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 A052102 The second of the three sequences associated with the polynomial x^3 - 2. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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). REFERENCES R. Schoof, Catalan's Conjecture, Springer-Verlag, 2008, pp. 17-18. LINKS A. Kumar Gupta, A. Kumar Mittal, Integer Sequences associated with Integer Monic Polynomial, arXiv:math.GM/0001112, Jan 2000. Index entries for linear recurrences with constant coefficients, signature (3,-3,3). FORMULA 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, Apr 01 2008 EXAMPLE x + 2*x^2 + 3*x^3 + 6*x^4 + 15*x^5 + 36*x^6 + 81*x^7 + 180*x^8 + ... MATHEMATICA LinearRecurrence[{3, -3, 3}, {0, 1, 2}, 32] (* Ray Chandler, Sep 23 2015 *) PROG (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 */ CROSSREFS Cf. A052101, A052103. Sequence in context: A100249 A138477 A182240 * A053561 A237585 A147773 Adjacent sequences:  A052099 A052100 A052101 * A052103 A052104 A052105 KEYWORD nonn,easy AUTHOR Ashok K. Gupta and Ashok K. Mittal (akgjkiapt(AT)hotmail.com), Jan 20 2000 STATUS approved

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Last modified June 4 17:14 EDT 2020. Contains 334828 sequences. (Running on oeis4.)