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

Author

Alois P. Heinz, Jun 06 2022

Keywords

• nonn

Examples

			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)}|.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..700
• Wikipedia, Infima and suprema of real numbers
• Wikipedia, Partition (number theory)

Crossrefs

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

Programs

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 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 *)