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A182978 Total number of parts that are the smallest part or the largest part in all partitions of n.

%I A182978 #15 Nov 03 2015 03:18:57
%S A182978 1,3,6,12,20,34,52,80,116,170,236,333,453,621,825,1111,1455,1923,2487,
%T A182978 3239,4149,5342,6770,8625,10852,13698,17107,21413,26567,33019,40721,
%U A182978 50270,61663,75665,92318,112686,136849,166173,200923,242836
%N A182978 Total number of parts that are the smallest part or the largest part in all partitions of n.
%H A182978 Alois P. Heinz, <a href="/A182978/b182978.txt">Table of n, a(n) for n = 1..1000</a>
%F A182978 a(n) = A006128(n) - A182977(n).
%e A182978 For n = 6 the partitions of 6 are
%e A182978 6
%e A182978 5 + 1
%e A182978 4 + 2
%e A182978 4 + 1 + 1
%e A182978 3 + 3
%e A182978 3 + (2) + 1 .... the "2" is the part that does not count.
%e A182978 3 + 1 + 1 + 1
%e A182978 2 + 2 + 2
%e A182978 2 + 2 + 1 + 1
%e A182978 2 + 1 + 1 + 1 + 1
%e A182978 1 + 1 + 1 + 1 + 1 + 1
%e A182978 The total number of parts in all partitions of 6 is equal to 35. All parts are the smallest part or the largest part, except the "2" in the partition (3 + 2 + 1), so a(6) = 35 - 1 = 34.
%p A182978 l:= proc(n, i) option remember; `if`(n=i, n, 0)+
%p A182978       `if`(i<1, 0, l(n, i-1) +`if`(n<i, 0, l(n-i, i)))
%p A182978     end:
%p A182978 s:= proc(n, i) option remember; `if`(n=0 or i=1, n,
%p A182978       `if`(irem(n, i, 'r')=0, r, 0)+add(s(n-i*j, i-1), j=0..n/i))
%p A182978     end:
%p A182978 a:= n-> l(n, n) +s(n, n) -numtheory[sigma](n):
%p A182978 seq(a(n), n=1..50);  # _Alois P. Heinz_, Jan 17 2013
%t A182978 l[n_, i_] := l[n, i] = If[n==i, n, 0] + If[i<1, 0, l[n, i-1] + If[n<i, 0, l[n-i, i]]]; s[n_, i_] := s[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r==0, q, 0] + Sum[s[n-i*j, i-1], {j, 0, n/i}] ]; a[n_] := l[n, n] + s[n, n] - DivisorSigma[1, n]; Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Nov 03 2015, after _Alois P. Heinz_ *)
%Y A182978 Cf. A006128, A046746, A092269, A116686, A182977, A182984.
%K A182978 nonn
%O A182978 1,2
%A A182978 _Omar E. Pol_, Jul 17 2011
%E A182978 a(12) corrected and more terms a(13)-a(40) from _David Scambler_, Jul 18 2011