A132103 Number of distinct Tsuro tiles which are digonal in shape and have Q points per side.
1, 1, 3, 8, 45, 283, 2847, 34518, 511209
Offset: 0
This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
(* derived from Joerg Arndt's PARI code *) f[n_] := f[n] = (2n-1)!! s[n_] := s[n] = f[n] - Sum[f[k] s[n - k], {k, 1, n - 1}] Table[f[k] - s[k], {k, 0, 22}] (* original brute force method *) GenerateDiagramsOfOrder[n_Integer /; n >= 0] := Diagrams[Range[2 n]] Diagrams[pool_List] := Block[{n = Count[pool, _]}, If[n <= 2, {{pool}}, Flatten[Map[ Flatten[ Outer[Join, {{{pool[[1]], pool[[#]]}}}, Diagrams[ Function[{poolset, droppos}, Drop[poolset, {droppos}] // Rest][pool, #]], 1], 1] &, Range[2, n]], 1]]] SelectDisconnected[diagrams_List] := Select[diagrams, IsDisconnected] IsDisconnected[{{}}] = False; IsDisconnected[pairs_List] := Block[{newPairs=Map[#~Append~(#[[2]] - #[[1]]) &, pairs], distanceList}, distanceList = Fold[ ReplacePart[#1, {#2[[1]] -> #2[[3]], #2[[2]] -> -#2[[3]]}] &, Range[2 Length[pairs]], newPairs]; Return[Length[Select[Drop[Accumulate[distanceList], -1], #<1 &]] > 0] ] Map[Length[SelectDisconnected[GenerateDiagramsOfOrder[#]]]&, Range[0,7]]
f(n)=(2*n)!/n!/2^n; \\ == (2n-1)!! s(n)=f(n) - sum(k=1, n-1, f(k)*s(n-k) ) a(n)=f(n)-s(n) \\ Joerg Arndt
from functools import cache
def a(n):
@cache
def h(n):
if n <= 1: return 1
return h(n - 1) * (2 * n - 1)
@cache
def c(n):
if n == 0: return 1
return h(n) - sum(h(k) * c(n - k) for k in range(1, n))
return h(n) - c(n)
print([a(n) for n in range(19)]) # Peter Luschny, Apr 16 2023
For n=3, there are a(3)=5 topologically distinct slicings from chords in general position. These exclude a sixth configuration found when the three chords meet at a point strictly internal to the pizza.
1; 1, 1; 2, 2, 2; 5, 8, 8, 5; 17, 39, 61, 39, 17; 79, 287, 556, 556, 287, 79;
C(p, d)={sum(k=0, p\2, binomial(p, 2*k) * (d*(1+y^d))^k * if(d%2, p==2*k, (1+y^(d/2))^(p-2*k)) * (2*k)!/(2^k*k!))}
R(n)={sum(k=0, n\2, binomial(n,2*k) * (1+y^2)^k * (1+y)^(n-2*k) * (2*k)!/k!)}
row(n)={Vec(if(n==0, 1, (sumdiv(2*n, d, eulerphi(d)*C(2*n/d, d))/n + R(n) + (1+y)*R(n-1))/4))}
{ for(n=0, 8, print(row(n))) } \\ Andrew Howroyd, Dec 13 2018
T(3,2)=2: two bracelets with 3 chords in two variants (i) one bracelet with one chord and the other with 2 noncrossing chords, (ii) one bracelet with one chord and the other with 2 crossing chords. 1 ; 2 1 ; 5 2 1 ; 17 8 2 1 ; 79 27 8 2 1 ; 554 128 31 8 2 1 ; 5283 797 143 31 8 2 1 ; 65346 6939 878 148 31 8 2 1 ; 966156 80025 7381 898 148 31 8 2 1 ; 16411700 1135841 83038 7494 904 148 31 8 2 1 ;
\\ See A339645 for combinatorial species functions. cycleIndexSeries(n)={1+sLog(sCartProd(sExp(dihedralGroupSeries(n)), sExp(symGroupCycleIndex(2)*x^2 + O(x*x^n))))} seq(n)={Vec(substpol(OgfSeries(cycleIndexSeries(2*n)), x^2, x))} \\ Andrew Howroyd, May 05 2023
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