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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.

%I A354800 #24 Jul 06 2022 13:35:23
%S A354800 1,1,2,3,5,7,11,13,20,26,33,41,55,63,77,93,111,129,160,180,209,240,
%T A354800 280,312,356,397,453,498,560,614,680,758,831,901,994,1087,1179,1280,
%U A354800 1389,1495,1629,1745,1868,2022,2159,2296,2485,2650,2809,2991,3181,3377,3600
%N 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.
%H A354800 Alois P. Heinz, <a href="/A354800/b354800.txt">Table of n, a(n) for n = 0..700</a>
%H A354800 Wikipedia, <a href="https://en.wikipedia.org/wiki/Infimum_and_supremum#Infima_and_suprema_of_real_numbers">Infima and suprema of real numbers</a>
%H A354800 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A354800 a(0) = 1 = |{(infinity,0)}|.
%e A354800 a(1) = 1 = |{(1,1)}|.
%e A354800 a(2) = 2 = |{(1,2), (2,4)}|.
%e A354800 a(3) = 3 = |{(1,3), (1,5), (3,9)}|.
%e A354800 a(4) = 5 = |{(1,4), (1,6), (1,10), (2,8), (4,16)}|.
%e A354800 a(5) = 7 = |{(1,5), (1,7), (1,9), (1,11), (1,17), (2,13), (5,25)}|.
%p A354800 a:= n-> nops({map(l-> [min(l), add(i^2, i=l)], combinat[partition](n))[]}):
%p A354800 seq(a(n), n=0..40);
%p A354800 # second Maple program:
%p A354800 b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(n<i,
%p A354800       {}, {b(n, i+1)[], map(x-> x+i^2, b(n-i, i))[]}))
%p A354800     end:
%p A354800 a:= n-> add(nops(b(n-i, i)), i=signum(n)..n):
%p A354800 seq(a(n), n=0..60);
%t A354800 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]}]];
%t A354800 a[n_] :=  Sum[Length[b[n - i, i]], {i, Sign[n], n}];
%t A354800 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Jul 06 2022, after _Alois P. Heinz_ *)
%Y A354800 Cf. A069999 (lower bound), A354468 (the same for supremum).
%K A354800 nonn
%O A354800 0,3
%A A354800 _Alois P. Heinz_, Jun 06 2022