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A196039
Total sum of the smallest part of every partition of every shell of n.
2
0, 1, 4, 9, 18, 30, 50, 75, 113, 162, 231, 318, 441, 593, 798, 1058, 1399, 1824, 2379, 3066, 3948, 5042, 6422, 8124, 10264, 12884, 16138, 20120, 25027, 30994, 38312, 47168, 57955, 70974, 86733, 105676, 128516, 155850, 188644, 227783, 274541
OFFSET
0,3
COMMENTS
Partial sums of A046746.
Total sum of parts of all regions of n that contain 1 as a part. - Omar E. Pol, Mar 11 2012
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
FORMULA
a(n) = A066186(n) - A196025(n).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*Pi*sqrt(2*n)). - Vaclav Kotesovec, Jul 06 2019
EXAMPLE
For n = 5 the seven partitions of 5 are:
5
3 + 2
4 + 1
2 + 2 + 1
3 + 1 + 1
2 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1
.
The five shells of 5 (see A135010 and also A138121), written as a triangle, are:
1
2, 1
3, 1, 1
4, (2, 2), 1, 1, 1
5, (3, 2), 1, 1, 1, 1, 1
.
The first "2" of row 4 does not count, also the "3" of row 5 does not count, so we have:
1
2, 1
3, 1, 1
4, 2, 1, 1, 1
5, 2, 1, 1, 1, 1, 1
.
thus a(5) = 1+2+1+3+1+1+4+2+1+1+1+5+2+1+1+1+1+1 = 30.
MAPLE
b:= proc(n, i) option remember;
`if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(n<i, 0, b(n-i, i)))
end:
a:= proc(n) option remember;
b(n, n) +`if`(n=0, 0, a(n-1))
end:
seq(a(n), n=0..50); # Alois P. Heinz, Apr 03 2012
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == i, n, 0] + If[i < 1, 0, b[n, i-1] + If[n < i, 0, b[n-i, i]]]; Accumulate[Table[b[n, n], {n, 0, 50}]] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)
KEYWORD
nonn
AUTHOR
Omar E. Pol, Oct 27 2011
STATUS
approved