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A051693 Square array read by antidiagonals: a(n,k) = sum of two positive regular n-polytopic numbers (i.e., a(n,k) = binomial(x,n) + binomial(y,n) for some x,y >= n).

%I A051693 #14 Jan 27 2018 06:23:00
%S A051693 2,2,3,2,4,4,2,5,6,5,2,6,8,7,6,2,7,10,11,9,7,2,8,12,16,14,11,8,2,9,14,
%T A051693 22,20,20,12,9,2,10,16,29,27,30,21,13,10,2,11,18,37,35,42,36,24,16,11,
%U A051693 2,12,20,46,44,56,57,40,30,18,12,2,13,22,56,54,72,85,62,50,36,20,13,2
%N A051693 Square array read by antidiagonals: a(n,k) = sum of two positive regular n-polytopic numbers (i.e., a(n,k) = binomial(x,n) + binomial(y,n) for some x,y >= n).
%e A051693 a(2,.) = 2,4,6,7,9,11,12,13,16,... = sum of two positive triangular numbers = A051533;
%e A051693 a(3,.) = 2,5,8,11,14,20,21,24,30,36,... = sum of two positive tetrahedral numbers;
%e A051693 First antidiagonals of the array are:
%e A051693 2;
%e A051693 2,3;
%e A051693 2,4,4;
%e A051693 2,5,6,5;
%e A051693 ...
%t A051693 nMax = 13; coeff = Floor[nMax/2]+1; row[n_] := Table[Binomial[x, n] + Binomial[y, n], {x, n, coeff*n}, {y, n, coeff*n}] // Flatten // Union; A0 = {}; While[A051693 = Table[row[n][[1 ;; nMax]], {n, 1, nMax}]; A051693 =!= A0, A0 = A051693; coeff++]; Table[A051693[[n-k+1, k]], {n, 1, nMax}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 14 2016 *)
%Y A051693 Cf. A051533.
%K A051693 nice,nonn,tabl
%O A051693 1,1
%A A051693 Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)