cp's OEIS Frontend

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)).

%I A354777 #11 Dec 10 2023 09:29:00
%S A354777 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,
%T A354777 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,
%U A354777 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
%N 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)).
%C A354777 Row n has width A307531(n).
%e A354777 The triangle begins:
%e A354777 [1],
%e A354777 [0, 4],
%e A354777 [12, 0, 6],
%e A354777 [0, 12, 0, 4],
%e A354777 [6, 0, 8, 0, 1],
%e A354777 [0, 12, 0, 12],
%e A354777 [24, 0, 24, 0, 12],
%e A354777 [0, 16, 0, 12, 0, 4],
%e A354777 [12, 0, 0, 0, 6],
%e A354777 [0, 24, 0, 16, 0, 12],
%e A354777 [24, 0, 30, 0, 24, 0, 6],
%e A354777 [0, 12, 0, 24, 0, 12],
%e A354777 [8, 0, 24, 0, 12, 0, 8],
%e A354777 [0, 24, 0, 12, 0, 16, 0, 4],
%e A354777 [48, 0, 24, 0, 24, 0, 24],
%e A354777 [0, 36, 0, 24, 0, 24, 0, 12],
%e A354777 [6, 0, 0, 0, 8, 0, 0, 0, 1],
%e A354777 [0, 12, 0, 36, 0, 12, 0, 12],
%e A354777 [36, 0, 48, 0, 48, 0, 30, 0, 12],
%e A354777 ...
%e A354777 T(4,2) = 8 from the solutions (u,v,w,x) = (2,0,0,0) (4 such) and (1,1,1,-1) (4 such).
%Y A354777 Cf. A000118, A307531.
%Y A354777 T(n^2,n) = A354778(n). See also A278085 and A354766.
%K A354777 nonn,tabf
%O A354777 0,3
%A A354777 _N. J. A. Sloane_, Jun 27 2022