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A059973
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Expansion of x*(1 + x - 2*x^2) / ( 1 - 4*x^2 - x^4).
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7
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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
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0,4
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COMMENTS
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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.
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LINKS
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FORMULA
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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
a(2n) = Sum_{k=0..2n-1} a(k).
a(2n+1) = A001076(n-1) + Sum_{k=0..2n} a(k), n>0. (End)
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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));
(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)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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H. Peter Aleff (hpaleff(AT)earthlink.net), Mar 05 2001
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EXTENSIONS
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I made the old definition into a comment and gave the g.f. as an explicit definition. - N. J. A. Sloane, Jan 05 2011
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STATUS
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approved
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