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A023531
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a(n) = 1 if n of form m(m+3)/2, otherwise 0.
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72
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1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0
(list;
table;
graph;
refs;
listen;
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internal format)
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OFFSET
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0,1
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COMMENTS
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Can read as table: a(n,m)= 1 if n=m >= 0, else 0 (unit matrix).
a(n) = number of 1's between successive 0's (see also A005614, A003589 and A007538) - Eric Angelini, Jul 06 2005
Triangle T(n,k), 0<=k<=n, read by rows, given by A000004 DELTA A000007 where DELTA is the operator defined in A084938. [Philippe DELEHAM, Jan 03 2009]
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.
A023531 is reverse reluctant sequence of sequence A000007. - Boris Putievskiy, Jan 11 2013
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LINKS
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Table of n, a(n) for n=0..98.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO].
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind
Index entries for characteristic functions
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FORMULA
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If (floor(sqrt(2*n))-(2*n/(floor(sqrt(2*n)))) = -1, 1, 0). - Gerald Hillier, Sep 11 2005
a(n)=1 - A023532(n); a(n)=1 - 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
a(n)=floor((sqrt(9+8n)-1)/2)-floor((sqrt(1+8n)-1)/2). - Paul Barry, May 25 2004
a(n)=round(sqrt(2n+3))-round(sqrt(2n+2)). - Hieronymus Fischer, Aug 06 2007
a(n)=ceiling(2*sqrt(2n+3))-floor(2*sqrt(2n+2))-1. - Hieronymus Fischer, Aug 06 2007
a(n)=Sum_{k=1..oo}{C((n+2^(2*k)-k^2/2-k/2-1)^(2*k),n+2^(2*k)-k^2/2-k/2+1) mod 2} - Paolo P. Lava, Sep 07 2007
Contribution from Franklin T. Adams-Watters, Jun 29 2009: (Start)
G.f. 1/2 x^{-1/8}theta_2(0,x^{1/2}), where theta_2 is a Jacobi theta function.
G.f. for triangle: Sum T(n,k) x^n y^k = 1/(1-x*y). Sum T(n,k) x^n y^k / n! = Sum T(n,k) x^n y^k / k! = exp(x*y). Sum T(n,k) x^n y^k / (n! k!) = I_0(2*sqrt(x*y)), where I is the modified Bessel function of the first kind. (End)
a(n)=A000007(m), where m=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Jan 11 2013
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EXAMPLE
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As a triangle:
......1
.....0.1
....0.0.1
...0.0.0.1
..0.0.0.0.1
.0.0.0.0.0.1
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MATHEMATICA
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If[IntegerQ[(Sqrt[9+8#]-3)/2], 1, 0]&/@Range[0, 100] (* Harvey P. Dale, Jul 27 2011 *)
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PROG
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(Haskell)
a023531 n = a023531_list !! n
a023531_list = concat $ iterate ([0, 1] *) [1]
instance Num a => Num [a] where
fromInteger k = [fromInteger k]
(p:ps) + (q:qs) = p + q : ps + qs
ps + qs = ps ++ qs
(p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
_ * _ = []
-- Reinhard Zumkeller, Apr 02 2011
(Sage)
def A023531_row(n) :
if n == 0: return [1]
return [0] + A023531_row(n-1)
for n in (0..9): print A023531_row(n) # Peter Luschny, Jul 22 2012
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CROSSREFS
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Cf. A000217, A010054, A000007, A023532.
Sequence in context: A179560 A128407 A134286 * A089495 A173857 A114482
Adjacent sequences: A023528 A023529 A023530 * A023532 A023533 A023534
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KEYWORD
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nonn,easy,tabl,nice
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AUTHOR
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Clark Kimberling
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STATUS
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approved
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