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%I #53 Feb 11 2023 11:23:00
%S 0,1,0,1,1,0,1,1,1,0,1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,1,0,1,1,1,1,1,1,
%T 1,0,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,0,1,1,
%U 1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1
%N a(n) = 0 if n is of the form m*(m+3)/2, otherwise 1.
%C 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.
%C "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.
%C "This contradicts the fact that there will always be zeros occurring after any given point in the expansion of alpha. Hence alpha is irrational."
%C a(A000096(n)) = 0; a(A007401(n)) = 1. - _Reinhard Zumkeller_, Dec 04 2012
%C 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
%C An example of a sequence with infinite critical exponent [Vaslet]. - _N. J. A. Sloane_, May 05 2013
%D Harold M. Stark, An Introduction to Number Theory, The MIT Press, Cambridge, Mass, eighth printing 1994, page 170.
%H Reinhard Zumkeller, <a href="/A023532/b023532.txt">Table of n, a(n) for n = 0..1000</a>
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%H Elise Vaslet, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p125">Critical exponents of words over 3 letters</a>, Electronic Journal of Combinatorics, 18 (2011), #P125.
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F a(n) = 0 if and only if 8n+9 is a square. - _Charles R Greathouse IV_, Jun 16 2011
%F Blocks of lengths 1, 2, 3, 4, ... of ones separated by a single zero.
%F a(n) = 1 - floor((sqrt(9+8n)-1)/2) + floor((sqrt(1+8n)-1)/2). - _Paul Barry_, May 25 2004
%F 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
%F a(n) = [A002262(n) < A003056(n)]. - _Yuchun Ji_, May 18 2020
%e From _Boris Putievskiy_, Jan 11 2013: (Start)
%e As a triangular array written by rows, the sequence begins:
%e 0;
%e 1, 0;
%e 1, 1, 0;
%e 1, 1, 1, 0;
%e 1, 1, 1, 1, 0;
%e 1, 1, 1, 1, 1, 0;
%e 1, 1, 1, 1, 1, 1, 0;
%e ...
%e (End)
%t a = {}; Do[a = Append[a, Join[ {0}, Table[1, {n} ] ] ], {n, 1, 13} ]; a = Flatten[a]
%t Table[PadLeft[{0},n,1],{n,0,20}]//Flatten (* _Harvey P. Dale_, Jul 10 2019 *)
%o (PARI) for(n=1,9,print1("0, ");for(i=1,n,print1("1, "))) \\ _Charles R Greathouse IV_, Jun 16 2011
%o (PARI) a(n)=!issquare(8*n+9) \\ _Charles R Greathouse IV_, Jun 16 2011
%o (Haskell)
%o a023532 = (1 -) . a010052 . (+ 9) . (* 8)
%o a023532_list = concat $ iterate (\rs -> 1 : rs) [0]
%o -- _Reinhard Zumkeller_, Dec 04 2012
%o (Python)
%o from sympy.ntheory.primetest import is_square
%o def A023532(n): return bool(is_square((n<<3)+9))^1 # _Chai Wah Wu_, Feb 10 2023
%Y Cf. A023531, A211666.
%Y Essentially the same sequence as A114607 and A123110. - _N. J. A. Sloane_, Feb 07 2020
%K nonn,easy
%O 0,1
%A _Clark Kimberling_
%E Additional comments from _Robert G. Wilson v_, Nov 06 2000
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