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Search: id:A052102
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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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+0
5
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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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COMMENT
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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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A. Kumar Gupta, A. Kumar Mittal, Integer Sequences associated with Integer Monic Polynomial, arXiv:math.GM/0001112, Jan 2000.
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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
Sum[0..[n/3], 2^k * C(n, 3k+1)]. - Ralf Stephan, Aug 30 2004
O.g.f.: (-x+x^2)/(-1+3*x-3*x^2+3*x^3). a(n+1)-a(n)=A052101(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 01 2008
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EXAMPLE
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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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PROGRAM
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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 */
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CROSSREFS
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Cf. A052101, A052103.
Sequence in context: A158027 A100249 A138477 this_sequence A053561 A147773 A006403
Adjacent sequences: A052099 A052100 A052101 this_sequence A052103 A052104 A052105
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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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More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 01 2008
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