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A111721
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a(n) = a(n-1) + a(n-2) + 5 where a(0) = a(1) = 1.
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1
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1, 1, 7, 13, 25, 43, 73, 121, 199, 325, 529, 859, 1393, 2257, 3655, 5917, 9577, 15499, 25081, 40585, 65671, 106261, 171937, 278203, 450145, 728353, 1178503, 1906861, 3085369, 4992235, 8077609, 13069849, 21147463, 34217317, 55364785, 89582107
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OFFSET
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0,3
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COMMENTS
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a(n+1)/a(n) converges to the golden ratio. - Stefan Steinerberger, Nov 19 2005
This is the sequence A(1,1;1,1;5) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. [From Wolfdieter Lang, Oct 17 2010]
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
W. Lang, Notes on certain inhomogeneous three term recurrences. [From Wolfdieter Lang, Oct 17 2010]
Index to sequences with linear recurrences with constant coefficients, signature (2,0,-1).
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FORMULA
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for n>1: a(n) = a(n-1) + 6*F(n-1). (a(n)-1)/6 = A000071(n+1) = F(n+1) - 1. Hence a(n) = 6*F(n+1) - 5. - Jonathan Vos Post, Nov 19 2005
G.f.: (5*x^2-x+1)/(x^3-2*x+1) - Stefan Steinerberger, Nov 19 2005
a(n) = 3/2^n*((1+sqrt(5))^n+(1-sqrt(5))^n)+3/(sqrt(5)*2^n)*((1+sqrt(5))^n-(1-sqrt(5))^n)-5. - Paolo P. Lava, Jul 27 2011
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EXAMPLE
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a(2) = a(0) + a(1) + 5 = 1 + 1 + 5 = 7
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MATHEMATICA
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Join[{a=1, b=1}, Table[c=a+b+5; a=b; b=c, {n, 50}]] (* From Vladimir Joseph Stephan Orlovsky, Apr 13 2011 *)
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PROG
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(MuPAD): a := 1; b := 1; for n from 1 to 50 do c := a+b+5; print(c); a := b; b := c; end_for; (Stefan Steinerberger)
(Sage) from sage.combinat.sloane_functions import recur_gen2b
it =recur_gen2b(1, 1, 1, 1, lambda n: 5)
[it.next() for i in range(38)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2008
(Haskell)
a111721 n = a111721_list !! n
a111721_list = 1 : 1 :
map (+ 5) (zipWith (+) a111721_list (tail a111721_list))
-- Reinhard Zumkeller, Nov 05 2011
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CROSSREFS
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Cf. A000045, A000071.
Sequence in context: A087195 A200270 A031887 * A213663 A060455 A205541
Adjacent sequences: A111718 A111719 A111720 * A111722 A111723 A111724
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KEYWORD
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nonn,easy
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AUTHOR
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Parthasarathy Nambi, Nov 17 2005
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EXTENSIONS
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More terms from Stefan Steinerberger, Nov 19 2005
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STATUS
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approved
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