A143461 -id:A143461 - cp's OEIS frontend

A143454 Expansion of 1/(x^k(1 - x - 3x^(k+1))) for k=3.

1, 4, 7, 10, 13, 25, 46, 76, 115, 190, 328, 556, 901, 1471, 2455, 4123, 6826, 11239, 18604, 30973, 51451, 85168, 140980, 233899, 388252, 643756, 1066696, 1768393, 2933149, 4864417, 8064505, 13369684, 22169131, 36762382, 60955897, 101064949, 167572342

Offset: 0

Alois P. Heinz, Aug 16 2008

a(n) is also the number of length n quaternary words with at least 3 0-digits between any other digits.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 7, 4*a(n-7) equals the number of 4-colored compositions of n with all parts >= 4, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 9.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,3).

3rd column of A143461.

[n le 4 select 3*n-2 else Self(n-1) +3*Self(n-4): n in [1..51]]; // G. C. Greubel, May 08 2021

a:= proc(k::nonnegint) local n,i,j; if k=0 then unapply(4^n,n) else unapply((Matrix(k+1, (i,j)-> if (i=j-1) or j=1 and i=1 then 1 elif j=1 and i=k+1 then 3 else 0 fi)^(n+k))[1,1], n) fi end(3): seq(a(n), n=0..50);

a[n_]:= Sum[3^j*Binomial[n-3*j+3, j], {j, 0, (n+3)/3}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 04 2014, after Vladimir Kruchinin *)
LinearRecurrence[{1,0,0,3}, {1,4,7,10}, 41] (* G. C. Greubel, May 08 2021 *)

a(n):= sum(3^j*binomial(n-3*j+3,j), j,0,(n+3)/3); /* Vladimir Kruchinin, May 24 2011 */

Vec(1/(x^3*(1-x-3*x^4))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

my(p=Mod('x,'x^4-'x^3-3)); a(n) = vecsum(Vec(lift(p^(n+3)))); \\ Kevin Ryde, May 11 2021

def a(n): return 3*n+1 if (n<4) else a(n-1) + 3*a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, May 08 2021

G.f.: (1 + 3*x + 3*x^2 + 3*x^3) / (1 - x - 3*x^4). - R. J. Mathar, Aug 04 2019

a(n) = Sum_{j=0..(n+3)/3} 3^j*C(n-3*j+3,j). - Vladimir Kruchinin, May 24 2011

A143455 Expansion of 1/(x^k(1-x-3x^(k+1))) for k=4.

Original entry on oeis.org

1, 4, 7, 10, 13, 16, 28, 49, 79, 118, 166, 250, 397, 634, 988, 1486, 2236, 3427, 5329, 8293, 12751, 19459, 29740, 45727, 70606, 108859, 167236, 256456, 393637, 605455, 932032, 1433740, 2203108, 3384019, 5200384, 7996480, 12297700, 18907024

Offset: 0

Alois P. Heinz, Aug 16 2008

a(n) is also the number of length n quaternary words with at least 4 0-digits between any other digits.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=9, 4*a(n-9) equals the number of 4-colored compositions of n with all parts >=5, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,3).

4th column of A143461.

a:= proc(k::nonnegint) local n,i,j; if k=0 then unapply(4^n,n) else unapply((Matrix(k+1, (i,j)-> if(i=j-1) or j=1 and i=1 then 1 elif j=1 and i=k+1 then 3 else 0 fi)^(n+k))[1,1], n) fi end(4): seq(a(n), n=0..50);

Series[1/(1-x-3*x^5), {x, 0, 50}] // CoefficientList[#, x]& // Drop[#, 4]& (* Jean-François Alcover, Feb 13 2014 *)

G.f.: 1/(x^4*(1-x-3*x^5)).

A143456 Expansion of 1/(x^k(1-x-3x^(k+1))) for k=5.

Original entry on oeis.org

1, 4, 7, 10, 13, 16, 19, 31, 52, 82, 121, 169, 226, 319, 475, 721, 1084, 1591, 2269, 3226, 4651, 6814, 10066, 14839, 21646, 31324, 45277, 65719, 95917, 140434, 205372, 299344, 435175, 632332, 920083, 1341385, 1957501, 2855533, 4161058, 6058054

Offset: 0

Alois P. Heinz, Aug 16 2008

a(n) is also the number of length n quaternary words with at least 5 0-digits between any other digits.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,3).

5th column of A143461.

a:= proc(k::nonnegint) local n,i,j; if k=0 then unapply(4^n,n) else unapply((Matrix(k+1, (i,j)-> if (i=j-1) or j=1 and i=1 then 1 elif j=1 and i=k+1 then 3 else 0 fi)^(n+k))[1,1], n) fi end(5): seq(a(n), n=0..52);

Series[1/(1-x-3*x^6), {x, 0, 52}] // CoefficientList[#, x]& // Drop[#, 5]& (* Jean-François Alcover, Feb 13 2014 *)

G.f.: 1/(x^5*(1-x-3*x^6)).

A143457 Expansion of 1/(x^k(1-x-3x^(k+1))) for k=6.

Original entry on oeis.org

1, 4, 7, 10, 13, 16, 19, 22, 34, 55, 85, 124, 172, 229, 295, 397, 562, 817, 1189, 1705, 2392, 3277, 4468, 6154, 8605, 12172, 17287, 24463, 34294, 47698, 66160, 91975, 128491, 180352, 253741, 356623, 499717, 698197, 974122, 1359595, 1900651

Offset: 0

Alois P. Heinz, Aug 16 2008

a(n) is also the number of length n quaternary words with at least 6 0-digits between any other digits.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,3).

6th column of A143461.

a := proc(k::nonnegint) local n,i,j; if k=0 then unapply (4^n,n) else unapply ((Matrix(k+1, (i,j)-> if (i=j-1) or j=1 and i=1 then 1 elif j=1 and i=k+1 then 3 else 0 fi)^(n+k))[1,1], n) fi end(6): seq (a(n), n=0..55);

Series[1/(1-x-3*x^7), {x, 0, 55}] // CoefficientList[#, x]& // Drop[#, 6]& (* Jean-François Alcover, Feb 13 2014 *)

G.f.: 1/(x^6*(1-x-3*x^7)).

A143458 Expansion of 1/(x^k(1-x-3x^(k+1))) for k=7.

Original entry on oeis.org

1, 4, 7, 10, 13, 16, 19, 22, 25, 37, 58, 88, 127, 175, 232, 298, 373, 484, 658, 922, 1303, 1828, 2524, 3418, 4537, 5989, 7963, 10729, 14638, 20122, 27694, 37948, 51559, 69526, 93415, 125602, 169516, 229882, 312964, 426808, 581485, 790063, 1070308, 1447114

Offset: 0

Alois P. Heinz, Aug 16 2008

a(n) is also the number of length n quaternary words with at least 7 0-digits between any other digits.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 3).

7th column of A143461.

a := proc(k::nonnegint) local n,i,j; if k=0 then unapply (4^n,n) else unapply ((Matrix(k+1, (i,j)-> if (i=j-1) or j=1 and i=1 then 1 elif j=1 and i=k+1 then 3 else 0 fi)^(n+k))[1,1], n) fi end(7): seq (a(n), n=0..60);

LinearRecurrence[{1,0,0,0,0,0,0,3},{1,4,7,10,13,16,19,22},50] (* Harvey P. Dale, Jul 22 2013 *)

G.f.: 1/(x^7*(1-x-3*x^8)).

a(0)=1, a(1)=4, a(2)=7, a(3)=10, a(4)=13, a(5)=16, a(6)=19, a(7)=22, a(n)=a(n-1)+3*a(n-8). - Harvey P. Dale, Jul 22 2013

A143459 Expansion of 1/(x^k(1-x-3x^(k+1))) for k=8.

Original entry on oeis.org

1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 40, 61, 91, 130, 178, 235, 301, 376, 460, 580, 763, 1036, 1426, 1960, 2665, 3568, 4696, 6076, 7816, 10105, 13213, 17491, 23371, 31366, 42070, 56158, 74386, 97834, 128149, 167788, 220261, 290374, 384472, 510682

Offset: 0

Alois P. Heinz, Aug 16 2008

a(n) is also the number of length n quaternary words with at least 8 0-digits between any other digits.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,3).

8th column of A143461.

a := proc(k::nonnegint) local n,i,j; if k=0 then unapply (4^n,n) else unapply ((Matrix(k+1, (i,j)-> if (i=j-1) or j=1 and i=1 then 1 elif j=1 and i=k+1 then 3 else 0 fi)^(n+k))[1,1], n) fi end(8): seq (a(n), n=0..58);

Series[1/(1-x-3*x^9), {x, 0, 58}] // CoefficientList[#, x]& // Drop[#, 8]& (* Jean-François Alcover, Feb 13 2014 *)

G.f.: 1/(x^8*(1-x-3*x^9)).

A143460 Expansion of 1/(x^k(1-x-3x^(k+1))) for k=9.

Original entry on oeis.org

1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 43, 64, 94, 133, 181, 238, 304, 379, 463, 556, 685, 877, 1159, 1558, 2101, 2815, 3727, 4864, 6253, 7921, 9976, 12607, 16084, 20758, 27061, 35506, 46687, 61279, 80038, 103801, 133729, 171550, 219802, 282076

Offset: 0

Alois P. Heinz, Aug 16 2008

a(n) is also the number of length n quaternary words with at least 9 0-digits between any other digits.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,3).

9th column of A143461.

a := proc(k::nonnegint) local n,i,j; if k=0 then unapply (4^n,n) else unapply ((Matrix(k+1, (i,j)-> if (i=j-1) or j=1 and i=1 then 1 elif j=1 and i=k+1 then 3 else 0 fi)^(n+k))[1,1], n) fi end(9): seq (a(n), n=0..61);

Series[1/(1-x-3*x^10), {x, 0, 61}] // CoefficientList[#, x]& // Drop[#, 9]& (* Jean-François Alcover, Feb 13 2014 *)

G.f.: 1/(x^9*(1-x-3*x^10)).

cp's OEIS Frontend

A143454 Expansion of 1/(x^k*(1 - x - 3*x^(k+1))) for k=3.

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A143455 Expansion of 1/(x^k*(1-x-3*x^(k+1))) for k=4.

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A143456 Expansion of 1/(x^k*(1-x-3*x^(k+1))) for k=5.

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A143457 Expansion of 1/(x^k*(1-x-3*x^(k+1))) for k=6.

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A143458 Expansion of 1/(x^k*(1-x-3*x^(k+1))) for k=7.

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A143459 Expansion of 1/(x^k*(1-x-3*x^(k+1))) for k=8.

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A143460 Expansion of 1/(x^k*(1-x-3*x^(k+1))) for k=9.

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A143454 Expansion of 1/(x^k(1 - x - 3x^(k+1))) for k=3.

A143455 Expansion of 1/(x^k(1-x-3x^(k+1))) for k=4.

A143456 Expansion of 1/(x^k(1-x-3x^(k+1))) for k=5.

A143457 Expansion of 1/(x^k(1-x-3x^(k+1))) for k=6.

A143458 Expansion of 1/(x^k(1-x-3x^(k+1))) for k=7.

A143459 Expansion of 1/(x^k(1-x-3x^(k+1))) for k=8.

A143460 Expansion of 1/(x^k(1-x-3x^(k+1))) for k=9.