A052102 The second of the three sequences associated with the polynomial x^3 - 2.

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

Offset: 0

Author

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

Keywords

• nonn
• easy

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).

Examples

			G.f.: = x + 2*x^2 + 3*x^3 + 6*x^4 + 15*x^5 + 36*x^6 + 81*x^7 + 180*x^8 + ...
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• A. Kumar Gupta and A. Kumar Mittal, Integer Sequences associated with Integer Monic Polynomial, arXiv:math/0001112 [math.GM], Jan 2000.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,3).

Crossrefs

Cf. A052101, A052103.

Programs

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

Maple

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

Mathematica

LinearRecurrence[{3,-3,3}, {0,1,2}, 32] (* Ray Chandler, Sep 23 2015 *)

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 */

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

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3), n > 2.

a(n) = Sum_{0..floor(n/3)}, 2^k * binomial(n, 3*k+1). - Ralf Stephan, Aug 30 2004

From R. J. Mathar, Apr 01 2008: (Start)

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

a(n+1) - a(n) = A052101(n). (End)