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%I #34 Sep 29 2020 04:19:45
%S 1,1,2,2,4,5,9,12,21,30,51,76,127,195,322,504,826,1309,2135,3410,5545,
%T 8900,14445,23256,37701,60813,98514,159094,257608,416325,673933,
%U 1089648,1763581,2852242,4615823,7466468,12082291,19546175,31628466
%N Antidiagonal sums of Losanitsch's triangle (A034851).
%C This is an interleaving of A005207 and A051450. Thus a(2*m) = A005207(m) = (F(2*m-1) + F(m+1)) / 2, a(2*m - 1) = A051450(m) = (F(2*m) + F(m)) / 2 where F() are Fibonacci numbers (A000045). - _Max Alekseyev_, Jun 28 2006
%C The Kn11(n) and Kn21(n) sums, see A180662 for their definitions, of Losanitsch's triangle A034851 equal a(n), while the Kn12(n) and Kn22(n) sums equal (a(n+2)-A000012(n)) and the Kn13(n) and Kn23(n) sums equal (a(n+4)-A008619(n+4)). - _Johannes W. Meijer_, Jul 14 2011
%C a(n) is the number of homeomorphically irreducible caterpillars with n + 3 edges. - _Christian Barrientos_, Sep 12 2020
%D Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World Scientific Press, 2007.
%H Johann Cigler, <a href="https://arxiv.org/abs/1711.03340">Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle</a>, arXiv:1711.03340 [math.CO], 2017.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,0,-1,-1).
%F G.f.: -(1+x)*(x^3+x-1) / ( (x^2+x-1)*(x^4+x^2-1) ). - _R. J. Mathar_, Nov 09 2013
%F a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-5) - a(n-6). - _Wesley Ivan Hurt_, Sep 17 2020
%p with(combinat): A102526 :=proc(n): if type(n, even) then (fibonacci(n+1)+fibonacci(n/2+2))/2 else (fibonacci(n+1)+fibonacci((n+1)/2))/2 fi: end: seq(A102526(n), n=0..38); # _Johannes W. Meijer_, Jul 14 2011
%t LinearRecurrence[{1, 2, -1, 0, -1, -1}, {1, 1, 2, 2, 4, 5}, 40] (* _Jean-François Alcover_, Nov 17 2017 *)
%o (PARI) Vec((1+x)*(1-x-x^3)/(x^2+x-1)/(x^4+x^2-1)+O(x^99)) \\ _Charles R Greathouse IV_, Nov 17 2017
%o (PARI) a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,-1,0,-1,2,1]^n*[1;1;2;2;4;5])[1,1] \\ _Charles R Greathouse IV_, Nov 17 2017
%Y Cf. A034851.
%Y Essentially the same as A001224, A060312 and A068928.
%K nonn,easy
%O 0,3
%A _Gerald McGarvey_, Feb 24 2005
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