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A182094 Total area of the bounding boxes of all integer partitions of n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) = A188814(n) + n*A000041(n) = A188814(n) + A066186(n).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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-Fran├žois Alcover, Feb 05 2017, translated from Maple *)

CROSSREFS

Cf. A000041, A066186, A188814.

Sequence in context: A083168 A143696 A058514 * A291949 A001979 A209970

Adjacent sequences:  A182091 A182092 A182093 * A182095 A182096 A182097

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Apr 11 2012

STATUS

approved

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Last modified May 25 17:53 EDT 2020. Contains 334595 sequences. (Running on oeis4.)