%I A010054
%S A010054 1,1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,
%T A010054 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,
%U A010054 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A010054 a(n) = 1 if n is a triangular number else 0.
%C A010054 Ramanujan's theta function f(a,b)=Sum a^{n*(n+1)/2}*b^{n*(n-1)/2}, n=-inf..inf.
%C A010054 Euler transform of period 2 sequence [1,-1,...].
%C A010054 This sequence is the concatenation of the base-b digits in the sequence 
               b^n, for any base b >= 2. - Davis Herring (herring(AT)lanl.gov), 
               Nov 16 2004
%C A010054 Number of partitions of n into distinct parts such that the greatest 
               part equals the number of all parts, see also A047993; a(n)=A117195(n,
               0) for n>0; a(n)=1-A117195(n,1) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 03 2006
%C A010054 Triangle T(n,k), 0<=k<=n, read by rows, given by A000007 DELTA A000004 
               where DELTA is the operator defined in A084938 . [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Jan 03 2009]
%C A010054 Convolved with A000041 = A022567, the convolution square of A000009 [From 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009]
%C A010054 A008441(n) = SUM(a(k)*a(n-k): 0<=k<=n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 03 2009]
%H A010054 <a href="Sindx_Ch.html#char_fns">Index entries for characteristic functions</
               a>
%H A010054 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A010054 G.f.: theta2(q)/(2*q^(1/4)) = f(q, q^3) where f is Ramanujan's theta 
               function.
%F A010054 G.f.: Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               May 02 2002
%F A010054 a(0)=1; for n>0, a(n)=A002024(n+1)-A002024(n). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jan 05 2004
%F A010054 G.f.: sum(j=0, oo, product(k=0, j, x^j)) - Jon Perry (perry(AT)globalnet.co.uk), 
               Mar 30 2004
%F A010054 Expansion of q^(-1/8)eta(q^2)^2/eta(q) in powers of q.
%F A010054 Given g.f. A(x), then B(x)=x*A(x^8) satisfies 0=f(B(x), B(x^2), B(x^3), 
               B(x^6)) where f(u1, u2, u3, u6)=u1*u6^3 +u2*u3^3 -u1*u2^2*u6. - Michael 
               Somos Apr 13 2005
%F A010054 a(n)=b(8n+1) where b(n) is multiplicative and b(2^e)=0^e, b(p^e)=(1+(-1)^e)/
               2 if p>2. - Michael Somos Jun 06 2005
%F A010054 a(n) = floor((1-cos(Pi*sqrt(8*n+1)))/2) - Carl R. White (oeisfan(AT)cyreksoft.yorks.com), 
               Mar 18 2006
%F A010054 a(n)=round(sqrt(2n+1))-round(sqrt(2n)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Aug 06 2007
%F A010054 a(n)=ceiling(2*sqrt(2n+1))-floor(2*sqrt(2n))-1. - Hieronymus Fischer 
               (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007
%F A010054 a(n) = f(n,0) with f(x,y) = if x>0 then f(x-y,y+1) else 0^(-x). [From 
               Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 27 2008]
%e A010054 Comment from Philippe DELEHAM, Jan 04 2008: As a triangle this begins:
%e A010054 .1;
%e A010054 .1, 0;
%e A010054 .1, 0, 0;
%e A010054 .1, 0, 0, 0;
%e A010054 .1, 0, 0, 0, 0;
%e A010054 .1, 0, 0, 0, 0, 0 ; ...
%o A010054 (PARI) a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X^2)^2/eta(X),
               n))
%o A010054 (PARI) a(n)=if(n<0,0,issquare(8*n+1))
%Y A010054 Cf. A000217, A023531.
%Y A010054 a(n) = A035214(n) - 1. Also a(n) = A005369(2n).
%Y A010054 A022567 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009]
%Y A010054 A052343. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 03 2009]
%Y A010054 Sequence in context: A113430 A113681 A155972 this_sequence A106459 A143433 
               A143434
%Y A010054 Adjacent sequences: A010051 A010052 A010053 this_sequence A010055 A010056 
               A010057
%K A010054 nonn,tabl
%O A010054 0,1
%A A010054 N. J. A. Sloane (njas(AT)research.att.com).
%E A010054 Additional comments from Michael Somos, Apr 27, 2000.