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A169918 Squares in carryless arithmetic mod 10 with addition of digits defined to be multiplication mod 10 and multiplication of digits defined to be addition mod 10.

%I A169918 #15 Mar 26 2015 14:14:51
%S A169918 0,2,4,6,8,0,2,4,6,8,210,242,294,266,258,260,292,244,216,208,440,492,
%T A169918 464,456,468,490,442,414,406,418,690,662,654,666,698,640,612,604,616,
%U A169918 648,860,852,864,896,848,810,802,814,846,898,50,62,94,46,18,0,12,44,96,68,260,292
%N A169918 Squares in carryless arithmetic mod 10 with addition of digits defined to be multiplication mod 10 and multiplication of digits defined to be addition mod 10.
%C A169918 The rules of arithmetic used in A169916, A169917, A169918 have very strange consequences. Many of the familiar laws fail. For instance, the arithmetic in A169916 is not associative: 10*(9*2) = 10*1 = 21 != (10*9)*2 = 9*2 = 1.
%H A169918 <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>
%e A169918 a(17) = 17*17 = 244:
%e A169918 ...17
%e A169918 ...17
%e A169918 -----
%e A169918 ...84 (7*7 = 7+7 mod 10 = 4, 7*1 = 7+1 mod 10 = 8)
%e A169918 ..28.
%e A169918 -----
%e A169918 ..244
%e A169918 (The rule for "adding" the columns is to multiply mod 10: 8+8 = 8 * 8 mod 10 = 4. Blanks are ignored)
%o A169918 (PARI) A169918(n)={u=vector(#n=digits(n),i,1);n=apply(d->n+d*u,n)%10;sum(i=0,2*#n-2,prod(j=max(1,#n-i),min(2*#n-1-i,#n),n[2*#n-i-j][j])%10*10^i)} \\ _M. F. Hasler_, Mar 26 2015
%Y A169918 See A048379, A169931-A169933, A169935 for other examples of calculations in this version of arithmetic.
%Y A169918 The four versions are A059729, A169916, A169917, A169918.
%K A169918 nonn,base
%O A169918 0,2
%A A169918 _David Applegate_, _Marc LeBrun_ and _N. J. A. Sloane_, Jul 20 2010
%E A169918 Thanks to _Rick L. Shepherd_ for pointing out a typo in the example. - _N. J. A. Sloane_, Nov 08 2014