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Symmetric array read by antidiagonals: A(n,k) is the number of carryless sums i + j with abs(i) <= n and abs(j) <= k.
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%I #24 Dec 22 2023 03:51:30

%S 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 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 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 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 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 Stefano Spezia, <a href="/A368310/b368310.txt">First 150 antidiagonals of the array, flattened</a>

%H 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 <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>

%F A(n,k) = A003991(n+1,k+1) for n + k < 10.

%F A(n,0) = A(0,n) = n + 1.

%F A(n,k) = A003991(n+1,k+1) - A368311(n,k).

%e Array begins:

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...

%e 2, 4, 6, 8, 10, 12, 14, 16, 18, 19, 21, ...

%e 3, 6, 9, 12, 15, 18, 21, 24, 26, 27, 30, ...

%e 4, 8, 12, 16, 20, 24, 28, 31, 33, 34, 38, ...

%e 5, 10, 15, 20, 25, 30, 34, 37, 39, 40, 45, ...

%e 6, 12, 18, 24, 30, 35, 39, 42, 44, 45, 51, ...

%e 7, 14, 21, 28, 34, 39, 43, 46, 48, 49, 56, ...

%e 8, 16, 24, 31, 37, 42, 46, 49, 51, 52, 60, ...

%e 9, 18, 26, 33, 39, 44, 48, 51, 53, 54, 63, ...

%e 10, 19, 27, 34, 40, 45, 49, 52, 54, 55, 65, ...

%e 11, 21, 30, 38, 45, 51, 56, 60, 63, 65, 76, ...

%e ...

%e 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 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 Cf. A003056, A003991, A059692, A169894, A368311 (sums with carries).

%K nonn,base,look,tabl

%O 0,2

%A _Stefano Spezia_, Dec 21 2023