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A010054
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a(n) = 1 if n is a triangular number else 0.
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210
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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, 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, 0, 0, 0, 0, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Ramanujan's theta function f(a, b) = Sum a^{n*(n+1)/2} * b^{n*(n-1)/2}, n=-inf..inf.
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
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
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]
Convolved with A000041 = A022567, the convolution square of A000009 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009]
A008441(n) = SUM(a(k)*a(n-k): 0<=k<=n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 03 2009]
Polcoeff inverse with alternate signs = A006950: (1, 1, 1, 2, 3, 4, 5, 7,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 15 2010]
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REFERENCES
| Cooper, S. and Hirschhorn, M. D., Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See psi(q).
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for characteristic functions
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FORMULA
| Expansion of f(x, x^3) in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-1/8) * eta(q^2)^2 / eta(q) in powers of q. - Michael Somos, Apr 13 2005
Euler transform of period 2 sequence [ 1, -1, ...]. - Michael Somos, Mar 24 2003
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
a(n) = b(8*n + 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
a(n) = A005369(2*n). - Michael Somos, Apr 29 2003
G.f.: theta_2(q) / (2 * q^(1/4)).
G.f.: Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic (vladeta(AT)eunet.rs), May 02 2002
a(0)=1; for n>0, a(n)=A002024(n+1)-A002024(n). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 05 2004
G.f.: sum(j=0, oo, product(k=0, j, x^j)) - Jon Perry (perry(AT)globalnet.co.uk), Mar 30 2004
a(n) = floor((1-cos(Pi*sqrt(8*n+1)))/2) - Carl R. White (oeisfan(AT)cyreksoft.yorks.com), Mar 18 2006
a(n)=round(sqrt(2n+1))-round(sqrt(2n)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007
a(n)=ceiling(2*sqrt(2n+1))-floor(2*sqrt(2n))-1. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007
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]
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EXAMPLE
| 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 + x^45 + x^55 + x^66 + ...
q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + q^225 + q^289 + q^361 + ...
Comment from Philippe DELEHAM, Jan 04 2008: As a triangle this begins:
.1;
.1, 0;
.1, 0, 0;
.1, 0, 0, 0;
.1, 0, 0, 0, 0;
.1, 0, 0, 0, 0, 0 ; ...
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MATHEMATICA
| 4[ n_] := If[ n < 0, 0, SquaresR[ 1, 8 n + 1] / 2] (* Michael Somos, Nov 15 2011 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), x^2], {x, 0, n + Floor @ Sqrt[n]}] // Normal // TrigToExp) /. {y -> x}, {x, 0, n}]] (* Michael Somos, Nov 15 2011 *)
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / eta(x + A), n))} /* Michael Somos, Mar 14 2011 */
(PARI) {a(n) = if( n<0, 0, issquare( 8*n + 1))} /* Michael Somos, Apr 27 2000 */
(Haskell)
a010054 = a010052 . (+ 1) . (* 8)
a010054_list = concatMap (\x -> 1 : replicate x 0) [0..]
-- Reinhard Zumkeller, Feb 12 2012, Oct 22 2011, Apr 02 2011
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CROSSREFS
| Cf. A000217, A005369, A023531.
a(n) = A035214(n) - 1.
A022567 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009]
A052343. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 03 2009]
A006950 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 15 2010]
Sequence in context: A106459 A143433 A143434 A197870 A033806 A033802 A033800
Adjacent sequences: A010051 A010052 A010053 * A010055 A010056 A010057
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KEYWORD
| nonn,tabl,changed
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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 27 2000.
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