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Write n in binary then square as if written in base 10.
5

%I #19 Dec 08 2022 20:46:36

%S 0,1,100,121,10000,10201,12100,12321,1000000,1002001,1020100,1022121,

%T 1210000,1212201,1232100,1234321,100000000,100020001,100200100,

%U 100220121,102010000,102030201,102212100,102232321,121000000,121022001

%N Write n in binary then square as if written in base 10.

%C The original description was: "Pre-carry binary squares: write n in binary then square as if written in a base large enough to avoid carries". But I changed it, since I prefer to work in base 10. There is no difference until a(1023). - _N. J. A. Sloane_, May 21 2002

%H Harry J. Smith, <a href="/A063009/b063009.txt">Table of n, a(n) for n=0,...,1000</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>.

%F a(2n) = 100*a(n); a(2n+1) = 100*a(n) + 20*A007088(n) + 1.

%e a(11)=1022121 since 11 written in binary is 1011 and 1011^2 = 1011000 + 0 + 10110 + 1011 = 1022121.

%e a(1023) = 1111111111^2 = 1234567900987654321.

%o (PARI) baseE(x, b)= { local(d, e, f); e=0; f=1; while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); return(e) } { for (n=0, 1000, write("b063009.txt", n, " ", baseE(n, 2)^2) ) } \\ _Harry J. Smith_, Aug 15 2009

%o (PARI) a(n) = fromdigits(binary(n))^2; \\ _Ruud H.G. van Tol_, Dec 08 2022

%Y Cf. A007088 for binary numbers, A001737 for binary squares (post-carry), A063010 for carryless binary squares.

%K base,easy,nonn

%O 0,3

%A _Henry Bottomley_, Jul 04 2001