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

1, 3, 6, 12, 20, 34, 52, 80, 116, 170, 236, 333, 453, 621, 825, 1111, 1455, 1923, 2487, 3239, 4149, 5342, 6770, 8625, 10852, 13698, 17107, 21413, 26567, 33019, 40721, 50270, 61663, 75665, 92318, 112686, 136849, 166173, 200923, 242836

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

a(n) = A006128(n) - A182977(n).

Example

For n = 6 the partitions of 6 are
6
5 + 1
4 + 2
4 + 1 + 1
3 + 3
3 + (2) + 1 .... the "2" is the part that does not count.
3 + 1 + 1 + 1
2 + 2 + 2
2 + 2 + 1 + 1
2 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1
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.

Maple

l:= proc(n, i) option remember; `if`(n=i, n, 0)+
      `if`(i<1, 0, l(n, i-1) +`if`(n<i, 0, l(n-i, i)))
    end:
s:= proc(n, i) option remember; `if`(n=0 or i=1, n,
      `if`(irem(n, i, 'r')=0, r, 0)+add(s(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> l(n, n) +s(n, n) -numtheory[sigma](n):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 17 2013

Mathematica

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 *)

Crossrefs

Cf. A006128, A046746, A092269, A116686, A182977, A182984.

Sequence in context: A096220 A034333 A348464 * A320609 A320610 A320611

Adjacent sequences: A182975 A182976 A182977 * A182979 A182980 A182981

Keyword

nonn

Author

Omar E. Pol, Jul 17 2011

Extensions

a(12) corrected and more terms a(13)-a(40) from David Scambler, Jul 18 2011

Status

approved