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