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A005369
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a(n) = 1 if n is of form m(m+1) else 0.
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3
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1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 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, 0, 1, 0, 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, 0, 0, 1, 0, 0, 0, 0, 0, 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, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Euler transform of period 4 sequence [0,1,0,-1,...].
Expansion of q^(-1/4)eta(q^4)^2/eta(q^2) in powers of q.
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LINKS
| Eric Weisstein's World of Mathematics, Jacobi Theta Functions [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 29 2009]
Index entries for characteristic functions
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FORMULA
| G.f.: Product_{k>0} (1-x^(4k))/(1-x^(4k-2)) = f(x^2, x^6) where f(a, b) is Ramanujan's theta function.
Given g.f. A(x), then B(x)=(x*A(x^4))^2 satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=v^3+4*v*w^2-u^2*w. - Michael Somos Apr 13 2005
Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u1*u2^2*u6-u1*u6^3-u3^3*u2. - Michael Somos Apr 13 2005
a(n)=b(4n+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
G.f. 1/2 x^{-1/4}theta_2(0,x), where theta_2 is a Jacobi theta function. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 29 2009]
a(n) = f(2*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), Mar 05 2010]
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EXAMPLE
| 1 + q^2 + q^6 + q^12 + q^20 + q^30 + q^42 + q^56 + ...
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PROG
| (PARI) a(n)=if(n<0, 0, issquare(4*n+1))
(PARI) a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff(eta(x^4+A)^2/eta(x^2+A), n))
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CROSSREFS
| Cf. A002378. Partial sums give A000194. A010054(n)=a(2n).
Sequence in context: A154271 A087032 A102242 * A108340 A088917 A014933
Adjacent sequences: A005366 A005367 A005368 * A005370 A005371 A005372
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Michael Somos, Apr 29 2003.
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