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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

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OFFSET

0,4

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.

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).

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)

EXAMPLE

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

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 *)

PROG

(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

(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

(SageMath)

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

CROSSREFS

Cf. A000045, A001076, A001077.

Cf. A179319, A183555, A183556.

Sequence in context: A316983 A136326 A362033 * A030035 A123431 A049961

Adjacent sequences: A059970 A059971 A059972 * A059974 A059975 A059976

KEYWORD