%I #39 Jan 11 2022 13:21:12
%S 1,3,6,11,17,32,47,66,105,162,198,376,451,634,1131,1405,1576,2487,
%T 2761,4216,7499,9207,9931,19328,25621,30701,48795,67006,70841,122397,
%U 128833,143082,270155,303385,493906,761627,789651,877486,1715085,2595269,2679136,4665660
%N a(n) is the number of decompositions of F(n,1) into disjoint unions of F(j,k) and G(q,r) where F(j,k) is the set of numbers { i*k, 0 <= i <= j } and where G(q,r) is the set of numbers { (2*p-1)*r, 1 <= p <= q }.
%H M. Znojil, <a href="https://arxiv.org/abs/2010.15014">Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities</a>, arXiv:2010.15014 [quant-ph], 2020.
%H M. Znojil, <a href="https://arxiv.org/abs/2102.12272">Quantum phase transitions mediated by clustered non-Hermitian degeneracies</a>, arXiv:2102.12272 [quant-ph], 2021.
%H M. Znojil, <a href="https://doi.org/10.1103/PhysRevE.103.032120">Quantum phase transitions mediated by clustered non-Hermitian degeneracies</a>, Physical Review E 103 (2021), 032120.
%H Miloslav Znojil, <a href="https://arxiv.org/abs/2108.07110">Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via PT-symmetric and exactly solvable linear-Bose-Hubbard building blocks</a>, arXiv:2108.07110 [quant-ph], 2021.
%e F(n,1) are the sets {0,1}, {0,1,2}, {0,1,2,3}, {0,1,2,3,4}, ...
%e F(n,2) are the sets {0,2}, {0,2,4}, {0,2,4,6}, {0,2,4,6,8}, ...
%e F(n,3) are the sets {0,3}, {0,3,6}, {0,3,6,9}, {0,3,6,9,12}, ...
%e etc., and
%e G(n,1) are the sets {1}, {1,3}, {1,3,5}, {1,3,5,7}, ...
%e G(n,2) are the sets {2}, {2,6}, {2,6,10}, {2,6,10,14}, ...
%e G(n,3) are the sets {3}, {3,9}, {3,9,15}, {3,9,15,21}, ...
%e etc.
%e a(2) = 3 because there are three decompositions of F(2,1) = {0,1,2}: the trivial F(2,1), F(1,2) + G(1,1) = {0,2} + {1}, and F(1,1) + G(1,2) = {0,1} + {2}.
%e The a(3) = 6 decompositions of {0,1,2,3} are:
%e {{0,1,2,3}},
%e {{0,1,2}, {3}},
%e {{0,1}, {2}, {3}},
%e {{0,2}, {1,3}},
%e {{0,2}, {1}, {3}},
%e {{0,3}, {1}, {2}}.
%o (PARI)
%o tiles(S, t, w)={((i, b)->w(b) + sum(i=1, i, if(!bitnegimply(S[i], b), self()(i-1, b-S[i]))))(#S, t)}
%o Q(j,k,t)={sum(i=0, j-1, 1<<(t+i*k))}
%o S(n)={concat(concat(vector(n\2+1, k, vector(n\k-1, j, Q(j+2,k,0)))), concat(vector(n\3+1, k, vector((n\k-1)\2, j, Q(j+1,2*k,k)))))}
%o a(n)={tiles(S(n), Q(n+1,1,0), b->if(bittest(b,0), hammingweight(b)-1, 1))} \\ _Andrew Howroyd_, Oct 29 2020
%Y Cf. A336739 (odd-number-sets decomposition).
%K nonn
%O 1,2
%A _Miloslav Znojil_, Oct 03 2020
%E a(10)-a(42) from _Andrew Howroyd_, Oct 29 2020