A354800 Cardinality of the set of ordered pairs (m(lambda),f(lambda)), where lambda ranges over all partitions of n and m gives the infimum and f gives the sum of the squares of the argument.

1, 1, 2, 3, 5, 7, 11, 13, 20, 26, 33, 41, 55, 63, 77, 93, 111, 129, 160, 180, 209, 240, 280, 312, 356, 397, 453, 498, 560, 614, 680, 758, 831, 901, 994, 1087, 1179, 1280, 1389, 1495, 1629, 1745, 1868, 2022, 2159, 2296, 2485, 2650, 2809, 2991, 3181, 3377, 3600

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..700

Wikipedia, Infima and suprema of real numbers

Wikipedia, Partition (number theory)

Example

a(0) = 1 = |{(infinity,0)}|.
a(1) = 1 = |{(1,1)}|.
a(2) = 2 = |{(1,2), (2,4)}|.
a(3) = 3 = |{(1,3), (1,5), (3,9)}|.
a(4) = 5 = |{(1,4), (1,6), (1,10), (2,8), (4,16)}|.
a(5) = 7 = |{(1,5), (1,7), (1,9), (1,11), (1,17), (2,13), (5,25)}|.

Maple

a:= n-> nops({map(l-> [min(l), add(i^2, i=l)], combinat[partition](n))[]}):
seq(a(n), n=0..40);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(n<i,
      {}, {b(n, i+1)[], map(x-> x+i^2, b(n-i, i))[]}))
    end:
a:= n-> add(nops(b(n-i, i)), i=signum(n)..n):
seq(a(n), n=0..60);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[n < i, {}, Union@ Flatten@ {b[n, i + 1], # + i^2& /@ b[n - i, i]}]];
a[n_] :=  Sum[Length[b[n - i, i]], {i, Sign[n], n}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jul 06 2022, after Alois P. Heinz *)

Crossrefs

Cf. A069999 (lower bound), A354468 (the same for supremum).

Sequence in context: A338396 A067910 A171574 * A280937 A296236 A357659

Adjacent sequences: A354797 A354798 A354799 * A354801 A354802 A354803

Keyword

nonn

Author

Alois P. Heinz, Jun 06 2022

Status

approved