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A182099 Total area of the largest inscribed rectangles of all integer partitions of n. 1
0, 1, 4, 8, 18, 29, 54, 82, 136, 202, 309, 441, 658, 915, 1303, 1790, 2479, 3337, 4541, 6022, 8045, 10554, 13876, 17996, 23409, 30055, 38634, 49208, 62650, 79116, 99898, 125213, 156848, 195339, 242964, 300707, 371770, 457493, 562292, 688451, 841707, 1025484 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) >= A000041(n)*A061017(n) for n>0 because the least largest inscribed rectangle of any integer partition of n is A061017(n) and A000041(n) is the number of partitions of n.

a(n) >= A116503(n), the sum of the areas of the Durfee squares of all partitions of n.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..175

FORMULA

a(n) = Sum_{k=1..n} k * A115723(n,k) for n>0, a(0) = 0.

a(n) = Sum_{k=1..n} k * (A182114(n,k) - A182114(n,k-1)).

EXAMPLE

a(4) = 18 = 4+3+4+3+4 because the partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4] and the largest inscribed rectangles have areas 4*1, 3*1, 2*2, 1*3, 1*4.

a(5) = 29 = 5+4+4+3+4+4+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, t, k) option remember; `if`(n=0, 1,

      `if`(i=1, `if`(t+n>k, 0, 1), `if`(i<1, 0, b(n, i-1, t, k)

      +add(`if`(t+j>k/i, 0, b(n-i*j, i-1, t+j, k)), j=1..n/i))))

    end:

a:= n-> add(k*(b(n, n, 0, k) -b(n, n, 0, k-1)), k=1..n):

seq(a(n), n=0..50);

CROSSREFS

Cf. A000041, A061017, A115723, A116503.

Sequence in context: A077474 A307817 A210433 * A318756 A195334 A009918

Adjacent sequences:  A182096 A182097 A182098 * A182100 A182101 A182102

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Apr 11 2012

STATUS

approved

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Last modified August 18 08:57 EDT 2019. Contains 326077 sequences. (Running on oeis4.)