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

Offset

1,1

Links

John Cerkan, Table of n, a(n) for n = 1..18

William Davidson, Introducing the peculiar 'Davidson Sequence', MathFest 2012; see p. 37.

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)

Example

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

Prog

(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 davidson(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

Crossrefs

Sequence in context: A174677 A073924 A362065 * A202360 A050472 A109457

Adjacent sequences: A061585 A061586 A061587 * A061589 A061590 A061591

Keyword

nonn,easy,base

Author

Amarnath Murthy, May 13 2001

Extensions

More terms from Larry Reeves (larryr(AT)acm.org) and Asher Auel, May 15 2001. Corrected by Matthew Vandermast, Apr 23 2003

Status

approved