cp's OEIS Frontend

A196039 Total sum of the smallest part of every partition of every shell of n.

%I A196039 #38 Jul 06 2019 06:40:00
%S A196039 0,1,4,9,18,30,50,75,113,162,231,318,441,593,798,1058,1399,1824,2379,
%T A196039 3066,3948,5042,6422,8124,10264,12884,16138,20120,25027,30994,38312,
%U A196039 47168,57955,70974,86733,105676,128516,155850,188644,227783,274541
%N A196039 Total sum of the smallest part of every partition of every shell of n.
%C A196039 Partial sums of A046746.
%C A196039 Total sum of parts of all regions of n that contain 1 as a part. - _Omar E. Pol_, Mar 11 2012
%H A196039 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 A196039 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>
%F A196039 a(n) = A066186(n) - A196025(n).
%F A196039 a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*Pi*sqrt(2*n)). - _Vaclav Kotesovec_, Jul 06 2019
%e A196039 For n = 5 the seven partitions of 5 are:
%e A196039 5
%e A196039 3         + 2
%e A196039 4             + 1
%e A196039 2     + 2     + 1
%e A196039 3         + 1 + 1
%e A196039 2     + 1 + 1 + 1
%e A196039 1 + 1 + 1 + 1 + 1
%e A196039 .
%e A196039 The five shells of 5 (see A135010 and also A138121), written as a triangle, are:
%e A196039 1
%e A196039 2, 1
%e A196039 3, 1, 1
%e A196039 4, (2, 2), 1, 1, 1
%e A196039 5, (3, 2), 1, 1, 1, 1, 1
%e A196039 .
%e A196039 The first "2" of row 4 does not count, also the "3" of row 5 does not count, so we have:
%e A196039 1
%e A196039 2, 1
%e A196039 3, 1, 1
%e A196039 4, 2, 1, 1, 1
%e A196039 5, 2, 1, 1, 1, 1, 1
%e A196039 .
%e A196039 thus a(5) = 1+2+1+3+1+1+4+2+1+1+1+5+2+1+1+1+1+1 = 30.
%p A196039 b:= proc(n, i) option remember;
%p A196039      `if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(n<i, 0, b(n-i, i)))
%p A196039     end:
%p A196039 a:= proc(n) option remember;
%p A196039       b(n, n) +`if`(n=0, 0, a(n-1))
%p A196039     end:
%p A196039 seq(a(n), n=0..50); # _Alois P. Heinz_, Apr 03 2012
%t A196039 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 A196039 Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196025, A196930, A196931, A198381, A206437.
%K A196039 nonn
%O A196039 0,3
%A A196039 _Omar E. Pol_, Oct 27 2011