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%I #55 Jul 16 2017 14:44:18
%S 1,2,10,59,397,2878,21266,162732,1253128,9839212,77644825,620377508,
%T 4981522538,40351448045,328421827064,2690586461296,22139293490054,
%U 183106636176023,1520309861062921,12675106437486945,106033283581264574,890035798660219755
%N 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).
%C 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.
%C Also the number of "island altitude-profiles" of length 2n-1, see examples, which satisfy the following requirements:
%C (1) Every profile starts and ends at sea-level (zero double-occupancies).
%C (2) The height increases and decreases with every step at most one unit.
%C (3) The maximum height does not go beyond floor(n/2).
%C (4) The minimum height does not fall below sea-level.
%C (5) Sea-level could only be reached after an even number of steps.
%C (6) For n even, no plateaus exist at maximum height (= n/2).
%C (7) For n even, two peaks at maximum height have an even distance.
%H Alois P. Heinz, <a href="/A240605/b240605.txt">Table of n, a(n) for n = 2..400</a>
%H E. Kalinowski and W. Gluza, <a href="http://arxiv.org/abs/1106.4938">Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit</a>, arXiv:1106.4938 [cond-mat.str-el], 2011 (Physical Review B 85, 045105, Jan 2012)
%H E. Kalinowski and M. Paech, <a href="/A240605/a240605.pdf">Table of island altitude-profiles I(n,k) up to order n = 6</a>.
%H M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, <a href="http://www.dpg-verhandlungen.de/year/2012/conference/berlin/part/tt/session/45/contribution/91">Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit</a>, DPG Spring Meeting, Berlin, TT 45.91 (2012)
%e n = 2
%e 0 1 0 |-> T_{+U} T_{-U} |-> /\
%e n = 3
%e __
%e 0 1 1 1 0 |-> T_{+U} T_{ 0} T_{ 0} T_{-U} |-> / \
%e 0 1 0 1 0 |-> T_{+U} T_{-U} T_{+U} T_{-U} |-> /\/\
%e n = 4
%e ____
%e 0 1 1 1 1 1 0 |-> T_{+U} T_{ 0} T_{ 0} T_{ 0} T_{ 0} T_{-U} |-> / \
%e __/\
%e 0 1 1 1 2 1 0 |-> T_{+U} T_{ 0} T_{ 0} T_{+U} T_{-U} T_{-U} |-> / \
%e __
%e 0 1 1 1 0 1 0 |-> T_{+U} T_{ 0} T_{ 0} T_{-U} T_{+U} T_{-U} |-> / \/\
%e _/\_
%e 0 1 1 2 1 1 0 |-> T_{+U} T_{ 0} T_{+U} T_{-U} T_{ 0} T_{-U} |-> / \
%e /\__
%e 0 1 2 1 1 1 0 |-> T_{+U} T_{+U} T_{-U} T_{ 0} T_{ 0} T_{-U} |-> / \
%e /\/\
%e 0 1 2 1 2 1 0 |-> T_{+U} T_{+U} T_{-U} T_{+U} T_{-U} T_{-U} |-> / \
%e /\
%e 0 1 2 1 0 1 0 |-> T_{+U} T_{+U} T_{-U} T_{-U} T_{+U} T_{-U} |-> / \/\
%e __
%e 0 1 0 1 1 1 0 |-> T_{+U} T_{-U} T_{+U} T_{ 0} T_{ 0} T_{-U} |-> /\/ \
%e /\
%e 0 1 0 1 2 1 0 |-> T_{+U} T_{-U} T_{+U} T_{+U} T_{-U} T_{-U} |-> /\/ \
%e 0 1 0 1 0 1 0 |-> T_{+U} T_{-U} T_{+U} T_{-U} T_{+U} T_{-U} |-> /\/\/\
%p b:= proc(x, y, m, v, d) option remember; `if`(y>x or y<0 or
%p y>m or v and y=m and d=1 or y=0 and irem(x, 2)=1, 0,
%p `if`(x=0, 1, `if`(v and y=m or y=0, 0, b(x-1, y, m, v,
%p `if`(d=2, 2, 1-d)))+ `if`(y=0 or y=1 and irem(x, 2)=0, 0,
%p b(x-1, y-1, m, v, `if`(d=2, `if`(v and y=m, 1, 2), 1-d)))+
%p b(x-1, y+1, m, v, `if`(d=2, 2, 1-d))))
%p end:
%p a:= n-> b(2*n-2, 0, iquo(n, 2, 'r'), r=0, 2):
%p seq(a(n), n=2..30); # _Alois P. Heinz_, May 09 2014
%t 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_ *)
%Y Cf. A198760, A198761, A225823.
%K nonn
%O 2,2
%A _Martin Paech_, Apr 09 2014
%E 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.)
%E a(18)-a(23) from _Alois P. Heinz_, May 08 2014
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