Revision History for A140735
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| A140735
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Triangle read by rows, X^n * [1,0,0,0,...]; where X = a tridiagonal matrix with (1,2,3,...) in the main diagonal and (1,1,1,...) in the sub and subsubdiagonals.
(history;
published version)
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#41 by Alois P. Heinz at Tue Apr 24 17:18:46 EDT 2018
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#40 by Alois P. Heinz at Tue Apr 24 17:18:08 EDT 2018
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| COMMENTS
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T[(m,k] ) is the number of achiral color patterns in a row or loop of length 2m-1 using exactly k different colors. Two color patterns are equivalent if we permute the colors. - Robert A. Russell, Mar 24 2018
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| EXAMPLE
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T[(3,3]=)=5 is the number of achiral color patterns of length five using exactly three colors. These are AABCC, ABACA, ABBBC, ABCAB, and ABCBA for both rows and loops. - Robert A. Russell, Mar 24 2018
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proposed
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#39 by Robert A. Russell at Tue Apr 24 14:09:43 EDT 2018
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#38 by Robert A. Russell at Tue Apr 24 14:09:30 EDT 2018
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| FORMULA
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T(m,k) = [m>1]*(k*Aodd[T(m-1,k]+Aodd[)+T(m-1,k-1]+Aodd[)+T(m-1,k-2]) + [)) + [m==1]*[k==1] - Robert A. Russell, Apr 24 2018
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proposed
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Discussion
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| Tue Apr 24
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| Robert A. Russell: Bad formula fixed.
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#37 by Robert A. Russell at Tue Apr 24 14:04:28 EDT 2018
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#36 by Robert A. Russell at Tue Apr 24 14:04:14 EDT 2018
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| FORMULA
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T(m,k) = [m>01]*(k*Aodd[m-1,k]+Aodd[m-1,k-1]+Aodd[m-1,k-2]) + [m==01]*[k==01] - Robert A. Russell, Apr 24 2018
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#35 by Robert A. Russell at Tue Apr 24 13:40:16 EDT 2018
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| FORMULA
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T(m,k) = [m>0]*(k*Aodd[m-1,k]+Aodd[m-1,k-1]+Aodd[m-1,k-2]) + [m==0]*[k==0] - Robert A. Russell, Apr 24 2018
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| MATHEMATICA
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Aodd[m_, k_] := Aodd[m, k] = If[m > 1, k Aodd[m-1, k] + Aodd[m-1, k-1]
+ Aodd[m-1, k-2], Boole[m==1 && k==1]]
Table[Aodd[m, k], {m, 1, 10}, {k, 1, 2m-1}] // Flatten (* Robert A. Russell, Apr 24 2018 *)
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approved
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#34 by N. J. A. Sloane at Tue Apr 17 00:09:01 EDT 2018
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#33 by Robert A. Russell at Sat Apr 14 08:42:38 EDT 2018
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#32 by Robert A. Russell at Sat Apr 14 08:40:47 EDT 2018
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| MATHEMATICA
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(* Ach[n, , k] is the number of achiral color patterns for a row or loop of n colors containing k different colors *)
colors containing k different colors *)
Ach[n_, k_] := Ach[n, k] = SwitchWhich[k, 0==k, IfBoole[0==n, 1, 0], 1==k, IfBoole[n>0, 1, 0],
(* else *) _, If[OddQ[n],
OddQ[n], Sum[Binomial[(n-1)/2, i] Ach[n-1-2i, k-1], {i, 0, (n-1)/2}],
True, Sum[Binomial[n/2-1, i] (Ach[n-2-2i, k-1]
+ + 2^i Ach[n-2-2i, k-2]), {i, 0, n/2-1}]]]}]]
{i, 1, 2 n - 1}, {j, 1, 2 n - 1}], n-1][[1]], {n, 1, 710}]
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| CROSSREFS
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Number of achiral color patterns of length even n in A293181.
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proposed
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Discussion
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| Sat Apr 14
| 08:42
| Robert A. Russell: Simplified Mathematica program, added cross reference.
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