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A002714
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Number of different keys with n cuts, depths between 1 and 7 and depth difference at most 1 between adjacent cut depths.
(Formerly M4366 N1832)
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5
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1, 7, 19, 53, 149, 421, 1193, 3387, 9627, 27383, 77923, 221805, 631469, 1797957, 5119593, 14578387, 41514003, 118218823, 336653331, 958698053, 2730124261, 7774706437, 22140438345, 63050541515, 179552587883, 511322221559
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OFFSET
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0,2
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COMMENTS
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Also number of base 7 n-digit numbers with adjacent digits differing by one or less.
[Empirical] a(base,n)=a(base-1,n)+3^(n-1) for base>=n; a(base,n)=a(base-1,n)+3^(n-1)-2 when base=n-1. - R. H. Hardin, Dec 26 2006
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REFERENCES
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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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FORMULA
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G.f.: (2*x^4 - 5*x^3 - 7*x^2 + 3*x + 1)/(-x^4 + 4*x^3 + 2*x^2 - 4*x + 1); (from the Knopfmacher et al. reference). - Joerg Arndt, Aug 10 2012
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MAPLE
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A002714:=-(7-9*z-9*z**2+3*z**3)/(-1+4*z-2*z**2-4*z**3+z**4); # conjectured by Simon Plouffe in his 1992 dissertation; correct up to offset
T := proc(d, n) option remember ; if n = 1 then 1; else if d = 7 then T(d, n-1)+T(d-1, n-1) ; elif d = 1 then T(d, n-1)+T(d+1, n-1) ; else T(d-1, n-1)+T(d, n-1)+T(d+1, n-1) ; fi ; fi ; end: A002714 := proc(n) local d ; add( T(d, n), d=1..7) ; end: seq(A002714(n), n=1..35) ; # R. J. Mathar, Jun 15 2008
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MATHEMATICA
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CoefficientList[Series[(2*x^4-5*x^3-7*x^2+3*x+1)/(-x^4+4*x^3+2*x^2-4*x+1), {x, 0, 200}], x] (* Vincenzo Librandi, Aug 13 2012 *)
Join[{1}, LinearRecurrence[{4, -2, -4, 1}, {7, 19, 53, 149}, 30]] (* Jean-François Alcover, Jan 07 2019 *)
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PROG
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(S/R) stvar $[N]:(0..M-1) init $[]:=0 asgn $[]->{*} kill +[i in 0..N-2](($[i]`-$[i+1]`>1)+($[i+1]`-$[i]`>1))
(PARI)
/* from the Knopfmacher et al. reference */
default(realprecision, 99); /* using floats */
sn(n, k)=1/n*sum(i=1, k, sumdiv(n, j, eulerphi(j)*(1+2*cos(i*Pi/(k+1)))^(n/j)));
vector(66, n, if (n==1, 1, round(sn(n-1, 7))) )
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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