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A368310 Symmetric array read by antidiagonals: A(n,k) is the number of carryless sums i + j with abs(i) <= n and abs(j) <= k.

%I A368310 #24 Dec 22 2023 03:51:30
%S A368310 1,2,2,3,4,3,4,6,6,4,5,8,9,8,5,6,10,12,12,10,6,7,12,15,16,15,12,7,8,
%T A368310 14,18,20,20,18,14,8,9,16,21,24,25,24,21,16,9,10,18,24,28,30,30,28,24,
%U A368310 18,10,11,19,26,31,34,35,34,31,26,19,11,12,21,27,33,37,39,39,37,33,27,21,12
%N A368310 Symmetric array read by antidiagonals: A(n,k) is the number of carryless sums i + j with abs(i) <= n and abs(j) <= k.
%C A368310 A(n,k) differs from A003991(n+1,k+1) starting at the second term of the 11th antidiagonal: A(9,1) = 19 <> A003991(10,2) = 20.
%H A368310 Stefano Spezia, <a href="/A368310/b368310.txt">First 150 antidiagonals of the array, flattened</a>
%H A368310 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 A368310 <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>
%F A368310 A(n,k) = A003991(n+1,k+1) for n + k < 10.
%F A368310 A(n,0) = A(0,n) = n + 1.
%F A368310 A(n,k) = A003991(n+1,k+1) - A368311(n,k).
%e A368310 Array begins:
%e A368310    1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, ...
%e A368310    2,  4,  6,  8, 10, 12, 14, 16, 18, 19, 21, ...
%e A368310    3,  6,  9, 12, 15, 18, 21, 24, 26, 27, 30, ...
%e A368310    4,  8, 12, 16, 20, 24, 28, 31, 33, 34, 38, ...
%e A368310    5, 10, 15, 20, 25, 30, 34, 37, 39, 40, 45, ...
%e A368310    6, 12, 18, 24, 30, 35, 39, 42, 44, 45, 51, ...
%e A368310    7, 14, 21, 28, 34, 39, 43, 46, 48, 49, 56, ...
%e A368310    8, 16, 24, 31, 37, 42, 46, 49, 51, 52, 60, ...
%e A368310    9, 18, 26, 33, 39, 44, 48, 51, 53, 54, 63, ...
%e A368310   10, 19, 27, 34, 40, 45, 49, 52, 54, 55, 65, ...
%e A368310   11, 21, 30, 38, 45, 51, 56, 60, 63, 65, 76, ...
%e A368310   ...
%e A368310 A(6,5) = A003991(7,6) - A368311(6,5) = (6 + 1)*(5 + 1) - 3 = 39 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 A368310 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]]; A[n_,k_] := Sum[Sum[Boole[Length[Select[B[i,j], #<10 &]] == IntegerLength[Max[i,j]]],{i,0,n}],{j,0,k}]; Table[A[i - j, j], {i, 0, 11}, {j, 0, i}]//Flatten
%Y A368310 Cf. A003056, A003991, A059692, A169894, A368311 (sums with carries).
%K A368310 nonn,base,look,tabl
%O A368310 0,2
%A A368310 _Stefano Spezia_, Dec 21 2023