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A368311 Symmetric array read by antidiagonals: A(n,k) is the number of sums with carries i + j with abs(i) <= n and abs(j) <= k.

%I A368311 #18 Dec 22 2023 03:51:06
%S A368311 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A368311 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,0,0,1,
%U A368311 3,3,3,3,3,3,3,3,1,0,0,1,3,6,6,6,6,6,6,6,3,1,0,0,1,3,6,10,10,10,10,10,10,6,3,1,0
%N A368311 Symmetric array read by antidiagonals: A(n,k) is the number of sums with carries i + j with abs(i) <= n and abs(j) <= k.
%H A368311 Stefano Spezia, <a href="/A368311/b368311.txt">First 150 antidiagonals of the array, flattened</a>
%H A368311 David Applegate, Marc LeBrun, and N. J. A. Sloane, <a href="http://neilsloane.com/doc/carry1.pdf">Carryless Arithmetic (I): The Mod 10 Version</a>.
%H A368311 <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>
%F A368311 A(n,0) = A(0,n) = 0.
%F A368311 A(n,k) = A003991(n+1,k+1) - A368310(n,k).
%e A368311 Array begins:
%e A368311   0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0, ...
%e A368311   0, 0, 0, 0,  0,  0,  0,  0,  0,  1,  1,  1, ...
%e A368311   0, 0, 0, 0,  0,  0,  0,  0,  1,  3,  3,  3, ...
%e A368311   0, 0, 0, 0,  0,  0,  0,  1,  3,  6,  6,  6, ...
%e A368311   0, 0, 0, 0,  0,  0,  1,  3,  6, 10, 10, 10, ...
%e A368311   0, 0, 0, 0,  0,  1,  3,  6, 10, 15, 15, 15, ...
%e A368311   0, 0, 0, 0,  1,  3,  6, 10, 15, 21, 21, 21, ...
%e A368311   0, 0, 0, 1,  3,  6, 10, 15, 21, 28, 28, 28, ...
%e A368311   0, 0, 1, 3,  6, 10, 15, 21, 28, 36, 36, 36, ...
%e A368311   0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 45, 46, ...
%e A368311   0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 45, 46, ...
%e A368311   0, 1, 3, 6, 10, 15, 21, 28, 36, 46, 46, 47, ...
%e A368311   ...
%e A368311 A(6,5) = 3 since there are three sums with carries having addends almost equal to 6 and 5, respectively: 5 + 5 = 10, 6 + 4 = 10, and 6 + 5 = 11.
%t A368311 len[num_]:=Length[IntegerDigits[num]]; digit[num_, d_] := Part[IntegerDigits[num], d];  B[i_, j_] := Reverse[CoefficientList[Sum[digit[i, c]*x^(len[i]-c), {c, len[i]}]+Sum[digit[j, r]*x^(len[j]-r), {r, len[j]}], x]]; F[n_,k_] := Sum[Sum[Boole[Length[Select[B[i,j], #<10 &]] == IntegerLength[Max[i,j]]],{i,0,n}],{j,0,k}]; A[i_,j_]:=(i+1)(j+1)-F[i,j]; Table[A[i - j, j], {i, 0, 13}, {j, 0, i}]//Flatten
%Y A368311 Cf. A003056, A003991, A059692, A169894, A368310 (carryless sums).
%K A368311 nonn,base,look,tabl
%O A368311 0,69
%A A368311 _Stefano Spezia_, Dec 21 2023