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%I #22 Apr 15 2021 23:46:41
%S 0,1,2,3,6,15,36,81,180,405,918,2079,4698,10611,23976,54189,122472,
%T 276777,625482,1413531,3194478,7219287,16315020,36870633,83324700,
%U 188307261,425559582,961731063,2173436226,4911794235,11100267216,25085727621
%N The second of the three sequences associated with the polynomial x^3 - 2.
%C 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).
%D R. Schoof, Catalan's Conjecture, Springer-Verlag, 2008, pp. 17-18.
%H G. C. Greubel, <a href="/A052102/b052102.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Kumar Gupta and A. Kumar Mittal, <a href="https://arxiv.org/abs/math/0001112">Integer Sequences associated with Integer Monic Polynomial</a>, arXiv:math/0001112 [math.GM], Jan 2000.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,3).
%F a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3), n > 2.
%F a(n) = Sum_{0..floor(n/3)}, 2^k * binomial(n, 3*k+1). - _Ralf Stephan_, Aug 30 2004
%F From _R. J. Mathar_, Apr 01 2008: (Start)
%F O.g.f.: x*(1 - x)/(1 - 3*x + 3*x^2 - 3*x^3).
%F a(n+1) - a(n) = A052101(n). (End)
%e G.f.: = x + 2*x^2 + 3*x^3 + 6*x^4 + 15*x^5 + 36*x^6 + 81*x^7 + 180*x^8 + ...
%p A052102:= n-> add(2^j*binomial(n, 3*j+1), j=0..floor(n/3)); seq(A052102(n), n=0..40); # _G. C. Greubel_, Apr 15 2021
%t LinearRecurrence[{3,-3,3}, {0,1,2}, 32] (* _Ray Chandler_, Sep 23 2015 *)
%o (PARI) {a(n) = polcoeff( lift( Mod(1 + x, x^3 - 2)^n ), 1)} /* _Michael Somos_, Aug 05 2009 */
%o (PARI) {a(n) = sum(k=0, n\3, 2^k * binomial(n, 3*k + 1))} /* _Michael Somos_, Aug 05 2009 */
%o (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 */
%o (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
%o (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
%Y Cf. A052101, A052103.
%K nonn,easy
%O 0,3
%A Ashok K. Gupta and Ashok K. Mittal (akgjkiapt(AT)hotmail.com), Jan 20 2000
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