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A059692 Table of carryless products i * j, i>=0, j>=0, read by antidiagonals.

%I A059692 #54 Sep 27 2022 09:05:51
%S A059692 0,0,0,0,1,0,0,2,2,0,0,3,4,3,0,0,4,6,6,4,0,0,5,8,9,8,5,0,0,6,0,2,2,0,
%T A059692 6,0,0,7,2,5,6,5,2,7,0,0,8,4,8,0,0,8,4,8,0,0,9,6,1,4,5,4,1,6,9,0,0,10,
%U A059692 8,4,8,0,0,8,4,8,10,0,0,11,20,7,2,5,6,5,2,7,20,11,0
%N A059692 Table of carryless products i * j, i>=0, j>=0, read by antidiagonals.
%H A059692 Stefano Spezia, <a href="/A059692/b059692.txt">First 140 antidiagonals of the table, flattened</a>
%H A059692 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 A059692 <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>
%e A059692 Table begins:
%e A059692   0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0 ...
%e A059692   0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ...
%e A059692   0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 20, 22, 24, 26, 28, 20 ...
%e A059692   0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 30, 33, 36, 39, 32, 35 ...
%e A059692   0, 4, 8, 2, 6, 0, 4, 8, 2, 6, 40, 44, 48, 42, 46, 40 ...
%e A059692   ...
%e A059692 T(12, 97) = 954 since we have 12 X 97 = carryless sum of 900, (180 mod 100=)80, 70 and (14 mod 10=)4 = 954.
%t A059692 len[num_]:=Length[IntegerDigits[num]]; digit[num_,d_]:=Part[IntegerDigits[num],d]; T[i_, j_] := FromDigits[Reverse[CoefficientList[PolynomialMod[Sum[digit[i,c]*x^(len[i]-c), {c, len[i]}]*Sum[digit[j,r]*x^(len[j]-r), {r, len[j]}], 10], x]]]; Flatten[Table[T[i - j, j], {i, 0, 12}, {j, 0, i}]] (* _Stefano Spezia_, Sep 26 2022 *)
%o A059692 (PARI) T(n,k) = fromdigits(lift(Vec( Mod(Pol(digits(n)),10) * Pol(digits(k))))); \\ _Kevin Ryde_, Sep 27 2022
%Y A059692 Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X 10 base 10, A055120 for carryless 9 X n, A008592 for carryless 10 X n.
%Y A059692 Cf. A048720 (binary), A325820 (ternary).
%K A059692 nonn,base,easy,tabl,look
%O A059692 0,8
%A A059692 _Henry Bottomley_, Feb 19 2001
%E A059692 Minor edits by _N. J. A. Sloane_, Aug 24 2010