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A052102
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The second of the three sequences associated with the polynomial x^3 - 2.
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6
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0, 1, 2, 3, 6, 15, 36, 81, 180, 405, 918, 2079, 4698, 10611, 23976, 54189, 122472, 276777, 625482, 1413531, 3194478, 7219287, 16315020, 36870633, 83324700, 188307261, 425559582, 961731063, 2173436226, 4911794235, 11100267216, 25085727621
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OFFSET
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0,3
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COMMENTS
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If x^3 = 2 and n >= 0, then there are unique integers a, b, c such that (1 + x)^n = a + b*x + c*x^2. The coefficient b is a(n).
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REFERENCES
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R. Schoof, Catalan's Conjecture, Springer-Verlag, 2008, pp. 17-18.
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LINKS
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FORMULA
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a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3), n > 2.
a(n) = Sum_{0..floor(n/3)}, 2^k * binomial(n, 3*k+1). - Ralf Stephan, Aug 30 2004
O.g.f.: x*(1 - x)/(1 - 3*x + 3*x^2 - 3*x^3).
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EXAMPLE
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G.f.: = x + 2*x^2 + 3*x^3 + 6*x^4 + 15*x^5 + 36*x^6 + 81*x^7 + 180*x^8 + ...
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MAPLE
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MATHEMATICA
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LinearRecurrence[{3, -3, 3}, {0, 1, 2}, 32] (* Ray Chandler, Sep 23 2015 *)
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PROG
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(PARI) {a(n) = polcoeff( lift( Mod(1 + x, x^3 - 2)^n ), 1)} /* Michael Somos, Aug 05 2009 */
(PARI) {a(n) = sum(k=0, n\3, 2^k * binomial(n, 3*k + 1))} /* Michael Somos, Aug 05 2009 */
(PARI) {a(n) = if( n<0, 0, polcoeff( (x - x^2) / (1 - 3*x + 3*x^2 - 3*x^3) + x * O(x^n), n))} /* Michael Somos, Aug 05 2009 */
(Magma) [n le 3 select n-1 else 3*(Self(n-1) -Self(n-2) +Self(n-3)): n in [1..40]]; // G. C. Greubel, Apr 15 2021
(Sage) [sum(2^j*binomial(n, 3*j+1) for j in (0..n//3)) for n in (0..40)] # G. C. Greubel, Apr 15 2021
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Ashok K. Gupta and Ashok K. Mittal (akgjkiapt(AT)hotmail.com), Jan 20 2000
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STATUS
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approved
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