A102543 Antidiagonal sums of the antidiagonals of Losanitsch's triangle.

1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 16, 24, 33, 49, 69, 102, 145, 214, 307, 452, 653, 960, 1393, 2046, 2978, 4371, 6376, 9354, 13665, 20041, 29307, 42972, 62884, 92191, 134974, 197858, 289772, 424746, 622198, 911970, 1336121, 1958319, 2869417, 4205538, 6162579, 9031996, 13235661, 19398240

Offset: 0

Author

Gerald McGarvey, Feb 24 2005

Keywords

• nonn

Comments

The Ca1 and Ca2 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 equal this sequence. - Johannes W. Meijer, Jul 14 2011

For n >= 5, a(n+1)-1 is the number of non-isomorphic snake polyominoes with n cells that can be inscribed in a rectangle of height 2. - Christian Barrientos and Sarah Minion, Jul 29 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• R. J. Mathar, Paving rectangular regions with rectangular tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 25.
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,1,-1,0,-1).

Crossrefs

Cf. A034851 A068927, A102451.

Programs

Maple

A102543 := proc(n): (A000930(n)+x(n)+x(n-1)+x(n-3))/2 end: A000930:=proc(n): sum(binomial(n-2*i, i), i=0..n/3) end: x:=proc(n): if type(n, even) then A000930(n/2) else 0 fi: end: seq(A102543(n), n=0..38); # Johannes W. Meijer, Jul 14 2011

Mathematica

CoefficientList[Series[(1 - x^2 - x^4 - x^6)/((x^3 + x - 1)*(x^6 + x^2 - 1)), {x, 0, 50}], x] (* G. C. Greubel, Apr 27 2017 *)
LinearRecurrence[{1,1,0,0,-1,1,-1,0,-1},{1,1,1,2,2,3,4,6,8},50] (* Harvey P. Dale, Dec 14 2023 *)

PARI

x='x+O('x^50); Vec((1 - x^2 - x^4 - x^6)/((x^3 + x - 1)*(x^6 + x^2 - 1))) \\ G. C. Greubel, Apr 27 2017

Formula

a(n) = A068927(n-1), n>3.

From Johannes W. Meijer, Jul 14 2011: (Start)

G.f.: (-1/2)*(1/(x^3+x-1)+(1+x+x^3)/(x^6+x^2-1))= ( 1-x^2-x^4-x^6 ) / ( (x^3+x-1)*(x^6+x^2-1) ).

a(n) = (A000930(n)+x(n)+x(n-1)+x(n-3))/2 with x(2*n) = A000930(n) and x(2*n+1) = 0. (End)