OFFSET
0,5
COMMENTS
The Heinz numbers of these partitions appear to be given by A296205.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,2,1,0,-1,-1).
FORMULA
From Colin Barker, Apr 08 2019: (Start)
G.f.: x^3*(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 4*x^5) / ((1 - x)^2*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = -a(n-1) + a(n-3) + 2*a(n-4) + a(n-5) - a(n-7) - a(n-8) for n>8. (End)
a(n) = (27*n + 3*(n - 7)*(-1)^n - 53 - 6*A056594(n) + 8*A061347(n))/24 for n > 0. - Stefano Spezia, Feb 20 2024
EXAMPLE
The a(3) = 1 through a(10) = 10 partitions:
(21) (31) (32) (42) (43) (53) (54) (64)
(211) (41) (51) (52) (62) (63) (73)
(221) (411) (61) (71) (72) (82)
(311) (2211) (322) (332) (81) (91)
(331) (422) (441) (433)
(511) (611) (522) (442)
(3311) (711) (622)
(811)
(3322)
(4411)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Union[#]]==2&&Max@@Length/@Split[#]<=2&]], {n, 0, 30}]
PROG
(PARI) concat([0, 0, 0], Vec(x^3*(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 4*x^5) / ((1 - x)^2*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Apr 08 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gus Wiseman, Apr 05 2019
STATUS
approved