A143453 -id:A143453 - cp's OEIS frontend

A052942 Expansion of 1/((1+x)(1-2x+2x^2-2x^3)).

1, 1, 1, 1, 3, 5, 7, 9, 15, 25, 39, 57, 87, 137, 215, 329, 503, 777, 1207, 1865, 2871, 4425, 6839, 10569, 16311, 25161, 38839, 59977, 92599, 142921, 220599, 340553, 525751, 811593, 1252791, 1933897, 2985399, 4608585, 7114167, 10981961

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

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 >= 4, 3*a(n-4) equals the number of 3-colored compositions of n with all parts >= 4, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

a(n+3) equals the number of ternary words of length n having at least 3 zeros between every two successive nonzero letters. - Milan Janjic, Mar 09 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 933
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2).

Column k=3 of A143453.

a:=[1,1,1,1];; for n in [5..40] do a[n]:=a[n-1]+2*a[n-4]; od; a; # G. C. Greubel, Jun 12 2019

I:=[1,1,1,1]; [n le 4 select I[n] else Self(n-1)+2*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Mar 10 2015

spec := [S,{S=Sequence(Union(Z,Prod(Union(Z,Z),Z,Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(add(binomial(n-3*k,k)*2^k, k=0..floor(n/3)), n=0..39); # Zerinvary Lajos, Apr 03 2007
with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S), S=Sequence(U, card >= 1), U=Sequence(Z, card >3)}, unlabeled]: seq(count(SeqSeqSeqL, size=j+4), j=0..39); # Zerinvary Lajos, Apr 04 2009
a := n -> `if`(n<9, [1, 1, 1, 1, 3, 5, 7, 9, 15][n+1], hypergeom([(1-n)/4,(2-n)/4,(3-n)/4,-n/4], [(1-n)/3,(2-n)/3,-n/3], -512/27)):
seq(simplify(a(n)),n=0..39); # Peter Luschny, Mar 09 2015

CoefficientList[Series[1/(1-x-2*x^4), {x,0,40}], x] (* Vincenzo Librandi, Mar 10 2015 *)
LinearRecurrence[{1,0,0,2},{1,1,1,1},50] (* Harvey P. Dale, Aug 17 2024 *)

Vec( 1/(1-x-2*x^4) + O(x^66)) \\ Joerg Arndt, Aug 28 2013

(1/(1-x-2*x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 12 2019

G.f.: 1/(1-x-2*x^4).
a(n) = a(n-1) + 2*a(n-4), with a(1)=1, a(0)=1, a(2)=1, a(3)=1.
a(n) = Sum_{alpha=RootOf(-1+_Z+2*_Z^4)} (1/539)*(27 + 72*alpha^3 + 96*alpha^2 + 128*alpha)*alpha^(-1-n)).
a(n) = Sum_{k=0..floor(n/3)} A128099(n-2*k, k). - Johannes W. Meijer, Aug 28 2013
a(n) = hypergeom([(1-n)/4,(2-n)/4,(3-n)/4,-n/4],[(1-n)/3,(2-n)/3,-n/3],-512/27) for n>=9. - Peter Luschny, Mar 09 2015

More terms from James Sellers, Jun 06 2000

A143447 Expansion of 1/(x^k(1-x-2x^(k+1))) for k=4.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 17, 27, 41, 59, 81, 115, 169, 251, 369, 531, 761, 1099, 1601, 2339, 3401, 4923, 7121, 10323, 15001, 21803, 31649, 45891, 66537, 96539, 140145, 203443, 295225, 428299, 621377, 901667, 1308553, 1899003, 2755601, 3998355, 5801689, 8418795

Offset: 0

Alois P. Heinz, Aug 16 2008

a(n) is also the number of length n ternary 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, 3*a(n-9) equals the number of 3-colored compositions of n with all parts >=5, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

Alois P. Heinz, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, 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 (1,0,0,0,2).

4th column of A143453.

a:= proc(k::nonnegint) local n,i,j; if k=0 then unapply(3^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 2 else 0 fi)^(n+k))[1,1], n) fi end(4): seq(a(n), n=0..54);

Series[1/(1-x-2*x^5), {x, 0, 54}] // CoefficientList[#, x]& // Drop[#, 4]& (* Jean-François Alcover, Feb 13 2014 *)

G.f.: ( -1-2*x-2*x^2-2*x^3-2*x^4 ) / ( -1+x+2*x^5 ). - R. J. Mathar, Aug 04 2019
G.f.: Q(0)/(2*x^4) -1/x -1/x^2 -1/x^3 -1/x^4, where Q(k) = 1 + 1/(1 - x*(2*k+1 + 2*x^4)/( x*(2*k+2 + 2*x^4) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 29 2013
a(n) = 2n+1 if n<=5, else a(n) = a(n-1) + 2a(n-5). - Milan Janjic, Mar 09 2015

A143448 Expansion of 1/(x^k(1-x-2x^(k+1))) for k=5.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 19, 29, 43, 61, 83, 109, 147, 205, 291, 413, 579, 797, 1091, 1501, 2083, 2909, 4067, 5661, 7843, 10845, 15011, 20829, 28963, 40285, 55971, 77661, 107683, 149341, 207267, 287837, 399779, 555101, 770467, 1069149, 1483683, 2059357, 2858915

Offset: 0

Alois P. Heinz, Aug 16 2008

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>=11, 3*a(n-11) equals the number of 3-colored compositions of n with all parts >=6, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

a(n) equals the number of ternary words of length n having at least 5 zeros between every two successive nonzero letters. - Milan Janjic, Mar 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,2).

5th column of A143453.

a:= proc(k::nonnegint) local n,i,j; if k=0 then unapply(3^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 2 else 0 fi)^(n+k))[1,1], n) fi end(5): seq(a(n), n=0..56);

Series[1/(1-x-2*x^6), {x, 0, 56}] // CoefficientList[#, x]& // Drop[#, 5]& (* Jean-François Alcover, Feb 13 2014 *)
LinearRecurrence[{1,0,0,0,0,2},{1,3,5,7,9,11},50] (* Harvey P. Dale, Aug 15 2021 *)

G.f.: (-1 - 2 x - 2 x^2 - 2 x^3 - 2 x^4 - 2 x^5)/(-1 + x + 2 x^6) - Harvey P. Dale, Aug 15 2021

a(n) = 2n+1 if n<=6, else a(n) = a(n-1) + 2a(n-6). - Milan Janjic, Mar 09 2015

Generating function corrected by Harvey P. Dale, Aug 15 2021

A143449 Expansion of 1/(x^k(1-x-2x^(k+1))) for k=6.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 21, 31, 45, 63, 85, 111, 141, 183, 245, 335, 461, 631, 853, 1135, 1501, 1991, 2661, 3583, 4845, 6551, 8821, 11823, 15805, 21127, 28293, 37983, 51085, 68727, 92373, 123983, 166237, 222823, 298789, 400959, 538413, 723159, 971125

Offset: 0

Alois P. Heinz, Aug 16 2008

a(n) is also the number of length n ternary words with at least 6 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>=13, 3*a(n-13) equals the number of 3-colored compositions of n with all parts >=7, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,2).

6th column of A143453.

a:= proc(k::nonnegint) local n,i,j; if k=0 then unapply(3^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 2 else 0 fi)^(n+k))[1,1], n) fi end(6): seq(a(n), n=0..58);

Series[1/(1-x-2*x^7), {x, 0, 58}] // CoefficientList[#, x]& // Drop[#, 6]& (* Jean-François Alcover, Feb 13 2014 *)

G.f.: 1/(x^6*(1-x-2*x^7)).

a(n) = 2n+1 if n<=7, else a(n) = a(n-1) + 2a(n-7). - Milan Janjic, Mar 09 2015

A143450 Expansion of 1/(x^k(1-x-2x^(k+1))) for k=7.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 23, 33, 47, 65, 87, 113, 143, 177, 223, 289, 383, 513, 687, 913, 1199, 1553, 1999, 2577, 3343, 4369, 5743, 7569, 9967, 13073, 17071, 22225, 28911, 37649, 49135, 64273, 84207, 110353, 144495, 188945, 246767, 322065, 420335, 548881

Offset: 0

Alois P. Heinz, Aug 16 2008

a(n) is also the number of length n ternary words with at least 7 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>=15, 3*a(n-15) equals the number of 3-colored compositions of n with all parts >=8, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,2).

7th column of A143453.

a:= proc(k::nonnegint) local n,i,j; if k=0 then unapply(3^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 2 else 0 fi)^(n+k))[1,1], n) fi end(7): seq(a(n), n=0..61);

Series[1/(1-x-2*x^8), {x, 0, 61}] // CoefficientList[#, x]& // Drop[#, 7]& (* Jean-François Alcover, Feb 13 2014 *)

G.f.: 1/(x^7*(1-x-2*x^8)).

a(n) = 2n+1 if n<=8, else a(n) = a(n-1) + 2a(n-8). - Milan Janjic, Mar 09 2015

A143451 Expansion of 1/(x^k(1-x-2x^(k+1))) for k=8.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 25, 35, 49, 67, 89, 115, 145, 179, 217, 267, 337, 435, 569, 747, 977, 1267, 1625, 2059, 2593, 3267, 4137, 5275, 6769, 8723, 11257, 14507, 18625, 23811, 30345, 38619, 49169, 62707, 80153, 102667, 131681, 168931, 216553, 277243

Offset: 0

Alois P. Heinz, Aug 16 2008

a(n) is also the number of length n ternary words with at least 8 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>=17, 3*a(n-17) equals the number of 3-colored compositions of n with all parts >=9, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,2).

8th column of A143453.

a:= proc(k::nonnegint) local n,i,j; if k=0 then unapply(3^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 2 else 0 fi)^(n+k))[1,1], n) fi end(8): seq(a(n), n=0..62);

Series[1/(1-x-2*x^9), {x, 0, 62}] // CoefficientList[#, x]& // Drop[#, 8]& (* Jean-François Alcover, Feb 13 2014 *)

Vec(1/(x^8*(1-x-2*x^9))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

G.f.: 1/(x^8*(1-x-2*x^9)).

a(n) = 2n+1 if n<=9, else a(n) = a(n-1) + 2a(n-9). - Milan Janjic, Mar 09 2015

A143452 Expansion of 1/(x^k(1-x-2x^(k+1))) for k=9.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 27, 37, 51, 69, 91, 117, 147, 181, 219, 261, 315, 389, 491, 629, 811, 1045, 1339, 1701, 2139, 2661, 3291, 4069, 5051, 6309, 7931, 10021, 12699, 16101, 20379, 25701, 32283, 40421, 50523, 63141, 79003, 99045, 124443, 156645

Offset: 0

Alois P. Heinz, Aug 16 2008

a(n) is also the number of length n ternary words with at least 9 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>=19, 3*a(n-19) equals the number of 3-colored compositions of n with all parts >=10, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,2).

9th column of A143453.

a:= proc(k::nonnegint) local n,i,j; if k=0 then unapply(3^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 2 else 0 fi)^(n+k))[1,1], n) fi end(9): seq(a(n), n=0..64);

Series[1/(1-x-2*x^10), {x, 0, 64}] // CoefficientList[#, x]& // Drop[#, 9]& (* Jean-François Alcover, Feb 13 2014 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,2},{1,3,5,7,9,11,13,15,17,19},50] (* Harvey P. Dale, Nov 28 2015 *)

G.f.: 1/(x^9*(1-x-2*x^10)).

a(n) = 2n+1 if n<=10, else a(n) = a(n-1) + 2a(n-10). - Milan Janjic, Mar 09 2015

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A052942 Expansion of 1/((1+x)*(1-2*x+2*x^2-2*x^3)).

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A143447 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=4.

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A143448 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=5.

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A143449 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=6.

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A143450 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=7.

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A143451 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=8.

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A143452 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=9.

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A052942 Expansion of 1/((1+x)(1-2x+2x^2-2x^3)).

A143447 Expansion of 1/(x^k(1-x-2x^(k+1))) for k=4.

A143448 Expansion of 1/(x^k(1-x-2x^(k+1))) for k=5.

A143449 Expansion of 1/(x^k(1-x-2x^(k+1))) for k=6.

A143450 Expansion of 1/(x^k(1-x-2x^(k+1))) for k=7.

A143451 Expansion of 1/(x^k(1-x-2x^(k+1))) for k=8.

A143452 Expansion of 1/(x^k(1-x-2x^(k+1))) for k=9.