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A001588
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a(n) = a(n-1) + a(n-2) - 1.
(Formerly M2279 N0901)
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1
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1, 3, 3, 5, 7, 11, 17, 27, 43, 69, 111, 179, 289, 467, 755, 1221, 1975, 3195, 5169, 8363, 13531, 21893, 35423, 57315, 92737, 150051, 242787, 392837, 635623, 1028459, 1664081, 2692539, 4356619, 7049157, 11405775, 18454931, 29860705, 48315635
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| Problem B-100, Fib. Quart., 5 (1967), 288.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| a(n) = 2*Fib(n)+1 = A000045(n)+A001611(n). G.f.: (1+x-3x^2)/(1-2*x+x^3) - Henry Bottomley (se16(AT)btinternet.com), Feb 20 2001
If n>=4, a(n) = floor(Phi*a(n-1)); Phi = (1 + sqrt(5))/2. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 08 2003
a(n)=F(n)+F(n+3)+1, n>=-2 {where F(n) is the n-th Fibonacci number} - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 01 2008
a(n)=1+(2/5)*[(1/2)+(1/2)*sqrt(5)]^n*sqrt(5)-(2/5)*sqrt(5)*[(1/2)-(1/2)*sqrt(5)]^n, with n>0=0 [From Paolo P. Lava (paoloplava(AT)gmail.com), Nov 21 2008]
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MAPLE
| A001588:=-(-1-z+3*z**2)/(z-1)/(z**2+z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
with(combinat): seq(fibonacci(n)+fibonacci(n+3)+1, n=-2..35); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 01 2008
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MATHEMATICA
| Fibonacci[Range[0, 5! ]]*2+1 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Mar 19 2010]
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CROSSREFS
| Sequence in context: A161834 A141867 A163646 * A107029 A169942 A147095
Adjacent sequences: A001585 A001586 A001587 * A001589 A001590 A001591
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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