A008937 -id:A008937 - cp's OEIS frontend

A077908 Expansion of (1-x)^(-1)/(1+x+x^2-x^3).

1, 0, 0, 2, -1, 0, 4, -4, 1, 8, -12, 6, 15, -32, 24, 24, -79, 80, 24, -182, 239, -32, -388, 660, -303, -744, 1708, -1266, -1185, 4160, -4240, -1104, 9505, -12640, 2032, 20114, -34785, 16704, 38196, -89684, 68193, 59688, -217564, 226070, 51183, -494816, 669704, -123704, -1040815, 1834224

Offset: 0

N. J. A. Sloane, Nov 17 2002

			G.f. = 1 + 2*x^3 - x^4 + 4*x^6 - 4*x^7 + x^8 + 8*x^9 - 12*x^10 + 6*x^11 + ...

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

Cf. A008937.

a:= n-> -(<<0|0|1|0>, <1|0|-1|0>, <0|1|-1|0>, <0|0|-1|1>>^(n+3))[4, 1]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 24 2008

a[n_] := -MatrixPower[{{0, 0, 1, 0}, {1, 0, -1, 0}, {0, 1, -1, 0}, {0, 0, -1, 1}}, n+3][[4, 1]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 14 2016, after Alois P. Heinz *)
LinearRecurrence[{0,0,2,-1},{1,0,0,2},50] (* Harvey P. Dale, Oct 25 2020 *)

Vec((1-x)^(-1)/(1+x+x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

{a(n) = if( n<0, polcoeff( - x^4 / (1 - 2*x + x^4) + x * O(x^-n), -n), polcoeff( 1 / (1 - 2*x^3 + x^4) + x * O(x^n), n))}; /* Michael Somos, Aug 19 2014 */

a(n) = -1 * term (4,1) in the 4x4 matrix [0,0,1,0; 1,0,-1,0; 0,1,-1,0; 0,0,-1,1]^(n+3) - Alois P. Heinz, Jul 24 2008
a(n) = -A008937(-n-3). - Alois P. Heinz, Jul 24 2008
G.f.: 1 / (1 - 2*x^3 + x^4). - Michael Somos, Aug 19 2014
a(n) = -a(n-1) - a(n-2) + a(n-3) + 1 = 2*a(n-3) - a(n-4) for all n in Z. - Michael Somos, Aug 19 2014
a(n) - a(n-1) = A057597(n+2). (first differences). - R. J. Mathar, Oct 16 2017

A118884 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0011 (n,k>=0).

Original entry on oeis.org

1, 2, 4, 8, 15, 1, 28, 4, 52, 12, 96, 32, 177, 78, 1, 326, 180, 6, 600, 400, 24, 1104, 864, 80, 2031, 1827, 237, 1, 3736, 3800, 648, 8, 6872, 7800, 1672, 40, 12640, 15840, 4128, 160, 23249, 31884, 9846, 556, 1, 42762, 63704, 22844, 1752, 10, 78652, 126480

Offset: 0

Emeric Deutsch, May 03 2006

Row n has 1+floor(n/4) terms. Sum of entries in row n is 2^n (A000079). T(n,0) = A008937(n+1). T(n,1) = A118885(n). Sum(k*T(n,k), k=0..n-1) = (n-3)*2^(n-4) (A001787).

			T(9,2) = 6 because we have aa0, aa1, a0a, a1a, 0aa and 1aa, where a=0011.
Triangle starts:
1;
2;
4;
8;
15, 1;
28, 4;
52, 12;
96, 32;

Alois P. Heinz, Rows n = 0..300, flattened

Cf. A000079, A008937, A118885, A001787.

G:=1/(1-2*z+(1-t)*z^4): Gser:=simplify(series(G,z=0,23)): P[0]:=1: for n from 1 to 19 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 19 do seq(coeff(P[n],t,j),j=0..floor(n/4)) od; # yields sequence in triangular form

nn=12;c=0;Map[Select[#,#>0&]&,CoefficientList[Series[1/(1-2x - (y-1)x^4/ (1-(y-1)c)), {x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Dec 25 2013 *)

G.f.: G(t,z) = 1/[1-2z+(1-t)z^4]. T(n,k) = 2T(n-1,k)-T(n-4,k)+T(n-4,k-1) (n>=4,k>=1).

A189155 Number of nX3 binary arrays without the pattern 0 0 1 1 diagonally, vertically or horizontally.

Original entry on oeis.org

8, 64, 512, 3375, 21952, 140608, 884736, 5545233, 34645976, 216000000, 1345572864, 8377795791, 52145952256, 324525966848, 2019487744000, 12566456507249, 78194107594728, 486552055503808, 3027480871826944, 18837870152811039

Offset: 1

R. H. Hardin Apr 17 2011

nonn

Column 3 of A189161

			Some solutions for 4X3
..0..0..1....1..0..1....0..0..0....0..1..1....1..1..0....1..0..0....0..0..1
..0..0..0....0..1..0....0..0..0....1..0..1....0..1..0....0..1..1....1..0..0
..0..1..0....0..0..1....0..0..0....0..0..0....1..0..0....1..1..0....0..0..1
..0..0..0....1..0..0....0..1..1....0..1..0....1..1..0....0..1..1....0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..200

A008937(n+1)^3

Empirical: a(n) = 7*a(n-1) -a(n-2) +a(n-3) -164*a(n-4) +28*a(n-5) +68*a(n-6) +684*a(n-7) +66*a(n-8) +66*a(n-9) -726*a(n-10) -138*a(n-11) -132*a(n-12) +220*a(n-13) +20*a(n-14) +28*a(n-15) -25*a(n-16) -a(n-17) -a(n-18) +a(n-19)

A192804 Constant term in the reduction of the polynomial 1+x+x^2+...+x^n by x^3->x^2+x+1. See Comments.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 16, 29, 53, 97, 178, 327, 601, 1105, 2032, 3737, 6873, 12641, 23250, 42763, 78653, 144665, 266080, 489397, 900141, 1655617, 3045154, 5600911, 10301681, 18947745, 34850336, 64099761, 117897841, 216847937, 398845538

Offset: 0

Clark Kimberling, Jul 10 2011

For discussions of polynomial reduction, see A192232 and A192744.

This sequence provides the most-significant place-values in the construction of a tribonacci code. - James Dow Allen, Jul 12 2021

			The first five polynomials p(n,x) and their reductions:
  p(1,x)=1 -> 1,
  p(2,x)=x+1 -> x+1,
  p(3,x)=x^2+x+1 -> x^2+x+1,
  p(4,x)=x^3+x^2+x+1 -> 2x^2+2x+2,
  p(5,x)=x^4+x^3+x^2+x+1 -> 4x^2+4*x+3, so that
A192804=(1,1,1,2,3,...), A000073=(0,1,1,2,4,...), A008937=(0,0,1,2,4,...).

James Dow Allen, Constructing the Tribonacci Code.
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).

Cf. A192744, A192232, A000073.

q = x^3; s = x^2 + x + 1; z = 40;
p[0, x_] := 1; p[n_, x_] := x^n + p[n - 1, x];
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
  (* A192804 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A000073 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A008937 *)

a(n) = 2*a(n-1) - a(n-4).
a(n) = a(n-1) + a(n-2) + a(n-3) - 1. - Alzhekeyev Ascar M, Feb 05 2012
G.f.: ( 1-x-x^2 ) / ( (x-1)*(x^3+x^2+x-1) ). - R. J. Mathar, May 06 2014
a(n) - a(n-1) = A000073(n-1). - R. J. Mathar, May 06 2014

A027083 a(n) = A027082(n, n+2).

Original entry on oeis.org

2, 6, 14, 28, 54, 102, 190, 352, 650, 1198, 2206, 4060, 7470, 13742, 25278, 46496, 85522, 157302, 289326, 532156, 978790, 1800278, 3311230, 6090304, 11201818, 20603358, 37895486, 69700668, 128199518, 235795678, 433695870

Offset: 2

Clark Kimberling

Paolo Xausa, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).

Cf. A008937, A027024, A027082.

LinearRecurrence[{2, 0, 0, -1}, {2, 6, 14, 28}, 50] (* Paolo Xausa, Sep 16 2024 *)

G.f.: (2x^2(1+x+x^2))/((1-x)(1-x-x^2-x^3)). Cf. A008937.
a(n) = A027024(n) + 1.
a(n) = A000213(n+3) -3. - R. J. Mathar, Jun 24 2020

A113301 Sum of odd-indexed terms of tribonacci numbers.

Original entry on oeis.org

0, 1, 5, 18, 62, 211, 715, 2420, 8188, 27701, 93713, 317030, 1072506, 3628263, 12274327, 41523752, 140473848, 475219625, 1607656477, 5438662906, 18398864822, 62242913851, 210566269283, 712340586524, 2409830942708, 8152399683933, 27579370581033, 93300342369742

Offset: 0

Jonathan Vos Post, Oct 24 2005

A000073 is the tribonacci numbers. A113300 is the sum of even-indexed terms of tribonacci numbers. A099463 is the bisection of the tribonacci numbers. A113300(n) + A113301(n) = cumulative sum of tribonacci numbers = A008937(n). Primes in A113300 include a(2) = 5, a(5) = 211, a(9) = 27701, .... A113300 is semiprime for n = 4, 10, 14, ...

			a(0) = 0 = A000073(1);
a(1) = 0+1 = A000073(1) + A000073(3) = 1;
a(2) = 0+1+4 = A000073(1) + A000073(3) + A000073(5) = 5, prime;
a(3) = 0+1+4+13 = A000073(1) + A000073(3) + A000073(5) + A000073(7) = 18;
a(4) = 0+1+4+13+44 = A000073(1) + A000073(3) + A000073(5) + A000073(7) + A000073(9) = 62 = 2 * 31, semiprime;
a(5) = 0+1+4+13+44+149 = A000073(1) + A000073(3) + A000073(5) + A000073(7) + A000073(9) + A000073(11) = 211, prime.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,0,-1).

Cf. A000073, A008937, A099463, A113300.

I:=[0,1,5,18]; [n le 4 select I[n] else 4*Self(n-1) - 2*Self(n-2) -Self(n-4): n in [1..41]]; // G. C. Greubel, Nov 20 2021

Accumulate[Take[LinearRecurrence[{1,1,1},{0,1,1},40],{1,-1,2}]] (* or *) LinearRecurrence[{4,-2,0,-1},{0,1,5,18},30] (* Harvey P. Dale, Apr 12 2013 *)

@CachedFunction
def T(n): # A000073
    if (n<2): return 0
    elif (n==2): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def A113301(n): return sum(T(2*j+1) for j in (0..n))
[A113301(n) for n in (0..40)] # G. C. Greubel, Nov 20 2021

a(n) = Sum_{j=0..n} A000073(2*j+1).
a(n) + A113300(n) = A008937(n).
a(n) = 4*a(n-1) - 2*a(n-2) - a(n-4), a(0)=0, a(1)=1, a(2)=5, a(3)=18. - Harvey P. Dale, Apr 12 2013
G.f.: x*(1+x) / ((1-x)*(1-3*x-x^2-x^3)). - Colin Barker, May 06 2013

More terms from Colin Barker, May 06 2013

A141579 Numbers k such that the arithmetic mean of the first k tribonacci numbers A000073 is an integer.

Original entry on oeis.org

1, 2, 47, 53, 94, 103, 106, 163, 199, 206, 257, 269, 311, 326, 397, 398, 401, 419, 421, 499, 514, 538, 587, 599, 617, 622, 683, 757, 773, 794, 802, 838, 842, 863, 883, 907, 911, 929, 991, 998, 1021, 1087, 1109, 1123, 1174, 1181, 1198, 1210, 1234, 1237, 1291

Offset: 1

R. J. Mathar, Aug 19 2008

nonn

Numbers in this sequence but not in A140973 are 2021 and 2090 (but no others below 8400). - Emeric Deutsch, Aug 19 2008.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1500 from Harvey P. Dale)

Cf. A000073, A008937, A140973.

A000073 := proc(n) option remember ; if n <= 1 then 0 ; elif n =2 then 1 ; else procname(n-1)+procname(n-2)+procname(n-3) ; fi; end: A008937 := proc(n) option remember ; add(A000073(i),i=0..n+1) ; end: isA := proc(n) if n = 1 then RETURN(true) ; fi; if A008937(n-2) mod n = 0 then true; else false ; fi; end: for n from 1 to 2000 do if isA(n) then printf("%d,",n) ; fi; od ;

Module[{nn=1300,tnos},tnos=LinearRecurrence[{1,1,1},{0,0,1},nn];Position[ Table[Mean[Take[tnos,n]],{n,nn}],?(IntegerQ[#]&)]]//Flatten (* _Harvey P. Dale, Oct 05 2020 *)

{k: k | A008937(k-2)}.

A189154 Number of n X 2 binary arrays without the pattern 0 0 1 1 diagonally, vertically or horizontally.

Original entry on oeis.org

4, 16, 64, 225, 784, 2704, 9216, 31329, 106276, 360000, 1218816, 4124961, 13957696, 47224384, 159769600, 540516001, 1828588644, 6186137104, 20927672896, 70798034241, 239508444816, 810252019600, 2741064339456, 9272956793409

Offset: 1

R. H. Hardin, Apr 17 2011

nonn

Column 2 of A189161.

			Some solutions for 4 X 2:
  1 0   1 0   0 1   1 1   0 1   1 0   1 1   1 1   0 0   1 1
  1 0   1 1   1 0   0 1   1 1   1 1   0 1   1 0   1 1   0 0
  0 1   1 1   1 1   1 0   1 1   1 1   1 1   0 0   1 0   1 1
  1 0   0 0   0 0   0 0   1 0   0 1   0 0   1 1   0 1   0 0

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A008937, A189161.

a(n) = A008937(n+1)^2.
Empirical: a(n) = 4*a(n-1) - a(n-2) - 14*a(n-4) + 4*a(n-5) + 2*a(n-6) + 8*a(n-7) - a(n-8) - a(n-10).
Empirical g.f.: x*(4 + 4*x^2 - 15*x^3 + 4*x^4 + x^5 + 8*x^6 - x^7 - x^9) / ((1 - x)*(1 + x + x^2 - x^3)*(1 - x - x^2 - x^3)*(1 - 3*x - x^2 - x^3)). - Colin Barker, Feb 28 2018

A260056 Irregular triangle read by rows: coefficients T(n, k) of certain polynomials p(n, x) with exponents in increasing order, n >= 0 and 0 <= k <= 2*n.

Original entry on oeis.org

1, 2, 1, 1, 3, 3, 4, 2, 1, 4, 6, 10, 9, 7, 3, 1, 5, 10, 20, 25, 26, 19, 11, 4, 1, 6, 15, 35, 55, 71, 70, 56, 34, 16, 5, 1, 7, 21, 56, 105, 161, 196, 197, 160, 106, 55, 22, 6, 1, 8, 28, 84, 182, 322, 462, 554, 553, 463, 321, 183, 83, 29, 7, 1, 9, 36, 120, 294, 588, 966, 1338, 1569, 1570, 1337, 967, 587, 295, 119, 37, 8, 1, 10, 45, 165, 450, 1002, 1848, 2892, 3873, 4477, 4476, 3874

Offset: 0

Werner Schulte, Nov 08 2015

The triangle is related to the triangle of trinomial coefficients.

			The irregular triangle T(n,k) begins:
n\k:  0   1   2    3    4    5    6    7    8    9   10  11  12  13  14  ...
0     1;
1     2   1   1;
2     3   3   4    2    1;
3     4   6  10    9    7    3    1;
4     5  10  20   25   26   19   11    4    1;
5     6  15  35   55   71   70   56   34   16    5    1;
6     7  21  56  105  161  196  197  160  106   55   22   6   1;
7     8  28  84  182  322  462  554  553  463  321  183  83  29   7   1;
etc.
The polynomial corresponding to row 2 is p(2,x) = 3+3*x+4*x^2+2*x^3+x^4.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A000027 (col 0), A000217 (col 1), A000292 (col 2), A001590, A002426, A004524, A005582 (col 3), A008937, A027907, A095662 (col 5), A113682, A246437.

A027907[n_, k_] := Sum[Binomial[n, j]*Binomial[j, k - j], {j, 0, n}]; Table[ Sum[A027907[j, k], {j, 0, n}], {n,0,10}, {k, 0, 2*n} ] // Flatten (* G. C. Greubel, Mar 07 2017 *)

T(n,0) = n+1, and T(n,k) = 0 for k < 0 or k > 2*n, and T(n+1,k) = T(n,k) + T(n,k-1) + T(n,k-2) for k > 0.
T(n,k) = Sum_{j=0..n} A027907(j,k) for 0 <= k <= 2*n.
T(n,k) = Sum_{j=0..k} (-1)^(k-j)*A027907(n+1,j+1) for 0 <= k <= 2*n.
T(n,k) = T(n,2*n-1-k) + (-1)^k for 0 <= k < 2*n.
p(n,x) = Sum_{k=0..2*n} T(n,k)*x^k = Sum_{k=0..n} (1+x+x^2)^k for n >= 0.
p(n,x) = ((1+x+x^2)^(n+1)-1)/(x+x^2), p(n,0) = p(n,-1) = n+1 for n >= 0.
p(n+1,x) = (1+x+x^2)*p(n,x)+1 for n >= 0.
Sum_{n>=0} p(n,x)*t^n = 1/((1-t)*(1-t*(1+x+x^2))).
T(n,2*n) = 1, and T(n,n) = A113682(n) for n >= 0.
T(n,n-1) = A246437(n+1), and T(n,n-1)+T(n,n) = A002426(n+1) for n > 0.
If d(n) is n-th antidiagonal sum of the triangle then: d(n) = A008937(n+1), and d(n+2)-d(n) = A001590(n+5) for n >= 0.
Conjecture: If a(n) is n-th antidiagonal alternating sum of the triangle then: a(n) = A004524(n+3).
Sum_{k=0..2*n} (-1)^k*T(n,k)^2 = (3^(n+1)-1)/2 for n >= 0.
Sum_{k=0..2*n} (-1)^k*(y*k+1)*T(n,k) = Sum{k=0..n} y*k+1 = (n+1)*(y*n+2)/2 for real y and n >= 0.
Conjecture of linear recurrence for column k: Sum_{m=0..k+2} (-1)^m*T(n+m,k)* binomial(k+2,m) = 0 for k >= 0 and n >= 0.

A317506 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-4,k-1) for 0 <= k <= floor(n/4); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, 16, -1, 32, -4, 64, -12, 128, -32, 256, -80, 1, 512, -192, 6, 1024, -448, 24, 2048, -1024, 80, 4096, -2304, 240, -1, 8192, -5120, 672, -8, 16384, -11264, 1792, -40, 32768, -24576, 4608, -160, 65536, -53248, 11520, -560, 1, 131072, -114688, 28160, -1792, 10

Offset: 0

Shara Lalo, Aug 31 2018

The numbers in rows of the triangle are along "third layer" skew diagonals pointing top-right in center-justified triangle given in A065109 ((2-x)^n) and along "third layer" skew diagonals pointing top-left in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1-2x+x^4) are given by the sequence generated by the row sums. The row sums give A008937. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.83928675521416113... (A058265: Decimal expansion of the tribonacci constant t, the real root of x^3-x^2-x-1), when n approaches infinity.

			Triangle begins:
       1;
       2;
       4;
       8;
      16,      -1;
      32,      -4;
      64,     -12;
     128,     -32;
     256,     -80,     1;
     512,    -192,     6;
    1024,    -448,    24;
    2048,   -1024,    80;
    4096,   -2304,   240,    -1;
    8192,   -5120,   672,    -8;
   16384,  -11264,  1792,   -40;
   32768,  -24576,  4608,  -160;
   65536,  -53248, 11520,  -560,  1;
  131072, -114688, 28160, -1792, 10;
  262144, -245760, 67584, -5376, 60;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Row sums give A008937.
Cf. A065109, A303872.
Cf. A133156, A305098.
Cf. A058265.

t[n_, k_] := t[n, k] = 2^(n - 4 k) * (-1)^k/((n - 4 k)! k!) * (n - 3 k)!; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 * t[n - 1, k] - t[n - 4, k - 1]]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]}] // Flatten

T(n,k) = 2^(n - 4*k) * (-1)^k / ((n - 4*k)! k!) * (n - 3*k)! where n >= 0 and 0 <= k <= floor(n/4).

cp's OEIS Frontend

A077908 Expansion of (1-x)^(-1)/(1+x+x^2-x^3).

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A118884 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0011 (n,k>=0).

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A189155 Number of nX3 binary arrays without the pattern 0 0 1 1 diagonally, vertically or horizontally.

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A192804 Constant term in the reduction of the polynomial 1+x+x^2+...+x^n by x^3->x^2+x+1. See Comments.

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A027083 a(n) = A027082(n, n+2).

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A113301 Sum of odd-indexed terms of tribonacci numbers.

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A141579 Numbers k such that the arithmetic mean of the first k tribonacci numbers A000073 is an integer.

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