A255463 -id:A255463 - cp's OEIS frontend

A005061 a(n) = 4^n - 3^n.

0, 1, 7, 37, 175, 781, 3367, 14197, 58975, 242461, 989527, 4017157, 16245775, 65514541, 263652487, 1059392917, 4251920575, 17050729021, 68332056247, 273715645477, 1096024843375, 4387586157901, 17560804984807, 70274600998837, 281192547174175, 1125052618233181

Offset: 0

N. J. A. Sloane

Number of 2 X n binary arrays with a path of adjacent 1's from top row to bottom row, see A359576. - R. H. Hardin, Mar 21 2002
Number of binary vectors (x_1, x_2, ..., x_{2n}) such that in at least one of the disjoint pairs (x_1, x_2), (x_3, x_4), ..., (x_{2n-1}, x_{2n}) both x_{2i-1} and x_{2i} are both 1. Equivalently, number of solutions (x_1, ..., x_n) to the equation x_1*x_2 + x_3*x_4 + x_5*x_6 + ... +x_{2n-1}*x_{2n} = 1 in base-2 lunar arithmetic. - N. J. A. Sloane, Apr 23 2011
a(n)/4^n is the probability that two randomly selected (with replacement) subsets of [n] will have at least one element in common if the probability of selection is equal for all subsets. - Geoffrey Critzer, May 09 2009
This sequence is also the second column of the Sheffer triangle A143495 (3-restricted Stirling2 numbers). (See the e.g.f. given below.) - Wolfdieter Lang, Oct 08 2011
Also, the number of numbers with at most n digits whose largest digit equals 3. See A255463 for the first differences (i.e., ...with exactly n digits...). - M. F. Hasler, May 03 2015
If 2^k | n then a(2^k) | a(n). - Bernard Schott, Oct 08 2020
a(n) is the number of ordered n-tuples with elements from {0,1,2,3} in which any of these elements, say 0, appears at least once. For example, a(2)=7 since 01,10,02,20,03,30,00 are the ordered 2-tuples that contain 0. - Enrique Navarrete, Apr 05 2021
a(n) is the number of n-digit numbers whose smallest decimal digit is 6. - Stefano Spezia, Nov 15 2023

			G.f. = x + 7*x^2 + 37*x^3 + 175*x^4 + 781*x^5 + 3367*x^6 + 14197*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..300
David Applegate, Marc LeBrun, and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Dominique Désérable, Versatile Topology for Two-Dimensional Cellular Automata, Advances in Cellular Automata, Emergence, Complexity and Computation (ECC Vol 52) Springer, Cham (2025), Ch. 6, pp. 151-186.
John Elias, Illustration of initial terms: Unfolded Sierpinski triangle or 3-branches tree in square configuration
Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.
Vladeta Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (Russian, translated in Discrete Mathematics and Applications, 9, (1999), no. 6).
Eric Weisstein's World of Mathematics, Power Fractional Parts
Index entries for linear recurrences with constant coefficients, signature (7,-12).
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A001047, A002250, A005060, A005062, A143495, A255463 (first differences), A359576.

Array column A047969(n-1, 3), or triangle's subdiagonal A047969(n+2, n-1), for n >= 1.

List([0..10^2], n->4*n - 3^n); # Muniru A Asiru, Feb 06 2018

[4^n - 3^n: n in [0..25]]; // Vincenzo Librandi, Jun 03 2011

seq(4^n - 3^n, n=0..10^2); # Muniru A Asiru, Feb 06 2018

Table[4^n - 3^n, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{7,-12},{0,1},30] (* Harvey P. Dale, May 04 2012 *)
Table[Numerator[1-(3/4)^n],{n,0,20}] (* see link Wolfram Mathworld - Fred Daniel Kline, Feb 05 2018 *)

a(n)=1<<(n+n)-3^n \\ Charles R Greathouse IV, Jun 16 2011

def a(n): return 4**n - 3**n
print([a(n) for n in range(23)]) # Michael S. Branicky, Sep 01 2021

a(n) = 4*a(n-1) + 3^(n-1) for n>=1. - Xavier Acloque, Oct 20 2003
Binomial transform of A001047. - Ross La Haye, Sep 17 2005
From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-4*x)-1/(1-3*x).
E.g.f.: exp(4*x)-exp(3*x). (End)
a(n) = 2^n * Sum_{i=0...n} binomial(n,i)*(2^i-1)/2^i. - Geoffrey Critzer, May 09 2009
a(n) = 7*a(n-1) - 12*a(n-2) for n>=2. - Bruno Berselli, Jan 25 2011
From Joe Slater, Jan 15 2017: (Start)
a(n) = 3*a(n-1) + 4^(n-1) for n>=0.
a(n+1) = Sum_{k=0..n} 4^(n-k) * 3^k. (End)
a(n) = -a(-n) * 12^n for all n in Z. - Michael Somos, Jan 22 2017

A257286 a(n) = 56^n - 45^n.

Original entry on oeis.org

1, 10, 80, 580, 3980, 26380, 170780, 1087180, 6835580, 42575980, 263268380, 1618672780, 9907349180, 60420657580, 367406757980, 2228854610380, 13495197974780, 81581539411180, 492540994279580, 2970504754739980, 17899322473752380

Offset: 0

M. F. Hasler, May 03 2015

First differences of 6^n - 5^n = A005062.

a(n-1) is the number of numbers with n digits having the largest digit equal to 5. Or, equivalently, number of n-letter words over a 6-letter alphabet {a,b,c,d,e,f}, 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 (11,-30).

Cf. A005062.
Coincides with A125373 only for the first terms.
See also A000225, A027649, A255463, A257285, A257286, A257287, A257288, A257289 and A088924.

[5*6^n-4*5^n: n in [0..20]]; // Vincenzo Librandi, May 04 2015

Table[5 6^n - 4 5^n, {n, 0, 30}] (* Vincenzo Librandi, May 04 2015 *)

PARI
```
a(n)=5*6^n-4*5^n
```

a(n) = 11 a(n-1) - 30 a(n-2).
G.f.: (1-x)/((1-5*x)*(1-6*x)). - Vincenzo Librandi, May 04 2015
E.g.f.: exp(5*x)*(5*exp(x) - 4). - Stefano Spezia, Nov 15 2023

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

A255462 Number of ON cells after n generations of the odd-rule cellular automaton defined by OddRule 365 when started with a single ON cell.

Original entry on oeis.org

1, 6, 6, 30, 6, 36, 30, 138, 6, 36, 36, 180, 30, 180, 138, 606, 6, 36, 36, 180, 36, 216, 180, 828, 30, 180, 180, 900, 138, 828, 606, 2586, 6, 36, 36, 180, 36, 216, 180, 828, 36, 216, 216, 1080, 180, 1080, 828, 3636, 30, 180, 180, 900, 180, 1080, 900, 4140, 138, 828, 828, 4140

Offset: 0

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

			From _Omar E. Pol_, Sep 08 2016: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
6;
6, 30;
6, 36, 30, 138;
6, 36, 36, 180, 30, 180, 138, 606;
6, 36, 36, 180, 36, 216, 180, 828, 30, 180, 180, 900, 138, 828, 606, 2586;
...
Right border gives A255463. (End)

Paul Tek, Table of n, a(n) for n = 0..10000
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
N. J. A. Sloane, Illustration of generations 0 to 15
N. J. A. Sloane, Illustration of generations 0 to 35
N. J. A. Sloane, Illustration of generation 7
N. J. A. Sloane, Illustration of generation 15
N. J. A. Sloane, Mathematica notebook to generate this cellular automaton
Index entries for sequences related to cellular automata

Run length transform of A255463.

(* See Mathematica notebook in link *)
(* or *)
A255462[n_] := Total[CellularAutomaton[{42, {2, {{0, 1, 1}, {1, 1, 0}, {1, 0, 1}}}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 2]; Array[A255462, 60, 0] (* JungHwan Min, Sep 06 2016 *)
A255462L[n_] := Total[#, 2] & /@ CellularAutomaton[{42, {2, {{0, 1, 1}, {1, 1, 0}, {1, 0, 1}}}, {1, 1}}, {{{1}}, 0}, n]; A255462L[59] (* JungHwan Min, Sep 06 2016 *)

It follows from Theorem 3 of the Fredkin.pdf (2015) paper that this satisfies the recurrence a(2t)=a(t), a(4t+1)=6*a(t), and a(4t+3)=7*a(2t+1)-12*a(t) for t>0, with a(0)=1. - N. J. A. Sloane, Mar 10 2015

A210381 Triangle by rows, derived from the beheaded Pascal's triangle, A074909.

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 1, 3, 4, 0, 1, 4, 6, 5, 0, 1, 5, 10, 10, 6, 0, 1, 6, 15, 20, 15, 7, 0, 1, 7, 21, 35, 35, 21, 8, 0, 1, 8, 28, 56, 70, 56, 28, 9, 0, 1, 9, 36, 84, 126, 126, 84, 36, 10, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 11

Offset: 0

Roger L. Bagula and Gary W. Adamson, Mar 20 2012

Row sums of the triangle = 2^n.
Let the triangle = an infinite lower triangular matrix, M.  Then M * The Bernoulli numbers, A027641/A027642  as a vector V = [1, -1, 0, 0, 0,...].  M * the Bernoulli sequence variant starting [1, 1/2, 1/6,...] = [1, 1, 1,...].  M * 2^n: [1, 2, 4, 8,...] = A027649.  M * 3^n = A255463; while M * [1, 2, 3,...] = A047859, and M * A027649 = A027650.
Row sums of powers of the triangle generate the Poly-Bernoulli number sequences shown in the array of A099594. - Gary W. Adamson, Mar 21 2012
Triangle T(n,k) given by (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

			{1},
{0, 2},
{0, 1, 3},
{0, 1, 3, 4},
{0, 1, 4, 6, 5},
{0, 1, 5, 10, 10, 6},
{0, 1, 6, 15, 20, 15, 7},
{0, 1, 7, 21, 35, 35, 21, 8},
{0, 1, 8, 28, 56, 70, 56, 28, 9},
{0, 1, 9, 36, 84, 126, 126, 84, 36, 10},
{0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 11}
...

Konrad Knopp, Elements of the Theory of Functions, Dover, 1952,pp 117-118.

Cf. A074909, A255463, A026749, A047859, A027650

Cf. A099594.

t2[n_, m_] = If[m - 1 <= n, Binomial[n, m - 1], 0];
O2 = Table[Table[If[n == m, t2[n, m] + 1, t2[n, m]], {m, 0, n}], {n, 0, 10}];
Flatten[O2]

Partial differences of the beheaded Pascal's triangle A074909 starting from the top, by columns.
G.f.: (1-x)/(1-x-2*y*x+y*x^2+y^2*x^2). - Philippe Deléham, Mar 25 2012
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(2,1) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,2) = 3 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2012

cp's OEIS Frontend

A005061 a(n) = 4^n - 3^n.

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A257286 a(n) = 5*6^n - 4*5^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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Magma

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

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A255462 Number of ON cells after n generations of the odd-rule cellular automaton defined by OddRule 365 when started with a single ON cell.

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Mathematica

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A210381 Triangle by rows, derived from the beheaded Pascal's triangle, A074909.

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A257286 a(n) = 56^n - 45^n.

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

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

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