A125122 -id:A125122 - cp's OEIS frontend

A125144 Increments in the number of decimal digits of 4^n.

1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jan 11 2007

This sequence is not periodic because log(4)/log(10) is an irrational number. - T. D. Noe, Jan 25 2007

			a(1)=1 because 4^(1+1)=16 (two digits) 4^1=4 (one digit) and the difference is 1.
a(2)=0 because 4^(2+1)=64 (two digits) 4^(2)=16 (two digits) and the difference is 0.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A125117, A125122.

First differences of A210434.

P:=proc(n) local i,j,k,w,old; k:=4; for i from 1 by 1 to n do j:=k^i; w:=0; while j>0 do w:=w+1; j:=trunc(j/10); od; if i>1 then print(w-old); old:=w; else old:=w; fi; od; end: P(1000);
# alternative:
H:= [seq(ilog10(4^i),i=1..1001)]:
H[2..-1]-H[1..-2]; # Robert Israel, Jul 12 2018

Differences[IntegerLength[4^Range[100]]] (* Paolo Xausa, Jun 08 2024 *)

a(n) = #digits(4^(n+1)) - #digits(4^n); \\ Michel Marcus, Jul 12 2018

a(n)=Number_of_digits{4^(n+1)}-Number_of digits{4^(n)} with n>=0 and where "Number_of digits" is a hypothetical function giving the number of digits of the argument.

Offset corrected by Robert Israel, Jul 11 2018

A271427 a(n) = 7^n - a(n-1) for n>0, a(0)=0.

Original entry on oeis.org

0, 7, 42, 301, 2100, 14707, 102942, 720601, 5044200, 35309407, 247165842, 1730160901, 12111126300, 84777884107, 593445188742, 4154116321201, 29078814248400, 203551699738807, 1424861898171642, 9974033287201501, 69818233010410500, 488727631072873507, 3421093417510114542

Offset: 0

Ilya Gutkovskiy, Apr 13 2016

In general, the ordinary generating function for the recurrence b(n) = k^n - b(n-1), where n>0 and b(0)=0, is k*x/((1 + x)*(1 - k*x)). This recurrence gives the closed form b(n) = k*(k^n - (-1)^n)/(k + 1).

			a(2) = 7^2 - a(2-1) = 49 - 7 = 42.
a(4) = 7^4 - a(4-1) = 2401 - 301 = 2100.

Index entries for linear recurrences with constant coefficients, signature (6,7).

Cf. A000420, A015552.

Cf. similar sequences with the recurrence b(n) = k^n - b(n-1): A125122 (k=1), A078008 (k=2), A054878 (k=3), A109499 (k=4), A109500 (k=5), A109501 (k=6), this sequence (k=7), A093134 (k=8), A001099 (k=n).

LinearRecurrence[{6, 7}, {0, 7}, 30]
Table[7 (7^n - (-1)^n)/8, {n, 0, 30}]

vector(50, n, n--; 7*(7^n-(-1)^n)/8) \\ Altug Alkan, Apr 13 2016

for n in range(0,10**2):print((int)((7*(7**n-(-1)**n))/8))
# Soumil Mandal, Apr 14 2016

O.g.f.: 7*x/(1 - 6*x - 7*x^2).
E.g.f.: (7/8)*(exp(7*x) - exp(-x)).
a(n) = 6*a(n-1) + 7*a(n-2).
a(n) = 7*(7^n - (-1)^n)/8.
a(n) = 7*A015552(n).
Sum_{n>0} 1/(a(n) + a(n-1)) = 1/6 = A020793.
Limit_{n->oo} a(n-1)/a(n) = 1/7 = A020806.

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A125144 Increments in the number of decimal digits of 4^n.

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