A002714 Number of different keys with n cuts, depths between 1 and 7 and depth difference at most 1 between adjacent cut depths.

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, 1456121982755, 4146683677885

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,1).

Programs

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 *)

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 */

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

Extensions

Information added from A126361, offset changed to 0 by Joerg Arndt, Aug 13 2012