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A023531 a(n) = 1 if n is of the form m(m+3)/2, otherwise 0. 73
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; history; text; internal format)
OFFSET

0,1

COMMENTS

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 Deléham, 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

LINKS

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

FORMULA

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

From Franklin T. Adams-Watters, Jun 29 2009: (Start)

G.f. (1/2 x^{-1/8}theta_2(0,x^{1/2}) - 1)/x, 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

The row polynomials are p(n,x) = x^n = (-1)^n n!Lag(n,-n,x), the normalized, associated Laguerre polynomials of order -n. As the prototypical Appell sequence with e.g.f. exp(x*y), its raising operator is R = x and lowering operator, L = d/dx, i.e., R p(n,x) = p(n+1,x), and L p(n,x) = n * p(n-1,x). - Tom Copeland, May 10 2014

a(n) = A010054(n+1) if n>=0. - Michael Somos, May 17 2014

EXAMPLE

As a triangle:

......1

.....0.1

....0.0.1

...0.0.0.1

..0.0.0.0.1

.0.0.0.0.0.1

G.f. = 1 + x^2 + x^5 + x^9 + x^14 + x^20 + x^27 + x^35 + x^44 + x^54 + ...

MATHEMATICA

If[IntegerQ[(Sqrt[9+8#]-3)/2], 1, 0]&/@Range[0, 100] (* Harvey P. Dale, Jul 27 2011 *)

a[ n_] := If[ n < 0, 0, Boole @ IntegerQ @ Sqrt[ 8 n + 9]]; (* Michael Somos, May 17 2014 *)

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)) - 1) / x, {x, 0, n}]; (* Michael Somos, May 17 2014 *)

PROG

(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

(PARI) {a(n) = if( n<0, 0, issquare(8*n + 9))}; /* Michael Somos, May 17 2014 */

CROSSREFS

Cf. A000217, A010054, A000007, A023532.

Sequence in context: A179560 A128407 A134286 * A089495 A173857 A114482

Adjacent sequences:  A023528 A023529 A023530 * A023532 A023533 A023534

KEYWORD

nonn,easy,tabl,nice

AUTHOR

Clark Kimberling, Jun 14 1998

STATUS

approved

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Last modified September 1 14:28 EDT 2014. Contains 246307 sequences.