cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A126528 Number of base 7 n-digit numbers with adjacent digits differing by five or less.

Original entry on oeis.org

1, 7, 47, 317, 2137, 14407, 97127, 654797, 4414417, 29760487, 200635007, 1352612477, 9118849897, 61476161767, 414451220087, 2794088129357, 18836784876577, 126991149906247, 856130823820367, 5771740692453437, 38911098273822457, 262325293105201927
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+11^(n-1) for base>=5n-4; a(base,n)=a(base-1,n)+11^(n-1)-2 when base=5n-5.
For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,6} containing no subwords 00 and 11. - Milan Janjic, Jan 31 2015

Links

Crossrefs

Cf. Base 7 differing by four or less A126502, three or less A126475, two or less A126394, one or less A126361.

Programs

  • Mathematica
    LinearRecurrence[{6, 5}, {1, 7}, 25] (* Paolo Xausa, Aug 08 2024 *)
  • PARI
    Vec((1+x)/(1-6*x-5*x^2) + O(x^30)) \\ Colin Barker, Sep 08 2016

Formula

From Philippe Deléham, Mar 24 2012: (Start)
G.f.: (1+x)/(1-6*x-5*x^2).
a(n) = 6*a(n-1) + 5*a(n-2), a(0) = 1, a(1) = 7 .
a(n) = Sum_{k=0..=n} A054458(n,k)*4^k.
(End)
a(n) = A091928(n+1)/5. - Philippe Deléham, Mar 27 2012
a(n) = (((3-sqrt(14))^n * (-4+sqrt(14)) + (3+sqrt(14))^n * (4+sqrt(14)))) / (2*sqrt(14)). - Colin Barker, Sep 08 2016

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

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

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

A126394 Number of base 7 n-digit numbers with adjacent digits differing by two or less.

Original entry on oeis.org

1, 7, 29, 125, 543, 2363, 10287, 44787, 194995, 848979, 3696331, 16093291, 70067867, 305065387, 1328210699, 5782837835, 25177641963, 109619822123, 477268896747, 2077959947307, 9047137938475, 39389933855019
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

a(base,n)=a(base-1,n)+5^(n-1) for base>=2n-1; a(base,n)=a(base-1,n)+5^(n-1)-2 when base=2n-2.

Links

Crossrefs

Cf. Base 7 differing by one or less A126361.

Formula

G.f.: 1 -x*(7-6*x-6*x^2+4*x^3) / ( (x-1)*(2*x^3-2*x^2-4*x+1) ). - R. J. Mathar, Jun 06 2013

A126475 Number of base 7 n-digit numbers with adjacent digits differing by three or less.

Original entry on oeis.org

1, 7, 37, 203, 1111, 6083, 33305, 182349, 998383, 5466269, 29928491, 163862147, 897165287, 4912089625, 26894291201, 147249532159, 806209189861, 4414093873755, 24167703582839, 132321131623579, 724474371929041
Offset: 0

Views

Author

R. H. Hardin, Dec 27 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+7^(n-1) for base>=3n-2; a(base,n)=a(base-1,n)+7^(n-1)-2 when base=3n-3.

Crossrefs

Cf. Base 7 differing by two or less A126394, one or less A126361.

Formula

Conjectures from Colin Barker, Jun 01 2017: (Start)
G.f.: (1 + 2*x - x^2 - x^3) / (1 - 5*x - 3*x^2 + 2*x^3 + x^4).
a(n) = 5*a(n-1) + 3*a(n-2) - 2*a(n-3) - a(n-4) for n>3.
(End)

A126502 Number of base 7 n-digit numbers with adjacent digits differing by four or less.

Original entry on oeis.org

1, 7, 43, 269, 1679, 10483, 65449, 408623, 2551187, 15928021, 99444631, 620870267, 3876326801, 24201367447, 151098247483, 943363239389, 5889771828959, 36772062710083, 229581830200249, 1433365791134783
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+9^(n-1) for base>=4n-3; a(base,n)=a(base-1,n)+9^(n-1)-2 when base=4n-4.

Crossrefs

Cf. Base 7 differing by three or less A126475, two or less A126394, one or less A126361.

Formula

Conjectures from Colin Barker, Jun 01 2017: (Start)
G.f.: (1 + x - x^2) / (1 - 6*x - 2*x^2 + 3*x^3).
a(n) = 6*a(n-1) + 2*a(n-2) - 3*a(n-3) for n>2.
(End)
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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