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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 (list; graph; refs; listen; history; text; internal format)
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)

Omar E. Pol, Illustration of the seven regions of 5

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 *)

CROSSREFS

Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196025, A196930, A196931, A198381, A206437.

Sequence in context: A038098 A299274 A111384 * A238091 A301017 A008219

Adjacent sequences:  A196036 A196037 A196038 * A196040 A196041 A196042

KEYWORD

nonn

AUTHOR

Omar E. Pol, Oct 27 2011

STATUS

approved

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Last modified December 3 14:43 EST 2021. Contains 349463 sequences. (Running on oeis4.)