A052103 The third of the three sequences associated with the polynomial x^3 - 2.

0, 0, 1, 3, 6, 12, 27, 63, 144, 324, 729, 1647, 3726, 8424, 19035, 43011, 97200, 219672, 496449, 1121931, 2535462, 5729940, 12949227, 29264247, 66134880, 149459580, 337766841, 763326423, 1725057486, 3898493712, 8810287947, 19910555163

Offset: 0

Author

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

Keywords

• nonn

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 c is a(n).

Examples

			G.f. = x^2 + 3*x^3 + 6*x^4 + 12*x^5 + 27*x^6 + 63*x^7 + 144*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, A052102.

Programs

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

Maple

A052103:= n-> add(2^j*binomial(n, 3*j+2), j = 0..floor(1/3*n));
seq(A052103(n), n = 0..40); # G. C. Greubel, Apr 15 2021

Mathematica

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

PARI

{a(n) = polcoeff( lift( Mod(1 + x, x^3 - 2)^n ), 2)} /* Michael Somos, Aug 05 2009 */

PARI

{a(n) = sum(k=0, n\3, 2^k * binomial(n, 3*k + 2))} /* Michael Somos, Aug 05 2009 */

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 */

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

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+2). - Ralf Stephan, Aug 30 2004

From Paul Curtz, Mar 10 2008: (Start)

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

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).

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

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

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

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

Extensions

More terms from R. J. Mathar, Apr 01 2008