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A052103
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The third of the three sequences associated with the polynomial x^3 - 2.
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9
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0, 0, 1, 3, 6, 12, 27, 63, 144, 324, 729, 1647, 3726, 8424, 19035, 43011, 97200, 219672, 496449, 1121931, 2535462, 5729940, 12949227, 29264247, 66134880, 149459580, 337766841, 763326423, 1725057486, 3898493712, 8810287947, 19910555163
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OFFSET
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0,4
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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 c 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+2). - Ralf Stephan, Aug 30 2004
a(n) = 4*a(n-1) - 6*a(n-2) + 6*a(n-3) - 3*a(n-4).
a(n) is binomial transform of 0, 0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32 (see A077958).
a(n) is a sequence identical to half its third differences. (End)
O.g.f.: x^2/(1 - 3*x + 3*x^2 - 3*x^3).
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EXAMPLE
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G.f. = x^2 + 3*x^3 + 6*x^4 + 12*x^5 + 27*x^6 + 63*x^7 + 144*x^8 + ...
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MAPLE
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A052103:= n-> add(2^j*binomial(n, 3*j+2), j = 0..floor(1/3*n));
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MATHEMATICA
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LinearRecurrence[{3, -3, 3}, {0, 0, 1}, 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 ), 2)} /* Michael Somos, Aug 05 2009 */
(PARI) {a(n) = sum(k=0, n\3, 2^k * binomial(n, 3*k + 2))} /* Michael Somos, Aug 05 2009 */
(PARI) {a(n) = if( n<0, 0, polcoeff( x^2 / (1 - 3*x + 3*x^2 - 3*x^3) + x * O(x^n), n))} /* Michael Somos, Aug 05 2009 */
(Magma) I:=[0, 0, 1]; [n le 3 select I[n] else 3*(Self(n-1) -Self(n-2) +Self(n-3)): n in [1..41]]; // G. C. Greubel, Apr 15 2021
(Sage) [sum(2^j*binomial(n, 3*j+2) 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
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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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EXTENSIONS
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STATUS
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approved
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