A330169 a(n) is the total area of all closed Deutsch paths of length n.
1, 3, 12, 39, 129, 411, 1300, 4065, 12633, 39046, 120204, 368844, 1128837, 3447303, 10508592, 31985085, 97226733, 295214316, 895502520, 2714106318, 8219809425, 24877611798, 75248738292, 227488953354, 687408882709, 2076269682831, 6268788729240, 18920387069731, 57086882549253
Offset: 2
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 2..2096
- Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020. See p. 8.
Programs
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Maple
a:= proc(n) option remember;`if`(n<4, [0$2, 1, 3][n+1], (4*n* a(n-1)+(2*n+4)*a(n-2)+12*(1-n)*a(n-3)+9*(1-n)*a(n-4))/(n+1)) end: seq(a(n), n=2..30); # Alois P. Heinz, Mar 05 2020 -
Mathematica
a = DifferenceRoot[Function[{y, n}, {9(n+3)y[n] + 12(n+3)y[n+1] - 2(n+6)y[n+2] - 4(n+4)y[n+3] + (n+5)y[n+4] == 0, y[2] == 1, y[3] == 3, y[4] == 12, y[5] == 39}]]; a /@ Range[2, 30] (* Jean-François Alcover, Mar 12 2020 *) -
PARI
my(z='z+O('z^30), v=(1-z-sqrt(1-2*z-3*z^2))/(2*z)); Vec(v^2*(1+v+v^2)^2/((1+v)^3*(1-v)^2))
Formula
G.f.: v^2*(1+v+v^2)^2/((1+v)^3*(1-v)^2) where v=(1-z-sqrt(1-2*z-3*z^2))/(2*z), that is, where v is the g.f. of A001006.
Comments