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Search: id:A059973
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| A059973 |
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Based on fact that third root ( 2 +/- 1 sqrt(5) ) = sixth root ( 9 +/- 4 sqrt(5) ) = ninth root (38 +/- 17 sqrt(5)) = ... = phi or 1/phi, where phi is the golden ratio. |
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+0
3
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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; listen)
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OFFSET
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0,4
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COMMENT
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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
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Thomas J. Osler, Cardan polynomials and the reduction of radicals, Mathematics Magazine, Vol. 47, No. 1, (2001), pp. 26-32.
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FORMULA
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G.f.: (1 + 2x + x^3)/(1 - 4x^2 - x^4).
G.f.: (x + x^2 - 2*x^3) / ( 1 - 4*x^2 - x^4). - Michael Somos Aug 11 2009
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EXAMPLE
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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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PROGRAM
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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 */
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CROSSREFS
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A000045 (Fibonacci Numbers).
A001076(n) = a(2*n). A001077(n) = a(2*n + 1). - Michael Somos Aug 11 2009
Sequence in context: A115451 A077931 A136326 this_sequence A030035 A123431 A049961
Adjacent sequences: A059970 A059971 A059972 this_sequence A059974 A059975 A059976
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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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Edited by Randall L. Rathbun, Jan 11, 2002
More terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Jan 31 2003
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