A336739 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, 2, 3, 4, 6, 7, 8, 13, 14, 15, 25, 26, 33, 50, 51, 52, 95, 152, 153, 295, 296, 297, 542, 543, 672, 1329, 1330, 2055, 4093, 4094, 4095, 6992, 10697, 10698, 21375, 21376, 21377, 39051, 55948, 55949, 86454, 86455, 130396, 260765, 260766, 365839, 731649, 1100442

Offset

1,2

Links

Table of n, a(n) for n=1..48.

Miloslav Znojil, Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities, arXiv:2010.15014 [quant-ph], 2020.

Miloslav Znojil, Quantum phase transitions mediated by clustered non-Hermitian degeneracies, arXiv:2102.12272 [quant-ph], 2021.

Miloslav Znojil, Quantum phase transitions mediated by clustered non-Hermitian degeneracies, Physical Review E 103 (2021), 032120.

Miloslav Znojil, Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via PT-symmetric and exactly solvable linear-Bose-Hubbard building blocks, arXiv:2108.07110 [quant-ph], 2021.

Example

H(n,1) are the sets {1}, {1,3}, {1,3,5}, {1,3,5,7}, ...
H(n,2) are the sets {3}, {3,9}, {3,9,15}, {3,9,15,21}, ...
H(n,3) are the sets {5}, {5,15}, {5,15,25}, {5,15,25,35}, ...
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}.
The a(5) = 6 decompositions of {1,3,5,7,9} are:
  {{1,3,5,7,9}},
  {{1,3,5,7}, {9}},
  {{1,3,5}, {7}, {9}},
  {{1,3}, {5}, {7}, {9}},
  {{1}, {3}, {5}, {7}, {9}},
  {{3,9}, {1}, {5}, {7}}.

Prog

(PARI)
tiles(S, t)={((i, b)->1 + sum(i=1, i, if(!bitnegimply(S[i], b), self()(i-1, b-S[i]))))(#S, t)}
H(j, k)={sum(i=1, j, 1<<((2*k-1)*(2*i-1)))}
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

Crossrefs

Cf. A335631.

Sequence in context: A283112 A174099 A099005 * A257648 A096360 A039087

Adjacent sequences: A336736 A336737 A336738 * A336740 A336741 A336742

Keyword

nonn

Author

Miloslav Znojil, Sep 30 2020

Extensions

Terms a(13) and beyond from Andrew Howroyd, Oct 02 2020

Status

approved