OFFSET
0,3
COMMENTS
From Gus Wiseman, Mar 31 2019: (Start)
A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. For example, the reversed binary expansions of 2, 5, and 8 are
{0,1}
{1,0,1}
{0,0,0,1}
and since there are no columns with more than one 1, the partition (8,5,2) is counted under a(15). The Heinz numbers of these partitions are given by A325097.
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
EXAMPLE
From Gus Wiseman, Mar 30 2019: (Start)
The a(1) = 1 through a(8) = 11 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (41) (33) (43) (44)
(111) (211) (221) (42) (52) (422)
(1111) (2111) (222) (61) (611)
(11111) (411) (421) (2222)
(2211) (2221) (4211)
(21111) (4111) (22211)
(111111) (22111) (41111)
(211111) (221111)
(1111111) (2111111)
(11111111)
(End)
MAPLE
with(Bits):
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, t) +`if`(i>n or And(t, i)>0, 0,
add(b(n-i*j, i-1, Or(t, i)), j=1..n/i))))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80); # Alois P. Heinz, Dec 28 2014
MATHEMATICA
binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[IntegerPartitions[n], stableQ[#, Intersection[binpos[#1], binpos[#2]]!={}&]&]], {n, 0, 20}] (* Gus Wiseman, Mar 30 2019 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, t] + If[i > n || BitAnd[t, i] > 0, 0, Sum[b[n - i*j, i - 1, BitOr[t, i]], {j, 1, n/i}]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 80] (* Jean-François Alcover, May 23 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David S. Newman, Sep 26 2014
EXTENSIONS
More terms from Alois P. Heinz, Oct 15 2014
Name edited by Gus Wiseman, Mar 31 2019
STATUS
approved