OFFSET
0,1
COMMENTS
From Stark: "alpha = 0.101101110111101111101111110 ... is irrational. For if alpha were rational, its decimal expansion would be periodic and have a period of length r starting with the k-th digit of the expansion.
"But by the very nature of alpha, there will be blocks of r digits, all 1, in this expansion after the k-th digit and the periodicity would then guarantee that everything after such a block of r digits would also be all ones.
"This contradicts the fact that there will always be zeros occurring after any given point in the expansion of alpha. Hence alpha is irrational."
Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. A023532 is reverse reluctant sequence of sequence A211666. - Boris Putievskiy, Jan 11 2013
An example of a sequence with infinite critical exponent [Vaslet]. - N. J. A. Sloane, May 05 2013
REFERENCES
Harold M. Stark, An Introduction to Number Theory, The MIT Press, Cambridge, Mass, eighth printing 1994, page 170.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Elise Vaslet, Critical exponents of words over 3 letters, Electronic Journal of Combinatorics, 18 (2011), #P125.
FORMULA
a(n) = 0 if and only if 8n+9 is a square. - Charles R Greathouse IV, Jun 16 2011
Blocks of lengths 1, 2, 3, 4, ... of ones separated by a single zero.
a(n) = 1 - floor((sqrt(9+8n)-1)/2) + floor((sqrt(1+8n)-1)/2). - Paul Barry, May 25 2004
a(n) = A211666(m), where m = (t^2 + 3*t + 4)/2n - n, t = floor((-1 + sqrt(8*n-7))/2). - Boris Putievskiy, Jan 11 2013
EXAMPLE
From Boris Putievskiy, Jan 11 2013: (Start)
As a triangular array written by rows, the sequence begins:
0;
1, 0;
1, 1, 0;
1, 1, 1, 0;
1, 1, 1, 1, 0;
1, 1, 1, 1, 1, 0;
1, 1, 1, 1, 1, 1, 0;
...
(End)
MATHEMATICA
a = {}; Do[a = Append[a, Join[ {0}, Table[1, {n} ] ] ], {n, 1, 13} ]; a = Flatten[a]
Table[PadLeft[{0}, n, 1], {n, 0, 20}]//Flatten (* Harvey P. Dale, Jul 10 2019 *)
PROG
(PARI) for(n=1, 9, print1("0, "); for(i=1, n, print1("1, "))) \\ Charles R Greathouse IV, Jun 16 2011
(PARI) a(n)=!issquare(8*n+9) \\ Charles R Greathouse IV, Jun 16 2011
(Haskell)
a023532 = (1 -) . a010052 . (+ 9) . (* 8)
a023532_list = concat $ iterate (\rs -> 1 : rs) [0]
-- Reinhard Zumkeller, Dec 04 2012
(Python)
from sympy.ntheory.primetest import is_square
def A023532(n): return bool(is_square((n<<3)+9))^1 # Chai Wah Wu, Feb 10 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Additional comments from Robert G. Wilson v, Nov 06 2000
STATUS
approved