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Search: id:A001053
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| A001053 |
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a(n+1) = n*a(n) + a(n-1) with a(0)=1, a(1)=0.
(Formerly M1783 N0704)
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+0
25
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| 1, 0, 1, 2, 7, 30, 157, 972, 6961, 56660, 516901, 5225670, 57999271, 701216922, 9173819257, 129134686520, 1946194117057, 31268240559432, 533506283627401, 9634381345852650, 183586751854827751, 3681369418442407670
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Denominator of continued fraction given by C(n) = [ 1; 2,3,4,...n ]. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 02 2001. Cf. A001040.
If initial 1 is omitted, CONTINUANT transform of 0, 1, 2, 3, 4, 5, ...
Number of deco polyominoes of height n having no 1-cell columns. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=1 because the vertical and horizontal dominoes are the deco polynomioes of height 2, of which only the vertical domino does not have 1-cell columns. a(n)=A121554(n,0). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 16 2006
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REFERENCES
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Archimedeans Problems Drive, Eureka, 20 (1957), 15.
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
S. B. Ekhad, Problem 10356, Amer. Math. Monthly, 101 (1994), 75. [From N. J. A. Sloane (njas(AT)research.att.com), Jan 29 2009]
M. E. Larsen, Summa Summarum, A. K. Peters, Wellesley, MA, 2007; see p. 35. [From N. J. A. Sloane (njas(AT)research.att.com), Jan 29 2009]
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
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T. D. Noe, Table of n, a(n) for n=0..100
N. J. A. Sloane, Transforms
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FORMULA
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a(-n) = a(n). - Michael Somos Sep 25 2005
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EXAMPLE
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a(5) = 4*a(4) + a(3) = 4*7+2 = 30.
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MAPLE
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a[0]:=1: a[1]:=0: for n from 2 to 23 do a[n]:=(n-1)*a[n-1]+a[n-2] od: seq(a[n], n=0..23); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 16 2006
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MATHEMATICA
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a[0] = 1; a[1] = 0; a[n_] := (n - 1)*a[n - 1] + a[n - 2]; Table[ a[n], {n, 0, 21}] (from Robert G. Wilson v Feb 24 2005)
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PROGRAM
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(PARI) a(n)=contfracpnqn(vector(abs(n), i, i))[2, 2] /* Michael Somos Sep 25 2005 */
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CROSSREFS
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A column of A058294.
The square roots of the terms of A144656.
See also the constant in A060997.
Cf. A121554.
Sequence in context: A154259 A066114 A088128 this_sequence A124552 A020045 A020135
Adjacent sequences: A001050 A001051 A001052 this_sequence A001054 A001055 A001056
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KEYWORD
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easy,nonn,nice,frac
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 19 2000
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