A102526 Antidiagonal sums of Losanitsch's triangle (A034851).

1, 1, 2, 2, 4, 5, 9, 12, 21, 30, 51, 76, 127, 195, 322, 504, 826, 1309, 2135, 3410, 5545, 8900, 14445, 23256, 37701, 60813, 98514, 159094, 257608, 416325, 673933, 1089648, 1763581, 2852242, 4615823, 7466468, 12082291, 19546175, 31628466

Offset: 0

Author

Gerald McGarvey, Feb 24 2005

Keywords

• nonn
• easy

Comments

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

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

a(n) is the number of homeomorphically irreducible caterpillars with n + 3 edges. - Christian Barrientos, Sep 12 2020

References

• Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World Scientific Press, 2007.

Links

• Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-1,0,-1,-1).

Crossrefs

Cf. A034851.

Essentially the same as A001224, A060312 and A068928.

Programs

Maple

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

Mathematica

LinearRecurrence[{1, 2, -1, 0, -1, -1}, {1, 1, 2, 2, 4, 5}, 40] (* Jean-François Alcover, Nov 17 2017 *)

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

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

Formula

G.f.: -(1+x)*(x^3+x-1) / ( (x^2+x-1)*(x^4+x^2-1) ). - R. J. Mathar, Nov 09 2013

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-5) - a(n-6). - Wesley Ivan Hurt, Sep 17 2020