A126475 -id:A126475 - cp's OEIS frontend

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

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

Offset: 0

R. H. Hardin, Dec 28 2006

[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

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,5).

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

LinearRecurrence[{6, 5}, {1, 7}, 25] (* Paolo Xausa, Aug 08 2024 *)

Vec((1+x)/(1-6*x-5*x^2) + O(x^30)) \\ Colin Barker, Sep 08 2016

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

A201042 T(n,k)=Number of -k..k arrays of n elements with adjacent element differences also in -k..k.

Original entry on oeis.org

3, 5, 7, 7, 19, 17, 9, 37, 75, 41, 11, 61, 203, 295, 99, 13, 91, 429, 1111, 1161, 239, 15, 127, 781, 3011, 6083, 4569, 577, 17, 169, 1287, 6691, 21141, 33305, 17981, 1393, 19, 217, 1975, 13021, 57343, 148433, 182349, 70763, 3363, 21, 271, 2873, 23045, 131781

Offset: 1

R. H. Hardin Nov 26 2011

Table starts
....3.......5........7.........9.........11..........13..........15
....7......19.......37........61.........91.........127.........169
...17......75......203.......429........781........1287........1975
...41.....295.....1111......3011.......6691.......13021.......23045
...99....1161.....6083.....21141......57343......131781......268983
..239....4569....33305....148433.....491429.....1333683.....3139529
..577...17981...182349...1042167....4211559....13497523....36644243
.1393...70763...998383...7317185...36093157...136601483...427707523
.3363..278483..5466269..51374875..309319197..1382473365..4992154799
.8119.1095951.29928491.360709449.2650872719.13991301963.58267877227

			Some solutions for n=4 k=7
.-5...-1....2....2...-3....4...-4....4....5....2...-6...-1....1....4....2....0
.-3....0....3....1....2....4...-3....4....2...-5....1....6....5....7....4....0
.-5...-5...-4...-3....2...-2....1....5....7...-7....0....2....4....1....1....2
.-7...-1....2...-6....1...-1...-5....7....0...-1...-5....6...-3....5...-1....2

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A001333(n+1)
Column 2 is A126392
Column 3 is A126475
Column 4 is A126504
Column 5 is A126532
Row 1 is A004273(n+1)
Row 2 is A003215
Row 3 is A063494(n+1)

Empirical for columns:
k=1: a(n) = 2*a(n-1) +a(n-2)
k=2: a(n) = 4*a(n-1) -a(n-3)
k=3: a(n) = 5*a(n-1) +3*a(n-2) -2*a(n-3) -a(n-4)
k=4: a(n) = 7*a(n-1) +a(n-2) -6*a(n-3) +a(n-5)
k=5: a(n) = 8*a(n-1) +6*a(n-2) -9*a(n-3) -5*a(n-4) +2*a(n-5) +a(n-6)
k=6: a(n) = 10*a(n-1) +3*a(n-2) -18*a(n-3) -a(n-4) +8*a(n-5) -a(n-7)
k=7: a(n) = 11*a(n-1) +10*a(n-2) -24*a(n-3) -15*a(n-4) +13*a(n-5) +7*a(n-6) -2*a(n-7) -a(n-8)
Empirical for rows:
n=1: a(k) = 2*k + 1
n=2: a(k) = 3*k^2 + 3*k + 1
n=3: a(k) = (14/3)*k^3 + 7*k^2 + (13/3)*k + 1
n=4: a(k) = (29/4)*k^4 + (29/2)*k^3 + (51/4)*k^2 + (11/2)*k + 1
n=5: a(k) = (169/15)*k^5 + (169/6)*k^4 + 32*k^3 + (119/6)*k^2 + (101/15)*k + 1
n=6: a(k) = (2101/120)*k^6 + (2101/40)*k^5 + (1753/24)*k^4 + (1405/24)*k^3 + (569/20)*k^2 + (119/15)*k + 1
n=7: a(k) = (17141/630)*k^7 + (17141/180)*k^6 + (28177/180)*k^5 + (2759/18)*k^4 + (17299/180)*k^3 + (6929/180)*k^2 + (1921/210)*k + 1

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

R. H. Hardin, Dec 28 2006

[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.

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

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)

cp's OEIS Frontend

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

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A201042 T(n,k)=Number of -k..k arrays of n elements with adjacent element differences also in -k..k.

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

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