A135851 -id:A135851 - cp's OEIS frontend

A078012 a(n) = a(n-1) + a(n-3) for n >= 3, with a(0) = 1, a(1) = a(2) = 0. This recurrence can also be used to define a(n) for n < 0.

1, 0, 0, 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964, 3914488, 5736961, 8407925

Offset: 0

N. J. A. Sloane, Nov 17 2002, Mar 08 2008

Number of compositions of n into parts >= 3. - Milan Janjic, Jun 28 2010
From Adi Dani, May 22 2011: (Start)
Number of compositions of number n into parts of the form 3*k+1, k >= 0.
For example, a(10)=19 and all compositions of 10 in parts 1,4,7 or 10 are
(1,1,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,4), (1,1,1,1,1,4,1), (1,1,1,1,4,1,1), (1,1,1,4,1,1,1), (1,1,4,1,1,1,1), (1,4,1,1,1,1,1), (4,1,1,1,1,1,1), (1,1,4,4), (1,4,1,4), (1,4,4,1), (4,1,1,4),(4,1,4,1), (4,4,1,1), (1,1,1,7), (1,1,7,1), (1,7,1,1), (7,1,1,1), (10). (End)
For n >= 0 a(n+1) is the number of 00's in the Narayana word NW(n); equivalently the number of two neighboring 0's at level n of the Narayana tree. See A257234. This implies that if a(0) is put to 0 then a(n) is the number of -1's in the Narayana word NW(n), and also at level n of the Narayana tree. - Wolfdieter Lang, Apr 24 2015

			G.f. = 1 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 6*x^9 + 9*x^10 + 13*x^11 + ...

Taylor L. Booth, Sequential Machines and Automata Theory, John Wiley and Sons, Inc., 1967, page 331ff.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Christian Ballot, On Functions Expressible as Words on a Pair of Beatty Sequences, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.2.
C. K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f_n.]
Taras Goy, On identities with multinomial coefficients for Fibonacci-Narayana sequence, Annales Mathematicae et Informaticae, Vol. 49 (2018), 75-84.
Bahar Kuloğlu, Engin Özkan, and Marin Marin, On the period of Pell-Narayana sequence in some groups, arXiv:2305.04786 [math.CO], 2023.
J. D. Opdyke, A unified approach to algorithms generating unrestricted.., J. Math. Model. Algor. 9 (2010) 53-97.
Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Cf. A000930, A065417, A068921, A077961.

Cf. also A135851, A257234.

a:=[1,0,0];; for n in [4..50] do a[n]:=a[n-1]+a[n-3]; od; a; # G. C. Greubel, Jun 28 2019

a078012 n = a078012_list !! n
a078012_list = 1 : 0 : 0 : 1 : zipWith (+) a078012_list
   (zipWith (+) (tail a078012_list) (drop 2 a078012_list))
-- Reinhard Zumkeller, Mar 23 2012

I:=[1,0,0]; [n le 3 select I[n] else Self(n-1) + Self(n-3): n in [1..50]]; // G. C. Greubel, Jan 19 2018

A078012 := proc(n): if n=0 then 1 else add(binomial(n-3-2*i,i),i=0..(n-3)/3) fi: end: seq(A078012(n), n=0..46); # Johannes W. Meijer, Aug 11 2011
# second Maple program:
a:= n-> (<<0|1|0>, <0|0|1>, <1|0|1>>^n)[1, 1]:
seq(a(n), n=0..46);  # Alois P. Heinz, May 08 2025

CoefficientList[ Series[(1-x)/(1-x-x^3), {x,0,50}], x] (* Robert G. Wilson v, May 25 2011 *)
LinearRecurrence[{1,0,1}, {1,0,0}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)
a[ n_]:= If[ n >= 0, SeriesCoefficient[ (1-x)/(1-x-x^3), {x, 0, n}], SeriesCoefficient[1/(1+x^2-x^3), {x, 0, -n}]]; (* Michael Somos, Feb 03 2018 *)

{a(n) = if( n<0, n = -n; polcoeff( 1 / (1 + x^2 - x^3) + x * O(x^n), n), polcoeff( (1 - x) / (1 - x - x^3) + x * O(x^n), n))}; /* Michael Somos, May 03 2011 */

((1-x)/(1-x-x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 28 2019

a(n) = Sum_{i=0..(n-3)/3} binomial(n-3-2*i, i), n >= 1, a(0) = 1.
From Michael Somos, May 03 2011: (Start)
Euler transform of A065417.
G.f.: (1 - x) / (1 - x - x^3).
a(-n) = A077961(n). a(n+3) = A000930(n).
a(n+5) = A068921(n). (End)
a(n+1) = A013979(n-3) + A135851(n) + A107458(n), n >= 3.
G.f.: 1/(1 - Sum_{k>=3} x^k). - Joerg Arndt, Aug 13 2012
G.f.: Q(0)*(1-x)/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + x^2)/( x*(4*k+3 + x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013
0 = -1 + a(n)*(a(n)*(a(n) + a(n+2)) + a(n+1)*(a(n+1) - 3*a(n+2))) + a(n+1)*(+a(n+1)*(+a(n+1) + a(n+2)) + a(n+2)*(-2*a(n+2))) + a(n+2)^3 for all n in Z. - Michael Somos, Feb 03 2018
a(-n) = a(n)*a(n+3) - a(n+1)*a(n+2) for all n in Z. - Greg Dresden, May 07 2025

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

Entry revised by N. J. A. Sloane, May 11 2025, making use of comments from Michael Somos, May 03 2011 and Greg Dresden, May 11 2025

A077868 Expansion of 1/((1-x)*(1-x-x^3)).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 18, 27, 40, 59, 87, 128, 188, 276, 405, 594, 871, 1277, 1872, 2744, 4022, 5895, 8640, 12663, 18559, 27200, 39864, 58424, 85625, 125490, 183915, 269541, 395032, 578948, 848490, 1243523, 1822472, 2670963, 3914487, 5736960, 8407924, 12322412

Offset: 0

N. J. A. Sloane, Nov 17 2002

Row sums of Riordan array (1/(1-x), x*(1+x^2)). - Paul Barry, Feb 16 2005

a(n) is the number of partitions of {1, ..., n+3} into two blocks in which only 1- or 3-strings of consecutive integers can appear in a block and there is at least one 3-string. E.g., a(3)=5 because the enumerated partitions of {1,2,3,4,5,6} are 1235/46, 1345/26, 15/2346, 13/2456, 123/456. - Augustine O. Munagi, Apr 11 2005

Chu, Hung Viet. "Various Sequences from Counting Subsets." Fib. Quart., 59:2 (May 2021), 150-157.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kassie Archer and Aaron Geary, Powers of permutations that avoid chains of patterns, arXiv:2312.14351 [math.CO], 2023. See p. 15.
Hung Viet Chu, Various sequences from counting subsets, arXiv:2005.10081 [math.CO], 2020-2021.
A. O. Munagi, Set Partitions with Successions and Separations, IJMMS 2005:3 (2005), 451-463.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1).

Cf. A000071, A077888, A077941, A105489.
Cf. A000930, A050228, A144898, A144899, A144900, A144901, A144902, A144903, A144904, A226405.
Cf. A078012, A099567, A135851.

A077868:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+, j+1): j in [0..Floor((n+1)/3)]]) >;
[A077868(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

a:= n-> (Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [2,-1,1,-1][i] else 0 fi)^n)[1,1]: seq(a(n), n=0..41); # Alois P. Heinz, Sep 05 2008
g:=(1+z+z^2)/(1-z-z^3): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-1, n=1..42); # Zerinvary Lajos, Jan 09 2009

LinearRecurrence[{1,1,0,0,-1}, {1,2,3,5,8,12}, 42] (* or *)
CoefficientList[Series[1/((1-x)(1-x-x^3)), {x, 0, 41}], x] (* Michael De Vlieger, Jun 06 2018 *)

Vec(1/(1-x)/(1-x-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

{a = vector(50);
a[1] = 1; a[2] = 2; a[3] = 3;
for(n=4,50,
a[n] = 1 + a[n-1] + a[n-3];
); a} \\ Gerry Martens, Jun 03 2018

{a(n) = if( n<0, n=-4-n; polcoeff( -1 / (1 - x) / (1 + x^2 - x^3) + x * O(x^n), n), polcoeff( 1 / (1 - x) / (1 - x - x^3) + x * O(x^n), n))}; /* Michael Somos, Jun 17 2018 */

def A077868(n): return sum(binomial(n-2*j+1, j+1) for j in (0..((n+1)//3)))
[A077868(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

Partial sums of A000930. a(n-1) = Sum_{k=0..floor(n/2)} binomial(n-2*k, k+1). - Paul Barry, Jul 07 2004
a(n-3) = Sum(binomial(n-r, r)), r=1, 2, ... which is the case t=3 and k=2 in the general case of t-strings and k blocks: a(n-3, k, t) = Sum(binomial(n-r*(t-1), r)*S2(n-r*(t-1)-1, k-1)), r=1, 2, ... - Augustine O. Munagi, Apr 11 2005
From Paul Weisenhorn, Oct 28 2011: (Start)
a(n) = a(n-1) + a(n-2) - a(n-5) for n > 4.
a(n) = a(n-2) + a(n-3) + a(n-4) + 2 for n > 3.
G.f.: 1/((1-x)*(1-x-x^3)). (End)
a(n) = 1 + a(n-1) + a(n-3), a(1)=1, a(2)=2, a(3)=3. - Gerry Martens, Jun 10 2018
a(n) = -A077888(-4-n) for all n in Z. - Michael Somos, Jun 17 2018
a(n) = A000930(n+3) - 1. - Greg Dresden, Jun 20 2021
a(n) = A099567(n+3, 4). - G. C. Greubel, Jul 27 2022

More terms from Augustine O. Munagi, Apr 11 2005

A077961 Expansion of 1 / (1 + x^2 - x^3) in powers of x.

Original entry on oeis.org

1, 0, -1, 1, 1, -2, 0, 3, -2, -3, 5, 1, -8, 4, 9, -12, -5, 21, -7, -26, 28, 19, -54, 9, 73, -63, -64, 136, 1, -200, 135, 201, -335, -66, 536, -269, -602, 805, 333, -1407, 472, 1740, -1879, -1268, 3619, -611, -4887, 4230, 4276, -9117, -46, 13393, -9071, -13439, 22464, 4368, -35903, 18096, 40271, -53999

Offset: 0

N. J. A. Sloane, Nov 17 2002

			G.f. = 1 - x^2 + x^3 + x^4 - 2*x^5 + 3*x^7 - 2*x^8 - 3*x^9 + 5*x^10 + x^11 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. Gogin and A. Mylläri, Padovan-like sequences and Bell polynomials, Proceedings of Applications of Computer Algebra ACA, 2013.
Michael D. Hirschhorn, Non-trivial intertwined second-order recurrence relations, Fibonacci Quart. 43 (2005), no. 4, 316-325. See L_n.
Index entries for linear recurrences with constant coefficients, signature (0,-1,1).

Cf. A000930, A078012, A135851.

I:=[1,0,-1]; [n le 3 select I[n] else -Self(n-2) +Self(n-3): n in [1..70]]; // G. C. Greubel, Mar 28 2024

a:= n-> (<<0|1|0>, <-1|0|1>, <1|0|0>>^n)[1, 1]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 20 2008

a[ n_] := If[ n >= 0, SeriesCoefficient[ 1 / (1 + x^2 - x^3), {x, 0, n}], SeriesCoefficient[ x^3 / (1 - x - x^3), {x, 0, -n}]] (* Michael Somos, Jan 08 2014 *)

{a(n) = if( n<0, polcoeff( x^3 / (1 - x - x^3) + x * O(x^-n), -n), polcoeff( 1 / (1 + x^2 - x^3) + x * O(x^n), n))} /* Michael Somos, Jan 08 2014 */

[sum((-1)^(n+k)*binomial(k,n-2*k) for k in range(1+n//2)) for n in range(71)] # G. C. Greubel, Mar 28 2024

a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*binomial(k, n-2*k). - Paul Barry, Jun 24 2005
From Alois P. Heinz, Jun 20 2008: (Start)
a(n) = term (1,1) in matrix [0,1,0; -1,0,1; 1,0,0]^n.
a(n) = A000930 (-3-n). (End)
a(-n) = A078012(n). - Michael Somos, May 03 2011
From Michael Somos, Jan 08 2014: (Start)
a(-n) = A135851(n+2).
a(n)^2 - a(n-1)*a(n+1) = A135851(n+5). (End)
G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x^2*(4*k+1 - x )/( x^2*(4*k+3 - x ) - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 09 2013
a(n)^2 + a(n-1)^2 + a(n-6)^2 = a(n-2)^2 + 3*a(n-3)^2 + a(n-4)^2 + a(n-5)^2. - Greg Dresden, Apr 12 2025

A144903 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of x/((1-x-x^3)*(1-x)^(k-1)).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 1, 0, 1, 3, 3, 2, 1, 0, 1, 4, 6, 5, 3, 1, 0, 1, 5, 10, 11, 8, 4, 2, 0, 1, 6, 15, 21, 19, 12, 6, 3, 0, 1, 7, 21, 36, 40, 31, 18, 9, 4, 0, 1, 8, 28, 57, 76, 71, 49, 27, 13, 6, 0, 1, 9, 36, 85, 133, 147, 120, 76, 40, 19, 9, 0, 1, 10, 45, 121, 218, 280, 267, 196, 116, 59, 28, 13

Offset: 0

Alois P. Heinz, Sep 24 2008

			Square array (A(n,k)) begins:
  0, 0,  0,  0,  0,   0,   0 ... A000004;
  1, 1,  1,  1,  1,   1,   1 ... A000012;
  0, 1,  2,  3,  4,   5,   6 ... A001477;
  0, 1,  3,  6, 10,  15,  21 ... A000217;
  1, 2,  5, 11, 21,  36,  57 ... A050407;
  1, 3,  8, 19, 40,  76, 133 ... ;
  1, 4, 12, 31, 71, 147, 200 ... A027658;
Antidiagonal triangle (T(n,k)) begins as:
  0;
  0,  1;
  0,  1,  0;
  0,  1,  1,  0;
  0,  1,  2,  1,  1;
  0,  1,  3,  3,  2,  1;
  0,  1,  4,  6,  5,  3,  1;
  0,  1,  5, 10, 11,  8,  4,  2;
  0,  1,  6, 15, 21, 19, 12,  6,  3;

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Rows 0-4, 6 give: A000004, A000012, A001477, A000217, A050407(n+3), A027658.
Columns 0-9 give: A078012 and A135851(n+2), A078012(n+2) and A135851(n+4), A077868(n-1) for n>0, A050228(n-1) for n>0, A226405, A144898, A144899, A144900, A144901, A144902.
Main diagonal gives: A144904.
Cf. A000930.

A000930:= func< n | (&+[Binomial(n-2*j,j): j in [0..Floor(n/3)]]) >;
A144903:= func< n,k | k eq 0 select 0 else (&+[Binomial(n-k+j-2,j)*A000930(k-j-1) : j in [0..k-1]]) >;
[A144903(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Aug 01 2022

A:= proc(n,k) coeftayl (x/ (1-x-x^3)/ (1-x)^(k-1), x=0, n) end:
seq(seq(A(n, d-n), n=0..d), d=0..13);

(* First program *)
a[n_, k_] := SeriesCoefficient[x/((1-x-x^3)*(1-x)^(k-1)), {x, 0, n}];
Table[a[n-k, k], {n,0,12}, {k,n,0,-1}]//Flatten (* Jean-François Alcover, Jan 15 2014 *)
(* Second Program *)
A000930[n_]:= A000930[n]= Sum[Binomial[n-2*j,j], {j,0,Floor[n/3]}];
T[n_, k_]:= T[n, k]= If[k==0, 0, Sum[Binomial[n-k+j-2,j]*A000930[k-j-1], {j,0,k- 1}]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 01 2022 *)

def A000930(n): return sum(binomial(n-2*j,j) for j in (0..(n//3)))
def A144903(n,k):
    if (k==0): return 0
    else: return sum(binomial(n-k+j-2,j)*A000930(k-j-1) for j in (0..k-1))
flatten([[A144903(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Aug 01 2022

G.f. of column k: x/((1-x-x^3)*(1-x)^(k-1)).
A(n, n) = A144904(n).
From G. C. Greubel, Aug 01 2022: (Start)
A(n, k) = Sum_{j=0..n-1} binomial(k+j-2, j)*A000930(n-j-1), with A(0, k) = 0.
T(n, k) = Sum_{j=0..k-1} binomial(n-k-j-2, j)*A000930(k-j-1), with T(n, 0) = 0.
T(2*n, n) = A144904(n). (End)

A050228 a(n) is the number of subsequences {s(k)} of {1,2,3,...n} such that s(k+1)-s(k) is 1 or 3.

Original entry on oeis.org

1, 3, 6, 11, 19, 31, 49, 76, 116, 175, 262, 390, 578, 854, 1259, 1853, 2724, 4001, 5873, 8617, 12639, 18534, 27174, 39837, 58396, 85596, 125460, 183884, 269509, 394999, 578914, 848455, 1243487, 1822435, 2670925, 3914448, 5736920, 8407883

Offset: 1

John W. Layman, Dec 20 1999

The second differences c(n) of {a(n)} satisfy c(n)=c(n-1)+c(n-3) and give A000930 with the first 5 terms deleted.

Partial sums of A077868. - Paul Barry, Sep 16 2004

Chu, Hung Viet. "Various Sequences from Counting Subsets." Fib. Quart., 59:2 (May 2021), 150-157.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Hung Viet Chu, Various sequences from counting subsets, arXiv:2005.10081 [math.CO], 2020-2021.
Z. Kasa, On scattered subword complexity, arXiv preprint arXiv:1104.4425 [cs.DM], 2011.
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-2,1).

Cf. A000930, A077868, A144898, A144899, A144900, A144901, A144902, A144903, A144904, A226405.

Cf. A078012, A099567, A135851.

A050228:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+1, j+2): j in [0..Floor((n+1)/3)]]) >;
[A050228(n): n in [1..40]]; // G. C. Greubel, Jul 27 2022

with(combstruct): SubSetSeqU := [T, {T=Subst(U,U), S=Set(U, card>=3), U=Sequence(Z, card>=3)}, unlabeled]: seq(count(SubSetSeqU, size=n), n=9..46); # Zerinvary Lajos, Mar 18 2008

Rest[CoefficientList[Series[1/((1-x)^2*(1-x-x^3)), {x, 0, 50}], x]] (* G. C. Greubel, Apr 27 2017 *)
LinearRecurrence[{3,-3,2,-2,1},{1,3,6,11,19},50] (* Harvey P. Dale, Apr 21 2020 *)

my(x='x+O('x^50)); Vec(x/((1-x)^3-x^3*(1-x)^2)) \\ G. C. Greubel, Apr 27 2017

def A050228(n): return sum(binomial(n-2*j+1, j+2) for j in (0..((n+1)//3)))
[A050228(n) for n in (1..40)] # G. C. Greubel, Jul 27 2022

From Paul Barry, Sep 16 2004: (Start)
G.f.: x/((1-x)^3 - x^3(1-x)^2).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 2*a(n-4) + a(n-5).
a(n-1) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k+2). (End)
G.f. = 1/((1-x)^2*(1-x-x^3)). - N. J. A. Sloane, Jun 02 2021
a(n) = A000930(n+5) - n - 4. - Greg Dresden, Jun 20 2021
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+1)/3)} binomial(n-2*j+1, j+2).
a(n) = A099567(n+1, 2). (End)

A144898 Expansion of x/((1-x-x^3)*(1-x)^4).

Original entry on oeis.org

0, 1, 5, 15, 36, 76, 147, 267, 463, 775, 1262, 2011, 3150, 4867, 7438, 11268, 16951, 25358, 37766, 56047, 82945, 122482, 180553, 265798, 390880, 574358, 843432, 1237966, 1816384, 2664311, 3907237, 5729077, 8399372, 12313154, 18049371, 26456513, 38778103

Offset: 0

Alois P. Heinz, Sep 24 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,11,-9,7,-4,1).

5th column of A144903.
Cf. A000930, A050228, A077868, A144899, A144900, A144901, A144902, A144903, A144904, A226405.
Cf. A078012, A099567, A135851.

A144898:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+3, j+4): j in [0..Floor((n+3)/3)]]) >;
[A144898(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

a:= n-> (Matrix(7, (i, j)-> if i=j-1 then 1 elif j=1 then [5, -10, 11, -9, 7, -4, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

CoefficientList[Series[ x/((1-x-x^3)(1-x)^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)

def A144898(n): return sum(binomial(n-2*j+3, j+4) for j in (0..((n+3)//3)))
[A144898(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

G.f.: x/((1-x-x^3)*(1-x)^4).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+3)/3)} binomial(n-2*j+3, j+4).
a(n) = A099567(n+3, 4). (End)

A144899 Expansion of x/((1-x-x^3)*(1-x)^5).

Original entry on oeis.org

0, 1, 6, 21, 57, 133, 280, 547, 1010, 1785, 3047, 5058, 8208, 13075, 20513, 31781, 48732, 74090, 111856, 167903, 250848, 373330, 553883, 819681, 1210561, 1784919, 2628351, 3866317, 5682701, 8347012, 12254249, 17983326, 26382698, 38695852, 56745223, 83201736

Offset: 0

Alois P. Heinz, Sep 24 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,21,-20,16,-11,5,-1).

6th column of A144903.
Cf. A000930, A050228, A077868, A144898, A144900, A144901, A144902, A144903, A144904, A226405.
Cf. A078012, A099567, A135851.

A144899:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+4, j+5): j in [0..Floor((n+4)/3)]]) >;
[A144899(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

a:= n-> (Matrix(8, (i, j)-> if i=j-1 then 1 elif j=1 then [6, -15, 21, -20, 16, -11, 5, -1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

CoefficientList[Series[x/((1-x-x^3)(1-x)^5), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)

def A144899(n): return sum(binomial(n-2*j+4, j+5) for j in (0..((n+4)//3)))
[A144899(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

G.f.: x/((1-x-x^3)*(1-x)^5).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+4)/3)} binomial(n-2*j+4, j+5).
a(n) = A099567(n+4, 5). (End)

A226405 Expansion of x/((1-x-x^3)*(1-x)^3).

Original entry on oeis.org

0, 1, 4, 10, 21, 40, 71, 120, 196, 312, 487, 749, 1139, 1717, 2571, 3830, 5683, 8407, 12408, 18281, 26898, 39537, 58071, 85245, 125082, 183478, 269074, 394534, 578418, 847927, 1242926, 1821840, 2670295, 3913782, 5736217, 8407142, 12321590, 18058510, 26466393

Offset: 0

Alois P. Heinz, Jun 06 2013

From Bruno Berselli, Jun 07 2013: (Start)
A050228(n)  = a(n)   -a(n-1), n>0.
A077868(n-1)= a(n) -2*a(n-1)   +a(n-2), n>1.
A000217(n)  = a(n)   -a(n-1)             -a(n-3), n>2.
A000930(n-1)= a(n) -3*a(n-1) +3*a(n-2)   -a(n-3), n>2.
n           = a(n) -2*a(n-1)   +a(n-2)   -a(n-3)   +a(n-4), n>3.
1           = a(n) -3*a(n-1) +3*a(n-2) -2*a(n-3) +2*a(n-4)   -a(n-5), n>4.
0           = a(n) -4*a(n-1) +6*a(n-2) -5*a(n-3) +4*a(n-4) -3*a(n-5) +a(n-6), n>5.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-4,3,-1).

4th column of A144903.
Cf. A000217, A078012, A099567, A135851.
Cf. A000930, A050228, A077868, A144898, A144899, A144900, A144901, A144902, A144903, A144904, A226405.

A226405:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+2, j+3): j in [0..Floor((n+2)/3)]]) >;
[A226405(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

a:= n-> (Matrix(6, (i, j)-> if i=j-1 then 1 elif j=1 then [4, -6, 5, -4, 3, -1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

LinearRecurrence[{4,-6,5,-4,3,-1}, {0,1,4,10,21,40}, 40]  (* Bruno Berselli, Jun 07 2013 *)
CoefficientList[Series[x/((1-x-x^3)*(1-x)^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 28 2017 *)

my(x='x+O('x^50)); Vec(x/((1-x-x^3)*(1-x)^3)) \\ G. C. Greubel, Apr 28 2017

def A226405(n): return sum(binomial(n-2*j+2, j+3) for j in (0..((n+2)//3)))
[A226405(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

G.f.: x/((1-x-x^3)*(1-x)^3).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+2)/3)} binomial(n-2*j+2, j+3).
a(n) = A099567(n+2, 3). (End)

A107458 Expansion of g.f.: (1-x^2-x^3)/( (1+x)*(1-x-x^3) ).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 5, 8, 11, 17, 24, 36, 52, 77, 112, 165, 241, 354, 518, 760, 1113, 1632, 2391, 3505, 5136, 7528, 11032, 16169, 23696, 34729, 50897, 74594, 109322, 160220, 234813, 344136, 504355, 739169, 1083304, 1587660, 2326828, 3410133, 4997792, 7324621

Offset: 0

N. J. A. Sloane, Mar 08 2008

The sequence can be interpreted as the top-left entry of the n-th power of a 4 X 4 (0,1) matrix. There are 12 different choices (out of 2^16) for that (0,1) matrix. - R. J. Mathar, Mar 19 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
C. Kenneth Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f^2_n.]
Renata Passos Machado Vieira, and Francisco Regis Vieira Alves, Sequences of Tridovan and their identities, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 3, 185-197. Sequence (T_n) is a subsequence of this sequence.
Renata Passos Machado Vieira, Francisco Regis Vieira Alves, and Paula Maria Machado Cruz Catarino, A note on the Tetrarrin sequence, Braz. Elect. J. Math. (2024).
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1).

Cf. A001634, A013979, A078012, A135851.

a:=[1,0,0,0];; for n in [5..50] do a[n]:=a[n-2]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 03 2020

a107458 n = a107458_list !! n
a107458_list = 1 : 0 : 0 : 0 : zipWith (+) a107458_list
   (zipWith (+) (tail a107458_list) (drop 2 a107458_list))
-- Reinhard Zumkeller, Mar 23 2012

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!(1-x^2-x^3)/( (1+x)*(1-x-x^3))); // Marius A. Burtea, Jan 02 2020

seq(coeff(series( (1-x^2-x^3)/( (1+x)*(1-x-x^3) ), x, n+1), x, n), n = 0..50); # G. C. Greubel, Jan 03 2020

CoefficientList[Series[(1-x^2-x^3)/(1-x^2-x^3-x^4),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,1,1},{1,0,0,0},50] (* Harvey P. Dale, Jun 20 2011 *)

my(x='x+O('x^50)); Vec((1-x^2-x^3)/((1+x)*(1-x-x^3))) \\ G. C. Greubel, Apr 27 2017

def A107458_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2-x^3)/((1+x)*(1-x-x^3)) ).list()
A107458_list(50) # G. C. Greubel, Jan 03 2020

a(n) = a(n-2) + a(n-3) + a(n-4); a(0)=1, a(1)=0, a(2)=0, a(3)=0. - Harvey P. Dale, Jun 20 2011

a(n) + a(n-1) = A000930(n-4). - R. J. Mathar, Mar 19 2014

cp's OEIS Frontend

A078012 a(n) = a(n-1) + a(n-3) for n >= 3, with a(0) = 1, a(1) = a(2) = 0. This recurrence can also be used to define a(n) for n < 0.

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A077868 Expansion of 1/((1-x)*(1-x-x^3)).

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A077961 Expansion of 1 / (1 + x^2 - x^3) in powers of x.

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A144903 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of x/((1-x-x^3)*(1-x)^(k-1)).

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A050228 a(n) is the number of subsequences {s(k)} of {1,2,3,...n} such that s(k+1)-s(k) is 1 or 3.

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