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A096655
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a(n) = F(n+1)*a(n-1) + F(n)*a(n-2), where F = A000045 (Fibonacci numbers), a(0)=1, a(1)=1.
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4
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1, 1, 3, 11, 64, 567, 7883, 172914, 6044619, 338333121, 30444101814, 4414062308985, 1032860468654721, 390416873200823322, 238543681049185056237, 235680767488198152732339
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Comments: If the initial values are changed to a(0)=1 and a(1)=2, the resulting sequence (p(0),p(1),...)=(1,2,5,19,....) is essentially A089126. Writing A096655 as (q(0),q(1),...), the quotients p(n)/q(n) are the self-convergents to the number 1.719525... whose self-continued fraction is (1,1,2,3,5,...)=A000045. For definitions, see A096654. Now writing A096655 as (p(0),p(1),...) and A096656 as (q(0),q(1),...), the quotients p(n)/q(n) are the self-convergents to the number 1.389805... whose self-continued fraction is (1,2,3,5,...).
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FORMULA
| a(n) is asymptotic to c*phi^(n(n+1)/2)/5^(n/2) where c=3.487197183858494166192... and phi is the golden ratio - Benoit Cloitre (benoit7848c(AT)orange.fr), Jul 02 2004
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EXAMPLE
| a(2)=F(3)*1+F(2)*1=3, a(3)=F(4)*3+F(3)*1=11.
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MATHEMATICA
| a[0] = 1; a[1] = 1; a[n_] := Fibonacci[n + 1]*a[n - 1] + Fibonacci[n]*a[n - 2]; Table[ a[n], {n, 0, 16}] (from Robert G. Wilson v Jul 09 2004)
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CROSSREFS
| Cf. A000045, A089126, A096654.
Sequence in context: A069725 A193499 A199135 * A030226 A132101 A077428
Adjacent sequences: A096652 A096653 A096654 * A096656 A096657 A096658
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KEYWORD
| nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), Jul 01 2004
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EXTENSIONS
| More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Jul 02 2004
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