%I #38 Jul 06 2019 06:40:00

%S 0,1,4,9,18,30,50,75,113,162,231,318,441,593,798,1058,1399,1824,2379,

%T 3066,3948,5042,6422,8124,10264,12884,16138,20120,25027,30994,38312,

%U 47168,57955,70974,86733,105676,128516,155850,188644,227783,274541

%N Total sum of the smallest part of every partition of every shell of n.

%C Partial sums of A046746.

%C Total sum of parts of all regions of n that contain 1 as a part. - _Omar E. Pol_, Mar 11 2012

%H Vaclav Kotesovec, <a href="/A196039/b196039.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>

%F a(n) = A066186(n) - A196025(n).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*Pi*sqrt(2*n)). - _Vaclav Kotesovec_, Jul 06 2019

%e For n = 5 the seven partitions of 5 are:

%e 5

%e 3 + 2

%e 4 + 1

%e 2 + 2 + 1

%e 3 + 1 + 1

%e 2 + 1 + 1 + 1

%e 1 + 1 + 1 + 1 + 1

%e .

%e The five shells of 5 (see A135010 and also A138121), written as a triangle, are:

%e 1

%e 2, 1

%e 3, 1, 1

%e 4, (2, 2), 1, 1, 1

%e 5, (3, 2), 1, 1, 1, 1, 1

%e .

%e The first "2" of row 4 does not count, also the "3" of row 5 does not count, so we have:

%e 1

%e 2, 1

%e 3, 1, 1

%e 4, 2, 1, 1, 1

%e 5, 2, 1, 1, 1, 1, 1

%e .

%e thus a(5) = 1+2+1+3+1+1+4+2+1+1+1+5+2+1+1+1+1+1 = 30.

%p b:= proc(n, i) option remember;

%p `if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(n<i, 0, b(n-i, i)))

%p end:

%p a:= proc(n) option remember;

%p b(n, n) +`if`(n=0, 0, a(n-1))

%p end:

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Apr 03 2012

%t 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_ *)

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

%K nonn

%O 0,3

%A _Omar E. Pol_, Oct 27 2011