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a(n) is the number of decompositions of H(n,1) into disjoint unions of H(j,k) where H(j,k) is the set of numbers { (2*i-1)*(2*k-1), 1 <= i <= j }.
1

%I #69 Jan 11 2022 13:21:18

%S 1,2,3,4,6,7,8,13,14,15,25,26,33,50,51,52,95,152,153,295,296,297,542,

%T 543,672,1329,1330,2055,4093,4094,4095,6992,10697,10698,21375,21376,

%U 21377,39051,55948,55949,86454,86455,130396,260765,260766,365839,731649,1100442

%N a(n) is the number of decompositions of H(n,1) into disjoint unions of H(j,k) where H(j,k) is the set of numbers { (2*i-1)*(2*k-1), 1 <= i <= j }.

%H Miloslav 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 Miloslav 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 Miloslav 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 H(n,1) are the sets {1}, {1,3}, {1,3,5}, {1,3,5,7}, ...

%e H(n,2) are the sets {3}, {3,9}, {3,9,15}, {3,9,15,21}, ...

%e H(n,3) are the sets {5}, {5,15}, {5,15,25}, {5,15,25,35}, ...

%e a(2) = 2 because there are two decompositions of H(2,1) = {1,3}: the trivial H(2,1) and H(1,1) + H(1,2) = {1} + {3}.

%e The a(5) = 6 decompositions of {1,3,5,7,9} are:

%e {{1,3,5,7,9}},

%e {{1,3,5,7}, {9}},

%e {{1,3,5}, {7}, {9}},

%e {{1,3}, {5}, {7}, {9}},

%e {{1}, {3}, {5}, {7}, {9}},

%e {{3,9}, {1}, {5}, {7}}.

%o (PARI)

%o tiles(S,t)={((i,b)->1 + sum(i=1, i, if(!bitnegimply(S[i],b), self()(i-1, b-S[i]))))(#S, t)}

%o H(j,k)={sum(i=1, j, 1<<((2*k-1)*(2*i-1)))}

%o a(n)={my(S=concat(vector(n, k, vector(n\(2*k-1), j, H(1+j,k))))); tiles(S, H(n,1))} \\ _Andrew Howroyd_, Oct 02 2020

%Y Cf. A335631.

%K nonn

%O 1,2

%A _Miloslav Znojil_, Sep 30 2020

%E Terms a(13) and beyond from _Andrew Howroyd_, Oct 02 2020