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A169917 Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be multiplication mod 10.

%I A169917 #9 Mar 26 2015 01:02:00
%S A169917 0,1,4,9,6,5,6,9,4,1,100,111,144,199,166,155,166,199,144,111,400,441,
%T A169917 464,469,446,405,446,469,464,441,900,991,964,919,946,955,946,919,964,
%U A169917 991,600,661,644,649,666,605,666,649,644,661,500,551,504,559,506,555,506,559,504
%N A169917 Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be multiplication mod 10.
%C A169917 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 A169917 <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>
%F A169917 a(n) = a(n') if the i-th digit of n' either equals the i-th digit of n or (10 - the i-th digit of n): e.g., a(12345) = a(18365), because the 2nd and 4th digit of 12345 equal 10-(the 2nd resp. 4th digit of 18365), and the other digits are the same. In particular, a(10k+5+m) = a(10k+5-m), for m=0,...,4. - _M. F. Hasler_, Mar 26 2015
%e A169917 a(24) = 24*24 = 446:
%e A169917 ...24
%e A169917 ...24
%e A169917 -----
%e A169917 ...86
%e A169917 ..48.
%e A169917 -----
%e A169917 ..446
%e A169917 (The rule for "adding" the columns is to multiply mod 10: 8+8 = 8 * 8 mod 10 = 4.)
%o A169917 (PARI) A169917(n)={#n=digits(n);n=apply(d->n*d,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 A169917 The four versions are A059729, A169916, A169917, A169918.
%K A169917 nonn,base
%O A169917 0,3
%A A169917 _David Applegate_, _Marc LeBrun_ and _N. J. A. Sloane_, Jul 20 2010