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.
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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
H. Peter Aleff (hpaleff(AT)earthlink.net), Mar 05 2001
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
STATUS
approved