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

Author

Martin Paech, Apr 09 2014

Keywords

• nonn

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.

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)

Crossrefs

Cf. A198760, A198761, A225823.

Programs

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