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a(n) is the number which, when concatenated with A003226(n), the n-th automorphic number, gives (A003226(n))^2.
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%I #53 May 20 2019 14:23:27

%S 0,0,2,3,6,57,141,390,8790,82128,11963,793212,835571,5054322,1661682,

%T 75880433,45322418,619541169,319375992,6745157241,3317093849,

%U 66891312600,843114912509,9837094694375,16065496578813,35901922360062,67557477392256,547721051611007

%N a(n) is the number which, when concatenated with A003226(n), the n-th automorphic number, gives (A003226(n))^2.

%C Let na and nb represent the indices of the preceding and next A003226(n)'s beginning with a 9, and where (na - nb) >= 3 (note that the first such 'zone' begins with an exception for which the index A003226(na) = 1). Then for na < n < nb and such that n == (na + 1) mod 2, it appears that A003226(n) - a(n) = A003226(n+1) - a(n+1) = k.

%C In such cases, it also appears that a(n)*a(n+1) = k^2 - k.

%F a(n) = (A003226(n)*(A003226(n) - 1) / 10^c(A003226(n)) where c(A003226(n)) is the number of digits in the n-th entry of A003226.

%e For n=4, A003226(4)=6, (A003226(4))^2=36. So a(4)=3.

%e For n=13, A003226(13)=2890625, (A003226(13))^2=8355712890625. So a(13)=835571.

%o (PARI) auto(n) = {n<3 & return(n-1); my(i=10, j=10, b=5, c=6, a=b); for( k=4, n, while(b<=a, b=b^2%i*=10); while(c<=a, c=(2-c)*c%j*=10); a=min(b, c)); a; } \\ A003226

%o a(n) = {my(m = auto(n), dm = digits(m), dm2 = digits(m^2)); fromdigits(vector(#dm2 - #dm, k, dm2[k]));} \\ _Michel Marcus_, May 18 2019

%Y Cf. A003226.

%K nonn,base

%O 1,3

%A _Christopher Hohl_, Mar 24 2019