A354777
Irregular triangle read by rows: T(n,k) is the number of integer quadruples (u,v,w,x) such that u^2+v^2+w^2+x^2 = n and u+v+w+x = k (n>=0, 0 <= k <= A307531(n)).
Original entry on oeis.org
1, 0, 4, 12, 0, 6, 0, 12, 0, 4, 6, 0, 8, 0, 1, 0, 12, 0, 12, 24, 0, 24, 0, 12, 0, 16, 0, 12, 0, 4, 12, 0, 0, 0, 6, 0, 24, 0, 16, 0, 12, 24, 0, 30, 0, 24, 0, 6, 0, 12, 0, 24, 0, 12, 8, 0, 24, 0, 12, 0, 8, 0, 24, 0, 12, 0, 16, 0, 4, 48, 0, 24, 0, 24, 0, 24, 0, 36, 0, 24, 0, 24, 0, 12, 6, 0, 0, 0, 8, 0, 0, 0, 1, 0, 12, 0, 36, 0, 12, 0, 12
Offset: 0
The triangle begins:
[1],
[0, 4],
[12, 0, 6],
[0, 12, 0, 4],
[6, 0, 8, 0, 1],
[0, 12, 0, 12],
[24, 0, 24, 0, 12],
[0, 16, 0, 12, 0, 4],
[12, 0, 0, 0, 6],
[0, 24, 0, 16, 0, 12],
[24, 0, 30, 0, 24, 0, 6],
[0, 12, 0, 24, 0, 12],
[8, 0, 24, 0, 12, 0, 8],
[0, 24, 0, 12, 0, 16, 0, 4],
[48, 0, 24, 0, 24, 0, 24],
[0, 36, 0, 24, 0, 24, 0, 12],
[6, 0, 0, 0, 8, 0, 0, 0, 1],
[0, 12, 0, 36, 0, 12, 0, 12],
[36, 0, 48, 0, 48, 0, 30, 0, 12],
...
T(4,2) = 8 from the solutions (u,v,w,x) = (2,0,0,0) (4 such) and (1,1,1,-1) (4 such).
A307510
a(n) is the greatest product i*j*k*l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.
Original entry on oeis.org
0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 4, 0, 3, 8, 0, 6, 16, 0, 12, 4, 9, 24, 8, 18, 0, 16, 36, 12, 32, 0, 24, 54, 0, 48, 20, 36, 81, 40, 72, 30, 64, 0, 60, 108, 45, 96, 40, 90, 48, 80, 144, 60, 135, 72, 120, 54, 0, 192, 108, 180, 96, 160, 72, 162, 256, 144, 240, 100
Offset: 0
For n = 34:
- 34 can be expressed in 4 ways as a sum of four squares:
i^2 + j^2 + k^2 + l^2 i*j*k*l
--------------------- -------
0^2 + 0^2 + 3^2 + 5^2 0
0^2 + 3^2 + 3^2 + 4^2 0
1^2 + 1^2 + 4^2 + 4^2 16
1^2 + 2^2 + 2^2 + 5^2 20
- a(34) = max(0, 16, 20) = 20.
See
A307531 for the additive variant.
-
See Links section.
-
g:= proc(n, k) option remember; local a;
if k = 1 then if issqr(n) then return sqrt(n) else return -infinity fi fi;
max(0,seq(a*procname(n-a^2, k-1), a=1..floor(sqrt(n))))
end proc:
seq(g(n, 4), n=0..100); # Robert Israel, Apr 15 2019
-
Array[Max[Times @@ # & /@ PowersRepresentations[#, 4, 2]] &, 68, 0] (* Michael De Vlieger, Apr 13 2019 *)
Showing 1-2 of 2 results.
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