A240605 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).
1, 2, 10, 59, 397, 2878, 21266, 162732, 1253128, 9839212, 77644825, 620377508, 4981522538, 40351448045, 328421827064, 2690586461296, 22139293490054, 183106636176023, 1520309861062921, 12675106437486945, 106033283581264574, 890035798660219755
Offset: 2
Keywords
Examples
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} |-> / \
__
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} |-> / \/\
__
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} |-> /\/\/\
Links
- Alois P. Heinz, Table of n, a(n) for n = 2..400
- E. Kalinowski and W. Gluza, Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit, arXiv:1106.4938 [cond-mat.str-el], 2011 (Physical Review B 85, 045105, Jan 2012)
- E. Kalinowski and M. Paech, Table of island altitude-profiles I(n,k) up to order n = 6.
- M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit, DPG Spring Meeting, Berlin, TT 45.91 (2012)
Programs
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Maple
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): seq(a(n), n=2..30); # Alois P. Heinz, May 09 2014 -
Mathematica
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 *)
Extensions
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.)
a(18)-a(23) from Alois P. Heinz, May 08 2014
Comments