OFFSET
2,2
COMMENTS
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.
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)
EXAMPLE
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} |-> /\/\/\
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Martin Paech, Apr 09 2014
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
STATUS
approved