A059973 Expansion of x*(1 + x - 2*x^2) / ( 1 - 4*x^2 - x^4).

0, 1, 1, 2, 4, 9, 17, 38, 72, 161, 305, 682, 1292, 2889, 5473, 12238, 23184, 51841, 98209, 219602, 416020, 930249, 1762289, 3940598, 7465176, 16692641, 31622993, 70711162, 133957148, 299537289, 567451585, 1268860318, 2403763488, 5374978561

Offset: 0

Author

H. Peter Aleff (hpaleff(AT)earthlink.net), Mar 05 2001

Keywords

• easy
• nonn

Comments

Based on fact that cube root of (2 +- 1 sqrt(5)) = sixth root of (9 +- 4 sqrt(5)) = ninth root of (38 +- 17 sqrt(5)) = ... = phi or 1/phi, where phi is the golden ratio.

Osler gives the first three of the above equalities with phi on page 27, stating they are simplified expressions from Ramanujan, but without hinting that the series continues.

Bisections: A001076 and A001077.

Examples

			G.f. = x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 17*x^6 + 38*x^7 + 72*x^8 + 161*x^9 + ... - _Michael Somos_, Aug 11 2009
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• T. J. Osler, Cardan polynomials and the reduction of radicals, Math. Mag., 74 (No. 1, 2001), 26-32.
• Index entries for linear recurrences with constant coefficients, signature (0,4,0,1).

Crossrefs

Cf. A000045, A001076, A001077.

Cf. A179319, A183555, A183556.

Programs

Magma

I:=[0,1,1,2]; [n le 4 select I[n] else 4*Self(n-2)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Oct 10 2015

Mathematica

CoefficientList[ Series[(x +x^2 -2x^3)/(1 -4x^2 -x^4), {x, 0, 33}], x]
LinearRecurrence[{0,4,0,1}, {0,1,1,2}, 50] (* Vincenzo Librandi, Oct 10 2015 *)

PARI

{a(n) = if( n<0, n = -n; polcoeff( (-2*x + x^2 + x^3) / (1 + 4*x^2 - x^4) + x*O(x^n), n), polcoeff( (x + x^2 - 2*x^3) / ( 1 - 4*x^2 - x^4) + x*O(x^n), n))} /* Michael Somos, Aug 11 2009 */

PARI

a(n) = if (n < 4, fibonacci(n), 4*a(n-2) + a(n-4));
vector(50, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

Sage

def a(n): return fibonacci(n) if (n<4) else 4*a(n-2) + a(n-4)
[a(n) for n in [0..40]] # G. C. Greubel, Jul 12 2021

Formula

From Michael Somos, Aug 11 2009: (Start)

a(2*n) = A001076(n).

a(2*n+1) = A001077(n). (End)

Recurrence: a(n) = 4*a(n-2) + a(n-4) for n >= 4; a(0)=0, a(1)=a(2)=1, a(3)=2. - Werner Schulte, Oct 03 2015

From Altug Alkan, Oct 06 2015: (Start)

a(2n) = Sum_{k=0..2n-1} a(k).

a(2n+1) = A001076(n-1) + Sum_{k=0..2n} a(k), n>0. (End)

Extensions

Edited by Randall L Rathbun, Jan 11 2002

More terms from Sascha Kurz, Jan 31 2003

I made the old definition into a comment and gave the g.f. as an explicit definition. - N. J. A. Sloane, Jan 05 2011

Moved g.f. from Michael Somos, into name to match terms. - Paul D. Hanna, Jan 12 2011