A257286 -id:A257286 - cp's OEIS frontend

A088924 Number of "9ish numbers" with n digits.

1, 18, 252, 3168, 37512, 427608, 4748472, 51736248, 555626232, 5900636088, 62105724792, 648951523128, 6740563708152, 69665073373368, 716985660360312, 7352870943242808, 75175838489185272, 766582546402667448

Offset: 1

Marc LeBrun, Oct 23 2003

First difference of A016189. ("9" can be replaced by any other nonzero digit, however only the 9ish numbers are closed under lunar multiplication.)

See A257285 - A257289 for first differences of 5^n-4^n, ..., 9^n-8^n. These also give the number of n-digit numbers whose largest digit is 5, 6, 7, 8, respectively. - M. F. Hasler, May 04 2015

			a(2) = 18 because 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98 and 99 are the eighteen two-digit 9ish numbers.

Stefano Spezia, Table of n, a(n) for n = 1..950
D. Applegate, C program for lunar arithmetic and number theory [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for linear recurrences with constant coefficients, signature (19,-90).

Cf. A016189, A011539, A257285, A257286, A257287, A257288, A257289.

[9*10^(n-1) - 8*9^(n-1): n in [1..30]]; // Vincenzo Librandi, May 04 2015

A088924:=n->9*10^(n-1) - 8*9^(n-1); seq(A088924(n), n=1..30); # Wesley Ivan Hurt, May 15 2014

Series[(x (1 - x))/(1 - 19 x + 90 x^2), {x, 0, 10}] (* Bobby Milazzo, May 02 2014 *)
Table[9*10^(n - 1) - 8*9^(n - 1), {n, 30}] (* Wesley Ivan Hurt, May 15 2014 *)

a(n)=9*10^n-8*9^n \\ M. F. Hasler, May 04 2015

def A088924(n): return 9*10**(n-1)-8*9**(n-1) # Chai Wah Wu, Jan 27 2025

a(n) = 9*10^(n-1) - 8*9^(n-1).
G.f.: x*(1 - x)/(1 - 19*x + 90*x^2). - Bobby Milazzo, May 02 2014
a(n) = 19*a(n-1) - 90*a(n-2). - Vincenzo Librandi, May 04 2015
E.g.f.: (81*exp(10*x) - 80*exp(9*x) - 1)/90. - Stefano Spezia, Nov 16 2023

A255463 a(n) = 34^n - 23^n.

Original entry on oeis.org

1, 6, 30, 138, 606, 2586, 10830, 44778, 183486, 747066, 3027630, 12228618, 49268766, 198137946, 795740430, 3192527658, 12798808446, 51281327226, 205383589230, 822309197898, 3291561314526, 13173218826906, 52713796014030, 210917946175338, 843860071059006, 3376005143308986, 13505715150454830

Offset: 0

N. J. A. Sloane and Doron Zeilberger, Feb 23 2015

a(n-1) is also the number of n-digit numbers whose largest decimal digit is 3. - Stefano Spezia, Nov 15 2023

Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (7,-12).

Cf. A255462.

First differences of 4^n - 3^n = A005061(n). See A257285, A257286, A257287, A257288, A257289 for first differences of 5^n - 4^n, ..., 9^n - 8^n. - M. F. Hasler, May 04 2015

[3*4^n-2*3^n: n in [0..30]]; // Vincenzo Librandi, May 04 2015

Table[3 4^n - 2 3^n, {n, 0, 30}] (* Vincenzo Librandi, May 04 2015 *)

a(n)=3*4^n-2*3^n \\ M. F. Hasler, May 04 2015

G.f.: (1-x)/((1-3*x)*(1-4*x)).
a(n+1) = 7*a(n) - 12*a(n-1) with a(0)=1, a(1)=6.
a(n) = A255462(2^n-1).
E.g.f.: exp(3*x)*(3*exp(x) - 2). - Stefano Spezia, Nov 15 2023

Simpler definition from N. J. A. Sloane, Mar 10 2015

A257285 a(n) = 45^n - 34^n.

Original entry on oeis.org

1, 8, 52, 308, 1732, 9428, 50212, 263348, 1365892, 7026068, 35916772, 182729588, 926230852, 4681485908, 23608756132, 118849087028, 597466660612, 3000218204948, 15052630632292, 75469311591668, 378171191679172, 1894154493279188, 9483966605929252

Offset: 0

M. F. Hasler, May 03 2015

First differences of 5^n - 4^n = A005060.

a(n-1) is the number of numbers with n digits having the largest digit equal to 4. Note that this is independent of the base b>4. Equivalently, number of n-letter words over a 5-letter alphabet {a,b,c,d,e}, which must not start with the first letter of the alphabet, and in which the last letter of the alphabet must appear.

Index entries for linear recurrences with constant coefficients, signature (9,-20).

Cf. A005060. See also A000225, A027649, A255463, A257286 - A257289 and A088924.

[4*5^n-3*4^n: n in [0..30]]; // Vincenzo Librandi, May 04 2015

Table[4 5^n - 3 4^n, {n, 0, 30}] (* Vincenzo Librandi, May 04 2015 *)

PARI
```
a(n)=4*5^n-3*4^n
```

From Vincenzo Librandi, May 04 2015: (Start)
G.f.: (1-x)/((1-4*x)*(1-5*x)).
a(n) = 9*a(n-1) - 20*a(n-2). - (End)
E.g.f.: exp(4*x)*(4*exp(x) - 3). - Stefano Spezia, Nov 15 2023

A257287 a(n) = 67^n - 56^n.

Original entry on oeis.org

1, 12, 114, 978, 7926, 61962, 472614, 3541578, 26190726, 191733162, 1392520614, 10049975178, 72163811526, 516030592362, 3677517616614, 26134444136778, 185292033880326, 1311149786699562, 9262681804120614, 65346572412186378

Offset: 0

M. F. Hasler, May 03 2015

First differences of 7^n - 6^n = A016169.
a(n-1) is the number of numbers with n digits having the largest digit equal to 6. Note that this is independent of the base b > 6.
Equivalently, number of n-letter words over a 7-letter alphabet {a,b,c,d,e,f,g}, which must not start with the first letter of the alphabet, and in which the last letter of the alphabet must appear.

Index entries for linear recurrences with constant coefficients, signature (13,-42).

Cf. A016169.

See also A000225, A027649, A255463, A257285, A257286, A257287, A257288, A257289, and A088924.

[6*7^n-5*6^n: n in [0..30]]; // Vincenzo Librandi, May 04 2015

Table[6 7^n - 5 6^n, {n, 0, 30}] (* Vincenzo Librandi, May 04 2015 *)
LinearRecurrence[{13,-42},{1,12},20] (* Harvey P. Dale, Dec 10 2023 *)

PARI
```
a(n)=6*7^n-5*6^n
```

From Vincenzo Librandi, May 04 2015: (Start)
G.f.: (1-x)/((1-6*x)*(1-7*x)).
a(n) = 13*a(n-1) - 42*a(n-2). (End)
E.g.f.: exp(6*x)*(6*exp(x) - 5). - Stefano Spezia, Nov 15 2023

A257289 a(n) = 89^n - 78^n.

Original entry on oeis.org

1, 16, 200, 2248, 23816, 243016, 2416520, 23583688, 226933256, 2159839816, 20378082440, 190918934728, 1778399954696, 16486635929416, 152228014061960, 1400838452135368, 12853836673840136, 117654854901535816, 1074656292809619080, 9798007424852945608

Offset: 0

M. F. Hasler, May 03 2015

First differences of 9^n - 8^n = A016185.
a(n-1) is the number of numbers with n digits having the largest digit equal to 8. Note that this is independent of the base b > 8.
Equivalently, number of n-letter words over a 9-letter alphabet, which must not start with the last letter of the alphabet, and in which the first letter of the alphabet must appear.

Index entries for linear recurrences with constant coefficients, signature (17,-72).

Cf. A016185.

See also A000225, A027649, A255463, A257285, A257286, A257287, A257288, and A088924.

[8*9^n-7*8^n: n in [0..20]]; // Vincenzo Librandi, May 04 2015

Table[8 9^n - 7 8^n, {n, 0, 20}] (* Vincenzo Librandi, May 04 2015 *)
LinearRecurrence[{17,-72},{1,16},30] (* Harvey P. Dale, May 26 2019 *)

PARI
```
a(n)=8*9^n-7*8^n
```

[8*9^n-7*8^n for n in (0..20)] # Bruno Berselli, May 04 2015

G.f.: (1-x)/((1-8*x)*(1-9*x)). - Vincenzo Librandi, May 04 2015

E.g.f.: exp(8*x)*(8*exp(x) - 7). - Stefano Spezia, Nov 15 2023

A125373 Number of base 10 circular n-digit numbers with adjacent digits differing by 5 or less.

Original entry on oeis.org

1, 10, 80, 580, 4660, 37960, 311378, 2559658, 21057948, 173287588, 1426133270, 11737272106, 96600478510, 795047628502, 6543462720560, 53854541701240, 443238127915788, 3647975524214452, 30023874009147704, 247105006940966092

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+F(5) for base>=5.int(n/2)+1 and F(d) is the largest coefficient in (1+x+...+x^(2d))^n

OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (11, -21, -19, 34, 8, -15, -1, 2).

LinearRecurrence[{11,-21,-19,34,8,-15,-1,2},{1,10,80,580,4660,37960,311378,2559658,21057948},30] (* Harvey P. Dale, May 14 2018 *)

G.f.: (1 - x - 9*x^2 - 71*x^3 + 116*x^4 + 52*x^5 - 87*x^6 - 9*x^7 + 16*x^8)/((1 + x)(1 - 3*x + x^3)(1 - 9*x + 6*x^2 + 3*x^3 - 2*x^4)). For n<4, a(n) = 5*6^n-4*5^n = A257286(n). - M. F. Hasler, May 03 2015

cp's OEIS Frontend

A088924 Number of "9ish numbers" with n digits.

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A255463 a(n) = 3*4^n - 2*3^n.

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A257285 a(n) = 4*5^n - 3*4^n.

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A257287 a(n) = 6*7^n - 5*6^n.

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A257289 a(n) = 8*9^n - 7*8^n.

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A125373 Number of base 10 circular n-digit numbers with adjacent digits differing by 5 or less.

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Mathematica

Formula

A255463 a(n) = 34^n - 23^n.

A257285 a(n) = 45^n - 34^n.

A257287 a(n) = 67^n - 56^n.

A257289 a(n) = 89^n - 78^n.