login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A182094
Total area of the bounding boxes of all integer partitions of n.
3
0, 1, 4, 10, 24, 47, 93, 162, 283, 462, 747, 1154, 1779, 2642, 3908, 5643, 8098, 11398, 15975, 22030, 30253, 41027, 55379, 73983, 98455, 129838, 170578, 222447, 289009, 373064, 479970, 613962, 782893, 993349, 1256546, 1582466, 1987365, 2485840, 3101146
OFFSET
0,3
LINKS
FORMULA
a(n) = A188814(n) + n*A000041(n) = A188814(n) + A066186(n).
EXAMPLE
a(4) = 24 = 4+6+4+6+4 because the partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4] and the bounding boxes have areas 4*1, 3*2, 2*2, 2*3, 1*4.
a(5) = 47 = 5+8+6+9+6+8+5 because the partitions of 5 are [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
MAPLE
b:= proc(n, i) option remember; local f, g;
if n=0 or i=1 then [1, n]
elif i<1 then [0, 0]
else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
[f[1]+g[1], f[2]+g[2]+g[1]]
fi
end:
a:= n-> add(add(i, i=b(n-j, min(j, n-j)))*j, j=1..n):
seq(a(n), n=0..40);
MATHEMATICA
b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, n}, If[i < 1, {0, 0}, f = b[n, i - 1]; g = If[i > n, {0, 0}, b[n - i, i]]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]}]]; a[n_] := Sum[Sum[i, {i, b[n - j, Min[j, n - j]]}]*j, {j, 1, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 05 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 11 2012
STATUS
approved