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A240605
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Total number of distinct sequences for the number of double occupancy in the underlying Fermion problem (see comment), i.e., the number of distinct hopping sequences (cf. A198761, A225823) in four-colored rooted trees with n nodes, starting and ending with the same coloring in two colors (cf. A198760, corresponding to zero double-occupancy).
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1
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1, 2, 10, 59, 397, 2878, 21266, 162732, 1253128, 9839212, 77644825, 620377508, 4981522538, 40351448045, 328421827064, 2690586461296, 22139293490054, 183106636176023, 1520309861062921, 12675106437486945, 106033283581264574, 890035798660219755
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OFFSET
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2,2
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COMMENTS
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The sequences of double-occupancy are generated by the operators T_{+U}, T_{-U}, and T_{0} defined in eq. (8) in Phys. Rev. B 85, 045105 (2012), see below.
Also the number of "island altitude-profiles" of length 2n-1, see examples, which satisfy the following requirements:
(1) Every profile starts and ends at sea-level (zero double-occupancies).
(2) The height increases and decreases with every step at most one unit.
(3) The maximum height does not go beyond floor(n/2).
(4) The minimum height does not fall below sea-level.
(5) Sea-level could only be reached after an even number of steps.
(6) For n even, no plateaus exist at maximum height (= n/2).
(7) For n even, two peaks at maximum height have an even distance.
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LINKS
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EXAMPLE
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n = 2
0 1 0 |-> T_{+U} T_{-U} |-> /\
n = 3
__
0 1 1 1 0 |-> T_{+U} T_{ 0} T_{ 0} T_{-U} |-> / \
0 1 0 1 0 |-> T_{+U} T_{-U} T_{+U} T_{-U} |-> /\/\
n = 4
____
0 1 1 1 1 1 0 |-> T_{+U} T_{ 0} T_{ 0} T_{ 0} T_{ 0} T_{-U} |-> / \
__/\
0 1 1 1 2 1 0 |-> T_{+U} T_{ 0} T_{ 0} T_{+U} T_{-U} T_{-U} |-> / \
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0 1 1 1 0 1 0 |-> T_{+U} T_{ 0} T_{ 0} T_{-U} T_{+U} T_{-U} |-> / \/\
_/\_
0 1 1 2 1 1 0 |-> T_{+U} T_{ 0} T_{+U} T_{-U} T_{ 0} T_{-U} |-> / \
/\__
0 1 2 1 1 1 0 |-> T_{+U} T_{+U} T_{-U} T_{ 0} T_{ 0} T_{-U} |-> / \
/\/\
0 1 2 1 2 1 0 |-> T_{+U} T_{+U} T_{-U} T_{+U} T_{-U} T_{-U} |-> / \
/\
0 1 2 1 0 1 0 |-> T_{+U} T_{+U} T_{-U} T_{-U} T_{+U} T_{-U} |-> / \/\
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0 1 0 1 1 1 0 |-> T_{+U} T_{-U} T_{+U} T_{ 0} T_{ 0} T_{-U} |-> /\/ \
/\
0 1 0 1 2 1 0 |-> T_{+U} T_{-U} T_{+U} T_{+U} T_{-U} T_{-U} |-> /\/ \
0 1 0 1 0 1 0 |-> T_{+U} T_{-U} T_{+U} T_{-U} T_{+U} T_{-U} |-> /\/\/\
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MAPLE
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b:= proc(x, y, m, v, d) option remember; `if`(y>x or y<0 or
y>m or v and y=m and d=1 or y=0 and irem(x, 2)=1, 0,
`if`(x=0, 1, `if`(v and y=m or y=0, 0, b(x-1, y, m, v,
`if`(d=2, 2, 1-d)))+ `if`(y=0 or y=1 and irem(x, 2)=0, 0,
b(x-1, y-1, m, v, `if`(d=2, `if`(v and y=m, 1, 2), 1-d)))+
b(x-1, y+1, m, v, `if`(d=2, 2, 1-d))))
end:
a:= n-> b(2*n-2, 0, iquo(n, 2, 'r'), r=0, 2):
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MATHEMATICA
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b[x_, y_, m_, v_, d_] := b[x, y, m, v, d] = If[y>x || y<0 || y>m || v && y == m && d==1 || y==0 && Mod[x, 2]==1, 0, If[x==0, 1, If[v && y==m || y==0, 0, b[x-1, y, m, v, If[d==2, 2, 1-d]]] + If[y==0 || y==1 && Mod[x, 2]==0, 0, b[x-1, y-1, m, v, If[d==2, If[v && y==m, 1, 2], 1-d]]] + b[x-1, y+1, m, v, If[d==2, 2, 1-d]]]]; a[n_] := b[2*n-2, 0, Quotient[n, 2], Mod[ n, 2]==0, 2]; Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Terms a(16) and a(17) are calculated on a HP Integrity Superdome 2-16s by courtesy of Hewlett-Packard Development Company, L.P., by Martin Paech, May 08 2014 (The used algorithm generates explicitly all distinct sequences of double-occupancy, i.e. all valid "island altitude-profiles", and counts them.)
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STATUS
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approved
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