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

2, 4, 16, 136, 1936, 181936, 164181936, 13616164181936, 193613613616164181936, 1819361936193613613616164181936, 1641819361819361819361936193613613616164181936, 136161641819361641819361641819361819361819361936193613613616164181936

Offset: 1

Author

Amarnath Murthy, May 13 2001

Keywords

• nonn
• easy
• base

Examples

			After 136: the squares of 1, 3, 6 are 1, 9, 36 respectively hence the next term is 1936.
a(11) = a(7)*10^L(a(6)+a(10))+a(6)*10^L(a(10))+a(10)
     = 13616164181936*10^55 + 164181936*10^46 +
      1641819361819361819361936193613613616164181936
     = 136161641819361641819361641819361819361819361936193613613616164181936
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
		

Links

• John Cerkan, Table of n, a(n) for n = 1..18
• William Davidson, Introducing the peculiar 'Davidson Sequence', MathFest 2012; see p. 37.

Programs

Mathematica

NestList[FromDigits[Flatten[IntegerDigits[IntegerDigits[#]^2]]] &, 2, 11] (* Paolo Xausa, Jan 10 2025 *)

Python

def digits(n):
    d = []
    while n > 0:
        d.append(n % 10)
        n = n // 10
    return d
def sqdig(n):
    new = 0
    num = digits(n)
    spacing = 0
    while num:
        k = num.pop(0)
        new += (10 ** (spacing)) * (k**2)
        if k > 3:
            spacing += 1
        spacing += 1
    return new
def a(n):
    i = 2
    while n > 1:
        i = sqdig(i)
        n -= 1
    return i
# David Nacin, Aug 19 2012

Python

from itertools import accumulate
def f(an, _): return int("".join(str(int(d)**2) for d in str(an)))
print(list(accumulate([2]*11, f))) # Michael S. Branicky, Jan 01 2022

Formula

From William Davidson, Aug 15 2012: (Start)

For integer n > 5,

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.

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.

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)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org) and Asher Auel, May 15 2001

Corrected by Matthew Vandermast, Apr 23 2003