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A102526 Antidiagonal sums of Losanitsch's triangle (A034851). 6
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 (list; graph; refs; listen; history; text; internal format)
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]

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

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

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Last modified October 17 05:23 EDT 2018. Contains 316275 sequences. (Running on oeis4.)