A052101 - OEIS

%I #27 Jun 17 2024 22:46:15

%S 1,1,1,3,9,21,45,99,225,513,1161,2619,5913,13365,30213,68283,154305,

%T 348705,788049,1780947,4024809,9095733,20555613,46454067,104982561,

%U 237252321,536171481,1211705163,2738358009,6188472981,13985460405

%N One 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 a 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="/A052101/b052101.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).

%F a(n)/a(n-1) tends to 2.259921049... = 1 + 2^(1/3) (a real root to (x - 1)^3 = 2 or x^3 - 3x^2 + 3x - 3 = 0). A 3 X 3 matrix corresponding to the latter polynomial is [0 1 0 / 0 0 1 / 3 -3 3]. Let the matrix = M. Then a(n) = the center term in M^n * [1, 1, 1]. M^[1, 1, 1] = [9, 21, 45], center term = a(4) - _Gary W. Adamson_, Mar 28 2004

%F a(n) = Sum_{0..floor(n/3)}, 2^k * binomial(n, 3*k). - _Ralf Stephan_, Aug 30 2004

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

%F Equals the first differences of A052102.

%F Equals the second differences of A052103.

%F Equals the binomial transform of A077959.

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

%F A052103 is binomial transform of c(n)=0, 1, 1, 0, 2, 2, 0, 4, 4, 0, 8, 8, ... b(n+1) - 2*b(n) is essentially 3*b(n). (End)

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

%e From the Schoof reference, pp. 17, 18: Set pi = 1 + sqrt[3]{2}. For every integer k >= 0, there are unique a_k,b_k,c_k in Q such that pi^k = a_k + b_k sqrt[3]{2} + c_k sqrt[3]{4}. The coefficients a_k,b_k,c_k are actually in Z:

%e      Coefficients a_k, b_k, c_k:

%e      k      0     1     2     3     4     5      6

%e      ----------------------------------------------

%e      a_k    1     1     1     3     9     21     45

%e      b_k    0     1     2     3     6     15     36

%e      c_k    0     0     1     3     6     12     27

%e      ----------------------------------------------

%e G.f. = 1 + x + x^2 + 3*x^3 + 9*x^4 + 21*x^5 + 45*x^6 + 99*x^7 + 225*x^8 + ...

%p A052101 := n -> add(2^j*binomial(n, 3*j), j = 0..floor(n/3));

%p seq(A052101(n), n = 0..40); # _G. C. Greubel_, Apr 15 2021

%t LinearRecurrence[{3, -3, 3},{1, 1, 1},31] (* _Ray Chandler_, Sep 23 2015 *)

%o (PARI) {a(n) = polcoeff( lift( Mod(1 + x, x^3 - 2)^n ), 0)} /* _Michael Somos_, Aug 05 2009 */

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

%o (PARI) {a(n) = if( n<0, 0, polcoeff( (1 - 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 1 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) for j in (0..n//3)) for n in (0..40)] # _G. C. Greubel_, Apr 15 2021

%Y Cf. A052102, A052103.

%K nonn,easy

%O 0,4

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