A298640
Number of compositions (ordered partitions) of n^2 into squares > 1.
Original entry on oeis.org
1, 0, 1, 1, 2, 8, 12, 129, 874, 9630, 167001, 3043147, 72844510, 2423789655, 106665874384, 6156805673648, 470151743582651, 47558937432498729, 6363358599941131580, 1126147544855148769425, 263646401550138303553708, 81649922556593759124887197
Offset: 0
a(5) = 8 because we have [25], [16, 9], [9, 16], [9, 4, 4, 4, 4], [4, 9, 4, 4, 4], [4, 4, 9, 4, 4], [4, 4, 4, 9, 4] and [4, 4, 4, 4, 9].
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b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-j^2), j=2..isqrt(n)))
end:
a:= n-> b(n^2):
seq(a(n), n=0..25); # Alois P. Heinz, Feb 05 2018
-
b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j^2], {j, 2, Floor @ Sqrt[n]}]];
a[n_] := b[n^2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 21 2018, after Alois P. Heinz *)
A298935
Number of partitions of n^3 into distinct squares.
Original entry on oeis.org
1, 1, 0, 0, 1, 5, 8, 40, 96, 297, 1269, 3456, 12839, 46691, 153111, 577167, 2054576, 7602937, 29000337, 110645967, 418889453, 1580667760, 6058528796, 23121913246, 89793473393, 350029321425, 1359919742613, 5340642744919, 20948242218543, 82505892314268
Offset: 0
a(5) = 5 because we have [121, 4], [100, 25], [100, 16, 9], [64, 36, 25] and [64, 36, 16, 9].
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Table[SeriesCoefficient[Product[1 + x^k^2, {k, 1, Floor[n^(3/2) + 1]}], {x, 0, n^3}], {n, 0, 29}]
A294071
Number of ordered ways of writing n^2 as a sum of n squares > 1.
Original entry on oeis.org
1, 0, 0, 0, 1, 5, 6, 7, 288, 262, 13702, 69531, 610567, 5356091, 51724960, 521956086, 5467658641, 59931636545, 690518644584, 8100858045744, 99142980567486, 1246972499954475, 16142015005905558, 215722810653380845, 2955759897694815985, 41614888439136252691
Offset: 0
a(5) = 5 because we have [9, 4, 4, 4, 4], [4, 9, 4, 4, 4], [4, 4, 9, 4, 4], [4, 4, 4, 9, 4] and [4, 4, 4, 4, 9].
Cf.
A000290,
A037444,
A066535,
A078134,
A092362,
A232173,
A280129,
A280542,
A281154,
A281155,
A298329,
A298330,
A298640,
A298642.
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Table[SeriesCoefficient[((-1 - 2 x + EllipticTheta[3, 0, x])/2)^n, {x, 0, n^2}], {n, 0, 25}]
Showing 1-3 of 3 results.