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A070047
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Number of partitions of n in which no part appears more than twice and no two parts differ by 1.
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9
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1, 1, 2, 1, 3, 3, 5, 5, 8, 8, 12, 12, 19, 19, 27, 28, 39, 41, 55, 58, 77, 82, 106, 113, 145, 156, 196, 210, 262, 283, 348, 376, 459, 497, 600, 651, 781, 849, 1009, 1097, 1298, 1413, 1660, 1807, 2113, 2302, 2676, 2916, 3377, 3681, 4242, 4623, 5309, 5787, 6619
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OFFSET
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0,3
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COMMENTS
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Coefficients in expansion of permanent of infinite tridiagonal matrix: matrix([[1, x, 0, 0, 0, ...], [1+x, 1, x^2, 0, 0, ...], [0, 1+x^2, 1, x^3, 0, ...], [0, 0, 1+x^3, 1, x^4, ...], ...]). - Vladeta Jovovic, Jul 18 2004
Number of partitions of n into non-multiples of 3 in which no two parts differ by 1 (see the Andrews-Lewis reference). Example: a(6)=5 because we have 51,42,411,222,111111. - Emeric Deutsch, May 19 2008
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REFERENCES
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D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.
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LINKS
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FORMULA
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Expansion of phi(-x^3) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions. - Michael Somos, Jun 02 2011
Expansion of q^(1/24) * eta(q^3)^2 / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Dec 04 2002
Euler transform of period 6 sequence [ 1, 1, -1, 1, 1, 0, ...]. - Michael Somos, Dec 04 2002
G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = (2/3)^(1/2) g(t) where q = exp(2 Pi i t) and g is the g.f. of A233006.
G.f.: Prod_{k>0} (1 - x^(6*k - 3))^2 * (1 - x^(6*k)) / (1 - x^k).
G.f.: Prod_{n>0}[(1-q^(6n-3))/[(1-q^(3n-2))(1-q^(3n-1))]]. - Emeric Deutsch, May 19 2008
a(n) ~ 2*Pi * BesselI(1, Pi/6 * sqrt((24*n-1)/2)) / sqrt(3*(24*n-1)) ~ exp(Pi*sqrt(n/3)) / (2*3^(3/4)*n^(3/4)) * (1 - (3*sqrt(3)/(8*Pi) + Pi/(48*sqrt(3)))/sqrt(n) + (Pi^2/13824 - 45/(128*Pi^2) + 5/128)/n). - Vaclav Kotesovec, Sep 02 2015, extended Jan 11 2017
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EXAMPLE
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G.f. = 1 + x + 2*x^2 + x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 5*x^7 + 8*x^8 + 8*x^9 + 12*x^10 + ...
G.f. = 1/q + q^23 + 2*q^47 + q^71 + 3*q^95 + 3*q^119 + 5*q^143 + 5*q^167 + 8*q^191 + ...
a(6)=5 because we have 6,51,42,411,33.
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +add(b(n-i*j, i-2), j=1..min(n/i, 2))))
end:
a:= n-> b(n, n):
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Product[ (1 - x^(6 k - 3))^2 (1 - x^(6 k)), {k, Ceiling[ n/6]}] / Product[ 1 - x^k, {k, n}], {x, 0, n}]]; (* Michael Somos, Jun 02 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Dec 03 2013 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-2], {j, 1, Min[n/i, 2]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 04 2015, after Alois P. Heinz *)
nmax = 100; CoefficientList[Series[Product[1 / ( (1-x^(3*k-2)) * (1-x^(3*k-1)) * (1 + x^(3*k)) ), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Jun 02 2011 */
(Haskell)
a070047 n = p 1 n where
p k m | m == 0 = 1 | m < k = 0 | otherwise = q k (m-k) + p (k+1) m
q k m | m == 0 = 1 | m < k = 0 | otherwise = p (k+2) (m-k) + p (k+2) m
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CROSSREFS
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KEYWORD
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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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