%I A023531
%S A023531 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,
%T A023531 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,
%U A023531 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
%N A023531 a(n) = 1 if n of form m(m+3)/2, otherwise 0.
%C A023531 Can read as table: a(n,m)= 1 if n=m >= 0, else 0 (unit matrix).
%C A023531 a(n) = number of 1's between successive 0's (see also A005614, A003589 
               and A007538) - Eric Angelini (eric.angelini(AT)kntv.be), Jul 06 2005
%C A023531 Triangle T(n,k), 0<=k<=n, read by rows, given by A000004 DELTA A000007 
               where DELTA is the operator defined in A084938 . [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Jan 03 2009]
%H A023531 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               On generalizations of Stirling number triangles</a>, J. Integer Seqs., 
               Vol. 3 (2000), #00.2.4.
%H A023531 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               JacobiThetaFunctions.html">Jacobi Theta Functions</a> [From Franklin 
               T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 29 2009]
%H A023531 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ModifiedBesselFunctionoftheFirstKind.html">Modified Bessel Function 
               of the First Kind</a> [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jun 29 2009]
%H A023531 <a href="Sindx_Ch.html#char_fns">Index entries for characteristic functions</
               a>
%F A023531 If (floor(sqrt(2*n))-(2*n/(floor(sqrt(2*n)))) = -1, 1, 0). - Gerald Hillier, 
               Sep 11 2005
%F A023531 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 (pbarry(AT)wit.ie), May 25 2004
%F A023531 a(n)=floor((sqrt(9+8n)-1)/2)-floor((sqrt(1+8n)-1)/2). - Paul Barry (pbarry(AT)wit.ie), 
               May 25 2004
%F A023531 a(n)=round(sqrt(2n+3))-round(sqrt(2n+2)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Aug 06 2007
%F A023531 a(n)=ceiling(2*sqrt(2n+3))-floor(2*sqrt(2n+2))-1. - Hieronymus Fischer 
               (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007
%F A023531 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 (ppl(AT)spl.at), Sep 07 2007
%F A023531 Contribution from Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jun 29 2009: (Start)
%F A023531 G.f. 1/2 x^{-1/8}theta_2(0,x^{1/2}), where theta_2 is a Jacobi theta 
               function.
%F A023531 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)
%e A023531 As a triangle:
%e A023531 ......1
%e A023531 .....0.1
%e A023531 ....0.0.1
%e A023531 ...0.0.0.1
%e A023531 ..0.0.0.0.1
%e A023531 .0.0.0.0.0.1
%Y A023531 Cf. A000217, A010054.
%Y A023531 Sequence in context: A074381 A128407 A134286 this_sequence A089495 A114482 
               A127829
%Y A023531 Adjacent sequences: A023528 A023529 A023530 this_sequence A023532 A023533 
               A023534
%K A023531 nonn,easy,tabl,nice
%O A023531 0,1
%A A023531 Clark Kimberling (ck6(AT)evansville.edu)