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A001612
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a(n) = a(n-1) + a(n-2) - 1.
(Formerly M0974 N0364)
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6
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3, 2, 4, 5, 8, 12, 19, 30, 48, 77, 124, 200, 323, 522, 844, 1365, 2208, 3572, 5779, 9350, 15128, 24477, 39604, 64080, 103683, 167762, 271444, 439205, 710648, 1149852, 1860499, 3010350, 4870848, 7881197, 12752044, 20633240, 33385283, 54018522
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| a(n+3)=A^(n)B^(2)(1), n>=0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g. 5=`00`, 8=`100`, 12=`1100`,..., in Wythoff code.
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REFERENCES
| F. Hazama, Spectra of graphs attached to the space of melodies, Discrete Math., 311 (2011), 2368-2383. See Table 5.2.
D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
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
| T. D. Noe, Table of n, a(n) for n = 0..500
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.
Index to sequences with linear recurrences with constant coefficients, signature (2,0,-1).
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FORMULA
| G.f.: (3-4*x)/((1-x)*(1-x-x^2)). a(n)=a(n-1)+a(n-2)-1.
a(n) = 1+A000032(n).
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MAPLE
| A001612:=-(-2+3*z**2)/(z-1)/(z**2+z-1); [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence except for the initial 3.]
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MATHEMATICA
| Join[{b=3}, a=0; Table[c=a+b-1; a=b; b=c, {n, 100}]] (*From Vladimir Joseph Stephan Orlovsky, Mar 15 2011*)
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PROG
| (PARI) a(n)=fibonacci(n+1)+fibonacci(n-1)+1
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CROSSREFS
| Sequence in context: A039882 A164287 A086962 * A097092 A059320 A129601
Adjacent sequences: A001609 A001610 A001611 * A001613 A001614 A001615
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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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EXTENSIONS
| Additional comments from Michael Somos, Jun 01 2000.
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