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A061588 a(1) = 2, a(n) = number obtained by replacing each digit of a(n-1) with its square. 4
2, 4, 16, 136, 1936, 181936, 164181936, 13616164181936, 193613613616164181936, 1819361936193613613616164181936, 1641819361819361819361936193613613616164181936 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

REFERENCES

William Davidson, Introducing the peculiar 'Davidson Sequence', MathFest 2012.

LINKS

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

FORMULA

From William Davidson, Aug 15 2012: (Start)

For integer n > 5,

a(n) = a(n-4)x10^(L(a(n-5))+L(a(n-1))) + a(n-5)x10^(L(a(n-1)))+a(n-1), where L(a(n)) is the number of digits in a(n).

L(a(n)) = (W^(n-1)*[s1]^T)^T*[d]^T, with the 5 x 5 square matrix W = [(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 is the transformation of a matrix.

To determine the starting integers of any a(n), for any integer n > 5, first let b = (((n + 2)mod(4)) + 2). Then using a(2), a(3), a(4), and a(5), any n > 5, a(n) begins with a(b). Let n = 100, b = 4, 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

CROSSREFS

Sequence in context: A217994 A174677 A073924 * 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 Natan Auel (auela(AT)reed.edu), May 15 2001. Corrected by Matthew Vandermast, Apr 23 2003

STATUS

approved

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Last modified September 26 20:34 EDT 2021. Contains 347672 sequences. (Running on oeis4.)