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%I #24 Dec 20 2023 05:22:06
%S 0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,
%T 7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,10,0,0,0,0,0,0,0,0,0,10,11,
%U 11,1,1,1,1,1,1,1,1,11,11,12,12,12,2,2,2,2,2,2,2,12,12,12
%N Table of carryless sums i + j, i>=0, j>=0, read by antidiagonals.
%H Stefano Spezia, <a href="/A169894/b169894.txt">First 151 antidiagonals of the table, flattened</a>
%H <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>
%e Table begins:
%e 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ...
%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16 ...
%e 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17 ...
%e 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18 ...
%e 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19 ...
%e ...
%p A169894 := proc(a,b)
%p local adigs,bdigs,cdigs ;
%p adigs := convert(a,base,10) ;
%p bdigs := convert(b,base,10) ;
%p len := max(nops(adigs),nops(bdigs)) ;
%p adigs := [op(adigs),seq(0,d=1..len-nops(adigs))] ;
%p bdigs := [op(bdigs),seq(0,d=1..len-nops(bdigs))] ;
%p cdigs := [] ;
%p for d from 1 to len do
%p cdigs := [op(cdigs),A010879(op(d,adigs)+op(d,bdigs))] ;
%p end do:
%p add(op(d,cdigs)*10^(d-1),d=1..len) ;
%p end proc: # _R. J. Mathar_, Jul 12 2013
%t 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]]]; Table[T[i - j, j], {i, 0, 12}, {j, 0, i}] (* _Stefano Spezia_, Dec 20 2023 *)
%Y Cf. A004520 (diagonal), A059692 (carryless products).
%K nonn,base,look,tabl
%O 0,4
%A _David Applegate_, _Marc LeBrun_ and _N. J. A. Sloane_, Jul 06 2010
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