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A354800
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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.
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3
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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
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OFFSET
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0,3
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LINKS
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EXAMPLE
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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)}|.
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MAPLE
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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);
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MATHEMATICA
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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}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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