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A245556 Irregular triangle read by rows: T(n,k) (n>=0, 0 <= k <= 2n) = number of triples (u,v,w) with entries in the range 0 to n which have some pair adding up to k.

%I A245556 #11 Aug 05 2014 13:34:01
%S A245556 1,4,6,4,7,12,19,12,7,10,18,28,36,28,18,10,13,24,37,48,61,48,37,24,13,
%T A245556 16,30,46,60,76,90,76,60,46,30,16,19,36,55,72,91,108,127,108,91,72,55,
%U A245556 36,19,22,42,64,84,106,126,148,168,148,126,106,84,64,42,22
%N A245556 Irregular triangle read by rows: T(n,k) (n>=0, 0 <= k <= 2n) = number of triples (u,v,w) with entries in the range 0 to n which have some pair adding up to k.
%e A245556 Triangle begins:
%e A245556 [1]
%e A245556 [4, 6, 4]
%e A245556 [7, 12, 19, 12, 7]
%e A245556 [10, 18, 28, 36, 28, 18, 10]
%e A245556 [13, 24, 37, 48, 61, 48, 37, 24, 13]
%e A245556 [16, 30, 46, 60, 76, 90, 76, 60, 46, 30, 16]
%e A245556 [19, 36, 55, 72, 91, 108, 127, 108, 91, 72, 55, 36, 19]
%e A245556 [22, 42, 64, 84, 106, 126, 148, 168, 148, 126, 106, 84, 64, 42, 22]
%e A245556 ...
%e A245556 See A245557 for specific examples; also the Example section of A090381 for some of the T(10,10)= 331 triples with n=k=10.
%p A245556 with(LinearAlgebra);
%p A245556 M:=10; A:=Array(0..M,0..2*M); B:=Array(0..M,0..2*M);
%p A245556 for n from 0 to M do
%p A245556 for i from 0 to n do for j from 0 to n do for k from 0 to n do
%p A245556   s1:={i+j,i+k,j+k}; s1:=convert(s1,list); m1:=max(i,j,k);
%p A245556     for r1 from 1 to nops(s1) do
%p A245556        s:=s1[r1]; A[n,s] := A[n,s]+1;
%p A245556        if (m1=n) then B[n,s] := B[n,s]+1; fi;
%p A245556                               od:
%p A245556 od: od: od: od:
%p A245556 lprint("A245556");
%p A245556 for i from 0 to M do lprint([seq(A[i,j],j=0..2*i)]); od:
%p A245556 lprint("A245557");
%p A245556 for i from 0 to M do lprint([seq(B[i,j],j=0..2*i)]); od:
%Y A245556 Rows are the partial sums of the rows of A245557.
%Y A245556 Main "spine" of triangle is A090381.
%Y A245556 Row sums are A005915.
%K A245556 nonn,tabf
%O A245556 0,2
%A A245556 _N. J. A. Sloane_, Aug 04 2014