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 A002714 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) 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 REFERENCES 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). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 C. A. Coulson, How Many different Keys?, Math. Gaz. vol 53 no 383 (1969), 7-13. C. A. Coulson, How many different keys?, Math. Gaz. vol 53 no 383 (1969), 7-13. [Annotated scanned copy] Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,1). FORMULA 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 MAPLE 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 MATHEMATICA 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 *) PROG (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))) ) /* Joerg Arndt, Aug 13 2012 */ CROSSREFS Sequence in context: A155272 A092053 A072630 * A126361 A069005 A000413 Adjacent sequences:  A002711 A002712 A002713 * A002715 A002716 A002717 KEYWORD nonn AUTHOR EXTENSIONS Information added from A126361, offset changed to 0 by Joerg Arndt, Aug 13 2012 STATUS approved

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Last modified October 19 03:34 EDT 2019. Contains 328211 sequences. (Running on oeis4.)