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A336251 a(1) = 15; for n > 1, a(n)^2 is the smallest square that begins with a(n-1) in base 6.

%I A336251 #27 Jul 22 2020 20:09:21
%S A336251 15,57,111,155,183,199,508,812,171,471,319,643,913,1088,2909,11650,
%T A336251 9518,21074,31357,93691,396693,55540,50905,119374,182804,226216,
%U A336251 251647,265415,111280,72055,142024,13567,25160,34262,39982,105795,172093,537634
%N A336251 a(1) = 15; for n > 1, a(n)^2 is the smallest square that begins with a(n-1) in base 6.
%C A336251 The sequence becomes periodic after 491 terms with a period of 10.
%C A336251 The only square in this sequence is 25 and the sequence becomes periodic two terms later.
%C A336251 The maximum value is a(226) = 325880259349618.
%H A336251 Sean Lipton, <a href="/A336251/b336251.txt">Table of n, a(n) for n = 1..600</a>
%F A336251 a(n+1) = ceiling(sqrt(a(n)*6^i)) such that floor(sqrt((a(n)+1)*6^i-1)) > floor(sqrt(a(n)*6^i-1)) where i is a whole number and minimized.
%F A336251 For n > 481, a(n) = a(n+10).
%e A336251 The first 10 a(n) alongside the base 6 representations of a(n) and squares of a(n+1):
%e A336251   n   a(n)  a(n) b6  a(n+1)^2 b6
%e A336251   --  ----  -------  -----------
%e A336251    1    15       23        23013
%e A336251    2    57      133       133013
%e A336251    3   111      303       303121
%e A336251    4   155      415       415013
%e A336251    5   183      503       503201
%e A336251    6   199      531      5310424
%e A336251    7   508     2204     22044304
%e A336251    8   812     3432       343213
%e A336251    9   171      443      4431013
%e A336251   10   471     2103      2103041
%o A336251 (PARI) lista(nn) = {my(a = 15); for (n=2, nn, print1(a, ", "); my(i=1); while((sqrtint((a+1)*6^i-1) <= sqrtint(a*6^i-1)), i++); a = ceil(sqrt(a*6^i)););} \\ _Michel Marcus_, Jul 15 2020
%Y A336251 Cf. A308055, A309123.
%K A336251 nonn,base
%O A336251 1,1
%A A336251 _Sean Lipton_, Jul 14 2020