024494(n-1). With initial zero, binomial transform of A011655 which is effectively A010892 unsigned. - _Henry Bottomley_, Jun 04 2001
%F a(2) = 1, a(3) = 3, a(n+2) = a(n+1) - a(n) + 2^n. - _Benoit Cloitre_, Sep 04 2002
%F a(n) = Sum_{k=0..n} 2^k*2*sin(Pi*(n-k)/3 + Pi/3)/sqrt(3) (offset 0). - _Paul Barry_, May 18 2004
%F G.f.: x^2/((1-x)^3 - x^3) = x^2 / ( (1-2*x)*(1-x+x^2) ).
%F a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3). - _Paul Curtz_, Nov 18 2007
%F a(n) + A024493(n-1) = A131577(n). - _Paul Curtz_, Jan 24 2008
%F From _Paul Curtz_, May 29 2011: (Start)
%F a(n) + a(n+3) = 3*2^n = A007283(n).
%F a(n+6) - a(n) = 21*2^n = A175805(n).
%F a(n) + a(n+9) = 171*2^n.
%F a(n+12) - a(n) = 1365*2^n. (End)
%F a(n) = A113405(n) + A113405(n+1). - _Paul Curtz_, Jun 05 2011
%F Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1) = x(n) + z(n), y(n+1) = y(n) + x(n), z(n+1) = z(n) + y(n). Then a(n) = z(n). - _Stanislav Sykora_, Jun 10 2012
%F G.f.: -x^2/( x^3 - 1 + 3*x/Q(0) ) where Q(k) = 1 + k*(x+1) + 3*x - x*(k+1)*(k+4)/Q(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Mar 15 2013
%F a(n) = 1/18*(-4*(-1)^floor((n - 1)/3) - 6*(-1)^floor(n/3) - 3*(-1)^floor((n + 1)/3) + (-1)^(1 + floor((n + 2)/3)) + 3*2^(n + 1)). - _John M. Campbell_, Dec 23 2016
%F a(n) = (1/63)*(-40 + 21*2^n - 42*floor(n/6) + 32*floor((n+3)/6) + 16*floor((n+ 4)/6) - 24*floor((n+5)/6) - 22*floor((n+7)/6) + 21*floor((n+8)/6) + 10*floor((n+9)/6) + 5*floor((n+10)/6) + 3*floor((n+11)/6) + floor((n+ 13)/6)). - _John M. Campbell_, Dec 24 2016
%F a(n+m) = a(n)*A024493(m) + A131708(n)*A131708(m) + A024493(n)*a(m). - _Vladimir Shevelev_, Aug 01 2017
%F From _Kevin Ryde_, Sep 24 2020: (Start)
%F a(n) = (1/3)*2^n - (1/3)*cos((1/3)*Pi*n) - (1/sqrt(3))*sin((1/3)*Pi*n). [Cournot]
%F a(n) + A111927(n) + A131708(n) = 2^n - 1. [Cournot, page 96 last formula, but misprint should be 2^x - 1 rather than 2^p - 1]
%F (End)
%p a:= proc(n) option remember; `if`(n=0, 0, 2*a(n-1)+
%p [-1, 0, 1, 1, 0, -1, -1][1+(n mod 6)])
%p end:
%p seq(a(n), n=0..33); # _Paul Weisenhorn_, May 17 2020
%t LinearRecurrence[{3,-3,2},{0,0,1},40] (* _Harvey P. Dale_, Sep 20 2016 *)
%o (PARI) a(n) = sum(k=0,n\3,binomial(n,3*k+2)) /* _Michael Somos_, Feb 14 2006 */
%o (PARI) a(n)=if(n<0, 0, ([1,0,1;1,1,0;0,1,1]^n)[3,1]) /* _Michael Somos_, Feb 14 2006 */
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!( x^2/((1-x)^3-x^3) )); // _G. C. Greubel_, Apr 11 2023
%o (SageMath)
%o def A024495(n): return (2^n - chebyshev_U(n, 1/2) - chebyshev_U(n-1, 1/2))/3
%o [A024495(n) for n in range(41)] # _G. C. Greubel_, Apr 11 2023
%Y Cf. A007283, A010892, A011655, A024493, A024494, A024495, A057079, A082365.
%Y Cf. A083322, A139469, A111927, A113405, A131577, A131708, A132804, A132805.
%Y Cf. A175805, A291000.
%Y Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), this sequence (m=3), A000749 (m=4), A049016 (m=5), A192080 (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).
%K nonn,easy
%O 0,4
%A _Clark Kimberling_
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