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A033538
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a(0)=1, a(1)=1, a(n)=3*a(n-1)+a(n-2)+1.
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4
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1, 1, 5, 17, 57, 189, 625, 2065, 6821, 22529, 74409, 245757, 811681, 2680801, 8854085, 29243057, 96583257, 318992829, 1053561745, 3479678065, 11492595941, 37957465889, 125364993609, 414052446717, 1367522333761, 4516619448001
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of times certain simple recursive programs (such as the Lisp program shown) call themselves on an input of length n.
This is the sequence A(1,1;3,1;1) 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 18 2010]
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REFERENCES
| E. Hyvonen and J. Seppanen, LISP-kurssi, Osa 6 (Funktionaalinen ohjelmointi), Prosessori 4/1983, pp. 48-50 (in Finnish).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
A. Karttunen, More information
W. Lang, Notes on certain inhomogeneous three term recurrences. [From Wolfdieter Lang, Oct 18 2010]
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FORMULA
| O.g.f.: (1-3*x+3*x^2)/((1-x)*(1-3*x-x^2)). a(n)=(4*A006190(n+1)-8*A006190(n)-1)/3. [From R. J. Mathar, Aug 22 2008]
a(n) = 4*a(n-1) -2*a(n-2) - a(n-3), a(0)=1=a(1), a(2)=5. Observed by G. Detlefs. See the W. Lang link. [From Wolfdieter Lang, Oct 18 2010]
a(n) = -1/3+2/39*(3/2-1/2*sqrt(13))^n*sqrt(13)-2/39*sqrt(13)*(3/2+1/2*sqrt(13))^n+2/3 *(3/2-1/2*sqrt(13))^n+2/3*(3/2+1/2*sqrt(13))^n, with n>=0 [From Paolo P. Lava, Sep 01 2008]
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MAPLE
| a := proc(n) option remember; if(n < 2) then RETURN(1); else RETURN(3*a(n-1)+a(n-2)+1); fi; end;
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MATHEMATICA
| CoefficientList[ Series[(1-3x+3x^2)/(1-4x+2x^2+x^3), {x, 0, 25}], x](* From Jean-François Alcover, Nov 30 2011 *)
RecurrenceTable[{a[0]==a[1]==1, a[n]==3a[n-1]+a[n-2]+1}, a, {n, 30}] (* or *) LinearRecurrence[{4, -2, -1}, {1, 1, 5}, 31] (* From Harvey P. Dale, Jan 05 2012 *)
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PROG
| (Lisp) (defun rewerse (lista) (cond ((null (cdr lista)) lista) (t (cons (car (rewerse (cdr lista))) (rewerse (cons (car lista) (rewerse (cdr (rewerse (cdr lista))))))))))
(Haskell)
a033538 n = a033538_list !! n
a033538_list =
1 : 1 : (map (+ 1) $ zipWith (+) a033538_list
$ map (3 *) $ tail a033538_list)
-- Reinhard Zumkeller, Aug 14 2011
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CROSSREFS
| Cf. A033539, A001595.
Sequence in context: A145371 A112044 A027030 * A027093 A027032 A027095
Adjacent sequences: A033535 A033536 A033537 * A033539 A033540 A033541
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KEYWORD
| nonn,nice,easy
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AUTHOR
| Antti Karttunen
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