OFFSET
1,3
COMMENTS
Also total number of smallest parts that are not on the right border in all partitions of n.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
FORMULA
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*Pi*sqrt(2*n)) * (1 - 25*Pi/(24*sqrt(6*n)) + (25/48 + 49*Pi^2/6912)/n). - Vaclav Kotesovec, Jul 31 2017
EXAMPLE
For n = 6 the partitions of 6 with the smallest parts that are not in the right border in brackets are
-----------------------------------------
. Partitions of 6 Value
-----------------------------------------
. 6 0
. [3]+ 3 1
. 4 + 2 0
. [2]+[2]+ 2 2
. 5 + 1 0
. 3 + 2 + 1 0
. 4 +[1]+ 1 1
. 2 + 2 +[1]+ 1 1
. 3 +[1]+[1]+ 1 2
. 2 +[1]+[1]+[1]+ 1 3
. [1]+[1]+[1]+[1]+[1]+ 1 5
--------------------------------------
. Total: 15
On the other hand the total number of smallest parts in all partitions of 6 is 26 and the number of partitions of 6 is 11, so a(6) = 26 - 11 = 15.
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r == 0, q, 0] + Sum[b[n - i*j, i - 1], {j, 0, n/i}]];
a[n_] := b[n, n] - PartitionsP[n];
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 13 2013
STATUS
approved