A168428 -id:A168428 - cp's OEIS frontend

A008565 Digits of powers of 4.

1, 4, 1, 6, 6, 4, 2, 5, 6, 1, 0, 2, 4, 4, 0, 9, 6, 1, 6, 3, 8, 4, 6, 5, 5, 3, 6, 2, 6, 2, 1, 4, 4, 1, 0, 4, 8, 5, 7, 6, 4, 1, 9, 4, 3, 0, 4, 1, 6, 7, 7, 7, 2, 1, 6, 6, 7, 1, 0, 8, 8, 6, 4, 2, 6, 8, 4, 3, 5, 4, 5, 6, 1, 0, 7, 3, 7, 4, 1, 8, 2, 4, 4, 2, 9, 4, 9, 6, 7, 2, 9, 6, 1, 7, 1, 7, 9, 8, 6, 9, 1, 8, 4

Offset: 0

N. J. A. Sloane

Irregular table with row length sequence A210434. - Jason Kimberley, Nov 26 2012

The constant whose decimal expansion is this sequence is irrational (Mahler, 1981). - Amiram Eldar, Mar 23 2025

			Triangle begins:
  1;
  4;
  1, 6;
  6, 4;
  2, 5, 6;
  1, 0, 2, 4;
  4, 0, 9, 6;
  1, 6, 3, 8, 4;
  6, 5, 5, 3, 6;
  2, 6, 2, 1, 4, 4;
  1, 0, 4, 8, 5, 7, 6;
  ...

Robert Israel, Table of n, a(n) for n = 0..10008 (rows 0 to 181, flattened)
Kurt Mahler, On some irrational decimal fractions, Journal of Number Theory, Vol. 13, No. 2 (1981), pp. 268-269.

Cf. A000302 (powers of 4), A210434.
Last elements of rows give A168428.
Cf. A000455, A008564, A008566, A008567, A008568, A008569, A008570, A008571, A008572, A008573, A010054.

R:= 1: t:= 1: count:= 1:
while count < 100 do
  t:= 4*t; L:= convert(t,base,10);
  count:= count+nops(L);
  R:= R, op(ListTools:-Reverse(L));
od:
R; # Robert Israel, May 05 2020

Table[IntegerDigits[4^i],{i,0,17}]//Flatten (* Stefano Spezia, Aug 06 2024 *)

A226294 Period 2: repeat [6, 4].

Original entry on oeis.org

6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6

Offset: 0

Richard R. Forberg, Jun 02 2013

A two number repeating sequence for constructing a summation sequence from negative to positive infinity containing all primes except 2 and 5.
Essentially the same as A168428, A101432 and A010711.
NOTE: This sequence has a shift in the starting value at index 0 relative to A010711. It is used here for the purpose stated with positive and negative indices making the formula in A010711 non-applicable.
This infinitely repeating sequence, a(n), of two numbers (6,4) starting with a(0) = 6, allows for the creation of an infinite summation sequence, s(n), extending from negative to positive infinity, using the formula below in parallel with how the same is done in A226276 using a different repeating sequence. Letting "s(n+)" be the set positive s(n) values, and "s(n-)" be the absolute value of the set of negative s(n) values, the following applies:
s(n+) includes all numbers with last digits of 1 and 7.
s(n-) includes all numbers with last digits of 3 and 9.
Therefore, s(n) includes all primes (except 2 and 5) without duplication.
This is one of only two such repeating patterns that accomplish this goal relative to the primes, while excluding all numbers with a last digit of 5. The other is (8,4,4,4) but with a different split between which primes occur as positive vs. negative numbers. See A226276 for details. Both patterns have the same density of primes relative to all s(n), and both, presumably, have the same average density of primes as positive vs. negative values of s(n).

			s(1) = 7, s(2) = 11, s(3) = 17, s(4) = 21, s(5) = 27, s(6) = 31;
s(-1) = -3, s(-2) = -9, s(-3) = -13, s(-4) = -19, s(-5) = -23, s(-6) = -29;

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010711, A101432, A168428, A226276.

&cat [[6, 4]^^50]; // Wesley Ivan Hurt, Jul 18 2016

A226294:=n->5+(-1)^n: seq(A226294(n), n=0..100); # Wesley Ivan Hurt, Mar 24 2014

Table[5+(-1)^n, {n,0,100}] (* Wesley Ivan Hurt, Mar 24 2014 *)

a(n)=6-n%2*2 \\ Charles R Greathouse IV, Jul 17 2016

a(n) = 5+(-1)^n = 2*A176059(n).
To generate the summation sequence s(n), start with s(0) = 1, and a(0) = 6.
For positive values of s(n):  s(n+1) = s(n) + a(n)
For negative values of s(n):  s(n-1) = s(n) - a(n-1). n is negative here.
See example values for s(n) below, for both positive and negative indices.
G.f.: ( 6+4*x ) / ( (1-x)*(1+x) ). - R. J. Mathar, Jun 12 2013
a(n) = a(n-2) for n>1. - Wesley Ivan Hurt, Jul 18 2016

A361390 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is carryless n^k base 10.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 6, 7, 6, 5, 1, 0, 1, 2, 1, 4, 5, 6, 1, 0, 1, 4, 3, 6, 5, 6, 7, 1, 0, 1, 8, 9, 4, 5, 6, 9, 8, 1, 0, 1, 6, 7, 6, 5, 6, 3, 4, 9, 1, 0, 1, 2, 1, 4, 5, 6, 1, 2, 1, 10, 1, 0, 1, 4, 3, 6, 5, 6, 7, 6, 9, 100, 11, 1, 0, 1, 8, 9, 4, 5, 6, 9, 8, 1, 1000, 121, 12, 1

Offset: 0

Seiichi Manyama, Mar 10 2023

			4 * 4 = 16, so T(4,2) = 6. 6 * 4 = 24, so T(4,3) = 4.
Square array begins:
  1, 0, 0, 0, 0, 0, 0, 0, ...
  1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 4, 8, 6, 2, 4, 8, ...
  1, 3, 9, 7, 1, 3, 9, 7, ...
  1, 4, 6, 4, 6, 4, 6, 4, ...
  1, 5, 5, 5, 5, 5, 5, 5, ...
  1, 6, 6, 6, 6, 6, 6, 6, ...
  1, 7, 9, 3, 1, 7, 9, 3, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

Columns k=0..4 give A000012, A001477, A059729, A169885, A169886.
Rows n=0..4 give A000007, A000012, A000689, A001148, A168428.
T(11,k) gives A059734.
Main diagonal gives A361351.
Cf. A004248, A359697.

T(n, k) = fromdigits(Vec(Pol(digits(n))^k)%10);

cp's OEIS Frontend

A008565 Digits of powers of 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A226294 Period 2: repeat [6, 4].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A361390 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is carryless n^k base 10.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI