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.

A235163 Number of positive integers with n digits in which adjacent digits differ by at most 1.

Original entry on oeis.org

9, 26, 75, 217, 629, 1826, 5307, 15438, 44941, 130900, 381444, 1111926, 3242224, 9455987, 27583372, 80472698, 234799873, 685149328, 1999414181, 5835044495, 17029601028, 49702671494, 145066398937, 423412132499, 1235854038791, 3607255734629, 10529101874491
Offset: 1

Views

Author

Gerry Leversha, Jan 04 2014

Keywords

Examples

			a(2) = 26: 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 98, 99.

Links

Crossrefs

Cf. A032981, A090994, A126364 (allowing leading zeros).

Programs

  • Maple
    u:= proc(n, r) option remember; `if`(n=1, `if`(r=0, 0, 1),
      add(`if`(r+i in [$0..9], u(n-1, r+i), 0), i=-1..1))
    end:
a:= n-> add(u(n, r), r = 0..9):
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 12 2014
  • Mathematica
    CoefficientList[Series[-x*(3*x^4-18*x^3-9*x^2+28*x-9)/(x^5-6*x^4-x^3+10*x^2-6*x+1),{x,0,30}],x]//Rest (* Harvey P. Dale, Aug 13 2019 *)
  • Python
    from functools import cache
@cache
def u(n, r):
    if r < 0 or r > 9: return 0
    if n == 1: return (r > 0)
    return u(n-1, r-1) + u(n-1, r) + u(n-1, r+1)
def a(n): return sum(u(n, r) for r in range(10))
print([a(n) for n in range(1, 28)]) # Michael S. Branicky, Sep 26 2021

Formula

a(n) = Sum_{r=0..9} u(n,r) where u(n,r) = 0 if r<0 or r>9, u(1,0) = 0, u(1,r) = 1 for 1<=r<=9, and otherwise u(n,r) = u(n-1,r-1) + u(n-1,r) + u(n-1,r+1).
G.f.: -x*(3*x^4-18*x^3-9*x^2+28*x-9)/(x^5-6*x^4-x^3+10*x^2-6*x+1). - Alois P. Heinz, Jan 12 2014

A126397 Number of base 10 n-digit numbers with adjacent digits differing by two or less.

Original entry on oeis.org

1, 10, 44, 200, 918, 4236, 19598, 90790, 420870, 1951636, 9051480, 41983154, 194736668, 903293618, 4190003458, 19435777562, 90155141564, 418195731384, 1939853565942, 8998257693932, 41739569773726, 193614349514214
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] 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.

Crossrefs

Cf. Base 10 differing by one or less A126364.

Formula

Conjectures from Colin Barker, May 31 2017: (Start)
G.f.: (1 + 3*x - 15*x^2 + 2*x^3 + x^4) / (1 - 7*x + 11*x^2 - x^4).
a(n) = 7*a(n-1) - 11*a(n-2) + a(n-4) for n>4.
(End)

A126478 Number of base 10 n-digit numbers with adjacent digits differing by three or less.

Original entry on oeis.org

1, 10, 58, 350, 2130, 12990, 79258, 483646, 2951370, 18010366, 109906170, 670689902, 4092809194, 24975905246, 152412637946, 930080892110, 5675713495690, 34635399953406, 211358612597658, 1289792038765742
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 10 differing by two or less A126397, one or less A126364.

Formula

Conjectures from Colin Barker, May 31 2017: (Start)
G.f.: (1 + 3*x - 8*x^2 - 6*x^3 + 6*x^4) / (1 - 7*x + 4*x^2 + 10*x^3 - 6*x^4).
a(n) = 7*a(n-1) - 4*a(n-2) - 10*a(n-3) + 6*a(n-4) for n>4.
(End)

A126505 Number of base 10 n-digit numbers with adjacent digits differing by four or less.

Original entry on oeis.org

1, 10, 70, 510, 3720, 27142, 198034, 1444904, 10542368, 76919662, 561224422, 4094828862, 29876859864, 217988781808, 1590498774316, 11604663002012, 84670422615182, 617775842744470, 4507441679052256, 32887382581680414
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

Base 10 differing by three or less A126478, two or less A126397, one or less A126364.

Formula

Conjectures from Colin Barker, May 31 2017: (Start)
G.f.: (1 + 3*x - 3*x^2 - 4*x^3 + x^4 + x^5) / (1 - 7*x - 3*x^2 + 6*x^3 + x^4 - x^5).
a(n) = 7*a(n-1) + 3*a(n-2) - 6*a(n-3) - a(n-4) + a(n-5) for n>5.
(End)

A126531 Number of base 10 n-digit numbers with adjacent digits differing by five or less.

Original entry on oeis.org

1, 10, 80, 660, 5430, 44692, 367826, 3027314, 24915652, 205062892, 1687725824, 13890462738, 114322452324, 940906242802, 7743925534706, 63734705924048, 524554726284436, 4317234336897124, 35532064407664766, 292438978880594940
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.

Crossrefs

Cf. Base 10 differing by four or less A126505, three or less A126478, two or less A126397, one or less A126364.

Formula

Conjectures from Colin Barker, May 31 2017: (Start)
G.f.: (1 - x)*(1 + 3*x - x^3) / ((1 + x)*(1 - 9*x + 6*x^2 + 3*x^3 - 2*x^4)).
a(n) = 8*a(n-1) + 3*a(n-2) - 9*a(n-3) - a(n-4) + 2*a(n-5) for n>4.
(End)
Showing 1-5 of 5 results.

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