A052103 - OEIS

%I #22 Jun 17 2024 22:47:08

%S 0,0,1,3,6,12,27,63,144,324,729,1647,3726,8424,19035,43011,97200,

%T 219672,496449,1121931,2535462,5729940,12949227,29264247,66134880,

%U 149459580,337766841,763326423,1725057486,3898493712,8810287947,19910555163

%N The third 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 c is a(n).

%D Maribel Díaz Noguera [Maribel Del Carmen Díaz Noguera], Rigoberto Flores, Jose L. Ramirez, and Martha Romero Rojas, Catalan identities for generalized Fibonacci polynomials, Fib. Q., 62:2 (2024), 100-111. See Table 3.

%D R. Schoof, Catalan's Conjecture, Springer-Verlag, 2008, pp. 17-18.

%H G. C. Greubel, <a href="/A052103/b052103.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+2). - _Ralf Stephan_, Aug 30 2004

%F From _Paul Curtz_, Mar 10 2008: (Start)

%F a(n) = 4*a(n-1) - 6*a(n-2) + 6*a(n-3) - 3*a(n-4).

%F 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).

%F a(n) is a sequence identical to half its third differences. (End)

%F From _R. J. Mathar_, Apr 01 2008: (Start)

%F O.g.f.: x^2/(1 - 3*x + 3*x^2 - 3*x^3).

%F a(n+1) - a(n) = A052102(n). (End)

%e G.f. = x^2 + 3*x^3 + 6*x^4 + 12*x^5 + 27*x^6 + 63*x^7 + 144*x^8 + ...

%p A052103:= n-> add(2^j*binomial(n, 3*j+2), j = 0..floor(1/3*n));

%p seq(A052103(n), n = 0..40); # _G. C. Greubel_, Apr 15 2021

%t LinearRecurrence[{3,-3,3}, {0,0,1}, 32] (* _Ray Chandler_, Sep 23 2015 *)

%o (PARI) {a(n) = polcoeff( lift( Mod(1 + x, x^3 - 2)^n ), 2)} /* _Michael Somos_, Aug 05 2009 */

%o (PARI) {a(n) = sum(k=0, n\3, 2^k * binomial(n, 3*k + 2))} /* _Michael Somos_, Aug 05 2009 */

%o (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 */

%o (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

%o (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

%Y Cf. A052101, A052102.

%K nonn

%O 0,4

%A Ashok K. Gupta and Ashok K. Mittal (akgjkiapt(AT)hotmail.com), Jan 20 2000

%E More terms from _R. J. Mathar_, Apr 01 2008