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A059973
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G.f.: (x + x^2 - 2*x^3) / ( 1 - 4*x^2 - x^4).
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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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.
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LINKS
| Thomas J. Osler, Cardan polynomials and the reduction of radicals, Mathematics Magazine, Vol. 47, No. 1, (2001), pp. 26-32.
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EXAMPLE
| 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
| CoefficientList[ Series[(x + x^2 - 2 x^3)/(1 - 4 x^2 - x^4), {x, 0, 33}], x]
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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 */
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CROSSREFS
| Cf. A000045 (Fibonacci Numbers).
A001076(n) = a(2*n), A001077(n) = a(2*n + 1). - Michael Somos Aug 11 2009
Cf. A179319, A183555, A183556.
Sequence in context: A077931 A115451 A136326 * A030035 A123431 A049961
Adjacent sequences: A059970 A059971 A059972 * A059974 A059975 A059976
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KEYWORD
| easy,nonn
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AUTHOR
| H. Peter Aleff (hpaleff(AT)earthlink.net), Mar 05 2001
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EXTENSIONS
| Edited by Randall L. Rathbun, Jan 11, 2002
More terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), 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.
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