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; 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).
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
Information added from A126361, offset changed to 0 by Joerg Arndt, Aug 13 2012
STATUS
approved