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

0,3

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

Table of n, a(n) for n=0..38.

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A034851.

Essentially the same as A001224, A060312 and A068928.

Sequence in context: A124280 A088518 A001224 * A050192 A191786 A007147

Adjacent sequences: A102523 A102524 A102525 * A102527 A102528 A102529

Keyword

nonn,easy

Author

Gerald McGarvey, Feb 24 2005

Status

approved