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 A023532 a(n) = 0 if n of form m(m+3)/2, otherwise 1. 56

%I

%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 of 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, ... ones separated by a single zero.

%F a(n) = mod(floor(((10^(n+2)-10)/9)10^(n+1-binomial(floor((1+sqrt(9+8n))/2), 2)- (1+floor(log((10^(n+2)-10)/9, 10))))), 10). - _Paul Barry_, May 25 2004

%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*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Jan 11 2013

%e The start of the sequence as triangle array read by rows:

%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 - _Boris Putievskiy_, Jan 11 2013

%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 a023532 = (1 -) . a010052 . (+ 9) . (* 8)

%o a023532_list = concat \$ iterate (\rs -> 1 : rs) [0]

%o -- _Reinhard Zumkeller_, Dec 04 2012

%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_