OFFSET
0,10
COMMENTS
A part x has a neighbor if either x - 1 or x + 1 is a part.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 301 terms from John Tyler Rascoe)
John Tyler Rascoe, Python program
FORMULA
G.f.: 1 + Sum_{i>0} A(x,i), where A(x,i) = x^((2*i)+1) * G(x,i+1) for i > 0, is the g.f. for partitions of this kind with least part i, and G(x,k) = 1 + x^(k+1) * G(x,k+1) + Sum_{m>=0} x^(2*(k+m)+5) * G(x,m+k+3). - John Tyler Rascoe, Feb 16 2024
EXAMPLE
The a(n) partitions for n = 0, 1, 3, 9, 15, 18, 20, 24 (A = 10, B = 11):
() . (21) (54) (87) (765) (7643) (987)
(432) (654) (6543) (8732) (8754)
(54321) (7632) (9821) (9843)
(8721) (65432) (A932)
(65421) (BA21)
(87432)
(87621)
(765321)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Function[ptn, UnsameQ@@ptn&&And@@Table[MemberQ[ptn, x-1]||MemberQ[ptn, x+1], {x, Union[ptn]}]]]], {n, 0, 30}]
PROG
(Python) # see linked program
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 24 2022
STATUS
approved