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A182094 Total area of the bounding boxes of all integer partitions of n.

%I A182094 #24 Jan 14 2024 15:18:24
%S A182094 0,1,4,10,24,47,93,162,283,462,747,1154,1779,2642,3908,5643,8098,
%T A182094 11398,15975,22030,30253,41027,55379,73983,98455,129838,170578,222447,
%U A182094 289009,373064,479970,613962,782893,993349,1256546,1582466,1987365,2485840,3101146
%N A182094 Total area of the bounding boxes of all integer partitions of n.
%H A182094 Alois P. Heinz, <a href="/A182094/b182094.txt">Table of n, a(n) for n = 0..1000</a>
%F A182094 a(n) = A188814(n) + n*A000041(n) = A188814(n) + A066186(n).
%e A182094 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.
%e A182094 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].
%p A182094 b:= proc(n, i) option remember; local f, g;
%p A182094       if n=0 or i=1 then [1, n]
%p A182094     elif i<1 then [0, 0]
%p A182094     else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
%p A182094          [f[1]+g[1], f[2]+g[2]+g[1]]
%p A182094       fi
%p A182094     end:
%p A182094 a:= n-> add(add(i, i=b(n-j, min(j, n-j)))*j, j=1..n):
%p A182094 seq(a(n), n=0..40);
%t A182094 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 *)
%Y A182094 Cf. A000041, A066186, A115995, A188814.
%K A182094 nonn
%O A182094 0,3
%A A182094 _Alois P. Heinz_, Apr 11 2012