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A092964
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Numbers > 1 in A051168, with a(0) = 1.
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8
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1, 2, 2, 2, 3, 2, 3, 5, 5, 3, 3, 7, 8, 7, 3, 4, 9, 14, 14, 9, 4, 4, 12, 20, 25, 20, 12, 4, 5, 15, 30, 42, 42, 30, 15, 5, 5, 18, 40, 66, 75, 66, 40, 18, 5, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 6, 26, 70, 143, 212, 245, 212, 143, 70, 26, 6, 7, 30, 91, 200, 333, 429, 429, 333, 200
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Original definition: Triangle read by rows in which row n gives the number of equal configurations under cyclic shift.
T(n,k)=A051168(n+3,k+1), if 0<k<=n. - Michael Somos, Jul 17 2004
Contribution from Paul Weisenhorn, Dec 21 2010 (Start):
T(n,k)=number of classes which have (k+1) ordered sums of (n+4) with (k+1) positive integers, that can be transformed into each other by a cyclic permutation.
m has 2^(m-1) ordered sums; for each sum one remove the first part z(1) and add 1 to the next z(1) parts to get a new ordered sum until a period is reached. T(n,k)=a(m) with m=(n+4)*(n+3)/2+k+1 gives for m the number of periods with length (n+4).
The numbers m=n*(n+3)/2 with 1<=n have one period with length (n+1).
The numbers m=n*(n+3)/2+2 with 1<=n have one period with length (n+2).
The triangle numbers n*(n+1)/2 with 1<=n have one period [(n+(n-1)+...+2+1)] with length 1 (End).
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REFERENCES
| Pieter Moree, <a href="http://dx.doi.org/10.1016/j.disc.2005.03.004">The formal series Witt transform</a>, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
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EXAMPLE
| 1; 2,2; 2,3,2; 3,5,5,3; 3,7,8,7,3; 4,9,14,14,9,4; 4,12,20,25,20,12,4; ...
Contribution from Paul Weisenhorn, Dec 21 2010 (Start):
T(2,2)=3 classes with 3 ordered sums of 6; [(1+1+4),(1+4+1),(4+1+1)]; [(1+2+3),(2+3+1),(3+1+2)]; [(1+3+2),(3+2+1),(2+1+3)].
T(2,2)=a(m)=3 periods with length 6 for m=6*5/2+3=18 [(5+5+4+3+1),(6+5+4+2+1),(6+5+3+2+1+1),(6+4+3+2+2+1),(5+4+3+3+2+1),(5+4+4+3+2)]; [5+5+3+3+2),(6+4+4+3+1),(5+5+4+2+1+1),(6+5+3+2+2),(6+4+3+3+1+1),(5+4+4+2+2+1)]; [(5+5+3+2+2+1),(6+4+3+3+2),(5+4+4+3+1+1),(5+5+4+2+2),(6+5+3+3+1),(6+4+4+2+1+1)] (End).
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PROG
| (PARI) T(n, k)=local(A, ps, c); n+=3; k++; if(k<1|k>=n-1, 0, A=x*O(x^n)+y*O(y^n); ps=1-x-y+A; for(m=1, n, for(i=0, m, c=polcoeff(polcoeff(ps, i, x), m-i, y); if(m==n&i==k, break(2), ps*=(1-y^(m-i)*x^i+A)^c))); -c) /* Michael Somos, Jul 17 2004 */
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CROSSREFS
| Row sums give A093210. Essentially the same as A051168. See A185158 for another version.
Sequence in context: A194312 A116997 A050142 * A183368 A156862 A076709
Adjacent sequences: A092961 A092962 A092963 * A092965 A092966 A092967
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KEYWORD
| nonn,tabl
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AUTHOR
| Thomas O. Hoffbauer (Thomas.Hoffbauer(AT)cibamberg.de), Apr 20 2004
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EXTENSIONS
| Edited with better definition by Omar E. Pol (info(AT)polprimos.com), Jan 05 2009
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