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A097451
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Number of partitions of n into parts congruent to {2, 3, 4} mod 6.
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9
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1, 0, 1, 1, 2, 1, 3, 2, 5, 4, 7, 6, 11, 9, 15, 14, 22, 20, 31, 29, 43, 41, 58, 57, 80, 78, 106, 107, 142, 143, 188, 191, 247, 253, 321, 332, 418, 432, 537, 561, 690, 721, 880, 924, 1118, 1178, 1412, 1493, 1781, 1884, 2231, 2370, 2789, 2965, 3472, 3698, 4309, 4596
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OFFSET
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0,5
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COMMENTS
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Number of partitions of n in which no part is 1, no part appears more than twice and no two parts differ by 1. Example: a(6)=3 because we have [6],[4,2] and [3,3]. - Emeric Deutsch, Feb 16 2006
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REFERENCES
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G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, Exercise 7.9.
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LINKS
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FORMULA
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Euler transform of period 6 sequence [ 0, 1, 1, 1, 0, 0, ...].
G.f.: 1/Product_{j>=0} ((1-x^(2+6j))(1-x^(3+6j))(1-x^(4+6j))). - Emeric Deutsch, Feb 16 2006
Expansion of psi(x^3) / f(-x^2) in powers of x where psi(), f() are Ramanujan theta functions. - Michael Somos, Sep 24 2013
Expansion of q^(-7/24) * eta(q^6)^2 / (eta(q^2) * eta(q^3) in powers of q. - Michael Somos, Sep 24 2013
Expansion of f(-x, -x^5) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 06 2015
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EXAMPLE
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a(8)=5 because we have [8],[44],[422],[332] and [2222].
G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 4*x^9 + ...
G.f. = q^7 + q^55 + q^79 + 2*q^103 + q^127 + 3*q^151 + 2*q^175 + 5*q^199 + ...
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MAPLE
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g:=1/product((1-x^(2+6*j))*(1-x^(3+6*j))*(1-x^(4+6*j)), j=0..15): gser:=series(g, x=0, 75): seq(coeff(gser, x, n), n=0..67); # Emeric Deutsch, Feb 16 2006
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - Boole[ OddQ[ Quotient[ k + 1, 3]]] x^k, {k, n}], {x, 0, n}; (* Michael Somos, Sep 24 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^3] QPochhammer[ x^6] / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Sep 24 2013 *)
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PROG
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(Haskell)
a097451 n = p a047228_list n where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - ( (k+1)\3 % 2) * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Sep 24 2013 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 24 2013 */
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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