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A061588 a(1) = 2; thereafter a(n) is the number obtained by replacing each digit of a(n-1) with its square.

%I A061588 #46 May 27 2025 14:53:49
%S A061588 2,4,16,136,1936,181936,164181936,13616164181936,
%T A061588 193613613616164181936,1819361936193613613616164181936,
%U A061588 1641819361819361819361936193613613616164181936,136161641819361641819361641819361819361819361936193613613616164181936
%N A061588 a(1) = 2; thereafter a(n) is the number obtained by replacing each digit of a(n-1) with its square.
%H A061588 John Cerkan, <a href="/A061588/b061588.txt">Table of n, a(n) for n = 1..18</a>
%H A061588 William Davidson, <a href="https://web.archive.org/web/20231214050824/https://www.maa.org/sites/default/files/pdf/abstracts/mf2012_abstracts.pdf">Introducing the peculiar 'Davidson Sequence'</a>, MathFest 2012; see p. 37.
%F A061588 From _William Davidson_, Aug 15 2012: (Start)
%F A061588 For integer n > 5,
%F A061588 a(n) = a(n-4)*10^(L(a(n-5))+L(a(n-1))) + a(n-5)*10^(L(a(n-1))) + a(n-1), where L(x) is the number of digits in x.
%F A061588 L(a(n)) = (W^(n-1)*[s1]^T)^T*[d]^T, where W is the 5 X 5 square matrix [(0 1 0 0 0) (0 0 1 0 0) (0 0 0 1 0) (0 0 0 0 1) (1 1 0 0 1)], [s1] = [1 2 3 4 6], [d] = [1 0 0 0 0], and T denotes transpose.
%F A061588 To determine the initial digits of a(n), n > 5, let b = ((n+2) mod 4) + 2. Then a(n) begins with a(b). E.g. let n = 100, b = 4, then a(100) = 1936... (End)
%e A061588 After 136: the squares of 1, 3, 6 are 1, 9, 36 respectively hence the next term is 1936.
%e A061588 a(11) = a(7)*10^L(a(6)+a(10))+a(6)*10^L(a(10))+a(10)
%e A061588      = 13616164181936*10^55 + 164181936*10^46 +
%e A061588       1641819361819361819361936193613613616164181936
%e A061588      = 136161641819361641819361641819361819361819361936193613613616164181936
%e A061588 a(100) = 1936...*10^L(a(96)+a(99))+136...*10^L(a(99))+136...936, where L(100) has approximately 2.74*10^17 digits. - _William Davidson_, Aug 15 2012
%t A061588 NestList[FromDigits[Flatten[IntegerDigits[IntegerDigits[#]^2]]] &, 2, 11] (* _Paolo Xausa_, Jan 10 2025 *)
%o A061588 (Python)
%o A061588 def digits(n):
%o A061588     d = []
%o A061588     while n > 0:
%o A061588         d.append(n % 10)
%o A061588         n = n // 10
%o A061588     return d
%o A061588 def sqdig(n):
%o A061588     new = 0
%o A061588     num = digits(n)
%o A061588     spacing = 0
%o A061588     while num:
%o A061588         k = num.pop(0)
%o A061588         new += (10 ** (spacing)) * (k**2)
%o A061588         if k > 3:
%o A061588             spacing += 1
%o A061588         spacing += 1
%o A061588     return new
%o A061588 def a(n):
%o A061588     i = 2
%o A061588     while n > 1:
%o A061588         i = sqdig(i)
%o A061588         n -= 1
%o A061588     return i
%o A061588 # _David Nacin_, Aug 19 2012
%o A061588 (Python)
%o A061588 from itertools import accumulate
%o A061588 def f(an, _): return int("".join(str(int(d)**2) for d in str(an)))
%o A061588 print(list(accumulate([2]*11, f))) # _Michael S. Branicky_, Jan 01 2022
%K A061588 nonn,easy,base
%O A061588 1,1
%A A061588 _Amarnath Murthy_, May 13 2001
%E A061588 More terms from Larry Reeves (larryr(AT)acm.org) and _Asher Auel_, May 15 2001
%E A061588 Corrected by _Matthew Vandermast_, Apr 23 2003