A115390 -id:A115390 - cp's OEIS frontend

A117189 Binomial transform of the tribonacci sequence A000073 (shifted left twice).

1, 2, 5, 14, 40, 114, 324, 920, 2612, 7416, 21056, 59784, 169744, 481952, 1368400, 3885280, 11031424, 31321376, 88930368, 252498816, 716916544, 2035531648, 5779458048, 16409538688, 46591385856, 132286304768, 375598753024, 1066432564736, 3027907856384

Offset: 0

Gary W. Adamson, Mar 01 2006

nonn

a(n)/a(n-1) tends to 2.83928675... = A058265 + 1.

Partial sums are in A073357. - R. J. Mathar, Apr 02 2008

			a(4) = 14 = 1*1 + 3*1 + 3*2 + 1*4;
a(6) = 324 = 2*114 + 1*40 + 2*14 + 3*5 + 4*2 + 5*1. - _Bob Selcoe_, Jun 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,2).

Cf. A000073, A115390.

CoefficientList[Series[-(x - 1)^2/(-1 + 4*x - 4*x^2 + 2*x^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jul 05 2014 *)
LinearRecurrence[{4,-4,2},{1,2,5},40] (* Harvey P. Dale, Oct 10 2016 *)

Binomial transform of A000073 starting with A000073(2): (1, 1, 2, 4, 7, 13, ...).
a(n) = 4*a(n-1)-4*a(n-2)+2*a(n-3), n>2. - T. D. Noe, Nov 07 2006
O.g.f.: -(x-1)^2/(-1+4*x-4*x^2+2*x^3). - R. J. Mathar, Apr 02 2008
a(n) = 2*a(n-1) + Sum_{j=1..n-1} j*a(n-j-1), n>=1; with a(0) = 1. - Bob Selcoe, Jun 28 2014

Corrected and extended by T. D. Noe, Nov 07 2006

A073357 Binomial transform of tribonacci numbers.

Original entry on oeis.org

0, 1, 3, 8, 22, 62, 176, 500, 1420, 4032, 11448, 32504, 92288, 262032, 743984, 2112384, 5997664, 17029088, 48350464, 137280832, 389779648, 1106696192, 3142227840, 8921685888

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jul 29 2002

For n-> infinity the ratio a(n)/a(n-1) approaches 1+c, where c is the real root of the cubic x^3-x^2-x-1=0; c=1.8392867...

a(n) = rightmost term of M^n *[100] where M = the 3X3 matrix [1 1 0 / 0 1 1 / 1 1 2]. Middle term of the vector = partial sums of A073357 through a(n-1). E.g., M^5*[1 0 0] = [18 34 62] where 62 = a(5) and 34 = partial sums of A073357 through a(4): 34 = 0+1+3+8+22. - Gary W. Adamson, Jul 24 2005

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Index entries for linear recurrences with constant coefficients, signature (4, -4, 2).

Cf. A000073, A073313. Trisection of A103685.

h[n_] := h[n]=4*h[n-1]-4*h[n-2]+2*h[n-3]; h[0]=0; h[1]=1; h[2]=3
LinearRecurrence[{4,-4,2},{0,1,3},30] (* Harvey P. Dale, Nov 13 2011 *)

a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-3), a(0)=0, a(1)=1, a(2)=3.
Generating function A(x)=(x-x^2)/(1-4x+4x^2-2x^3).
a(n) = A115390(n+1) - A115390(n). - R. J. Mathar, Apr 16 2009

A103685 Consider the morphism 1->{1,2}, 2->{1,3}, 3->{1}; a(n) is the total number of '3' after n substitutions.

Original entry on oeis.org

0, 0, 1, 5, 17, 51, 147, 419, 1191, 3383, 9607, 27279, 77455, 219919, 624415, 1772895, 5033759, 14292287, 40579903, 115217983, 327136895, 928835455, 2637230207, 7487852799, 21260161279, 60363694335, 171389837823, 486624896511, 1381667623423, 3922950583295

Offset: 0

Roger L. Bagula, Mar 26 2005

Examples of the morphism starting with {1} are shown in A103684. Counting the total number of '1' in rows 1 to n of A103684 yields 1, 3, 8,... = A073357(n+1),
counting the total number of '2' in rows 1 to n yields 0, 1, 4,.. = A115390(n+1),
and counting the total number '3' in rows 1 to n yields a(n), the sequence here.
Inverse binomial transform yields 0, 0, 1, 2, 3, 6, 11, 20,..., a variant of A001590 [Nov 18 2010]

Index entries for linear recurrences with constant coefficients, signature (5,-8,6,-2).

Cf. A073058, A103684.

LinearRecurrence[{5,-8,6,-2},{0,0,1,5},30] (* Harvey P. Dale, Nov 10 2011 *)

a(n)= +5*a(n-1) -8*a(n-2) +6*a(n-3) -2*a(n-4) = a(n-1)+A115390(n). [Nov 18 2010]

G.f.: x^2 / ( (x-1)*(2*x^3-4*x^2+4*x-1) ). [Nov 18 2010]

Depleted by the information already in A073357 and A115390; corrected image of {2} in the defn. - The Assoc. Eds. of the OEIS, Nov 18 2010

A192801 Constant term in the reduction of the polynomial (x+2)^n by x^3->x^2+x+1. See Comments.

Original entry on oeis.org

1, 2, 4, 9, 25, 84, 312, 1199, 4637, 17906, 68976, 265249, 1019069, 3913484, 15026092, 57690143, 221487945, 850350482, 3264725772, 12534190569, 48122302705, 184755243892, 709328262928, 2723314511871, 10455585321989, 40141990468066

Offset: 0

Clark Kimberling, Jul 10 2011

For discussions of polynomial reduction, see A192232 and A192744.
If the same reduction is applied to the sequence (x+1)^n instead of (x+2)^n, the resulting three coefficient sequences are essentially as follows:
A078484: constants
A099216: coefficients of x
A115390: coefficients of x^2.

			The first five polynomials p(n,x) and their reductions:
p(1,x)=1 -> 1
p(2,x)=x+2 -> x+2
p(3,x)=x^2+4x+4 -> x^2+1
p(4,x)=x^3+6x^2+12x+8 -> x^2+4x+4
p(5,x)=x^4+8x^3+24x^2+32x+16 -> 7x^2+13*x+9, so that
A192798=(1,2,4,9,25,...), A192799=(0,1,4,13,42,...), A192800=(0,0,1,7,34,...).

Index entries for linear recurrences with constant coefficients, signature (7,-15,11).

Cf. A192744, A192232, A192616, A192802, A192803.

q = x^3; s = x^2 + x + 1; z = 40;
p[n_, x_] := (x + 2)^n;
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}]  (* A192801 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192802 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A192803 *)

a(n) = 7*a(n-1)-15*a(n-2)+11*a(n-3).

G.f.: -(5*x^2-5*x+1)/(11*x^3-15*x^2+7*x-1). [Colin Barker, Jul 27 2012]

Recurrence corrected by Colin Barker, Jul 27 2012

A116521 Binomial transform of tetranacci sequence A000078.

Original entry on oeis.org

0, 0, 0, 1, 5, 17, 51, 148, 429, 1250, 3655, 10701, 31336, 91752, 268623, 786414, 2302262, 6739984, 19731685, 57765711, 169112717, 495088023, 1449400960, 4243211207, 12422263776, 36366946961, 106466490879, 311687250156

Offset: 0

Jonathan Vos Post, Mar 10 2006

See also A115390, the binomial transform of tribonacci sequence A000073. Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4), with a(0) = a(1) = a(2) = 0, a(3) = 1.

			Table shows the tetranacci numbers multiplied into rows of Pascal's triangle.
1*0 = 0.
1*0 + 1*0 = 0.
1*0 + 2*0 + 1*0 = 0.
1*0 + 3*0 + 3*0 + 1* 1 = 1.
1*0 + 4*0 + 6*0 + 4*1 + 1*1 = 5.
1*0 + 5*0 + 10*0 + 10*1 + 5*1 + 1*2 = 17.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,6,-1).

Cf. A000073, A000078, A115390.

t[0]:=0: t[1]:=0: t[2]:=0: t[3]:=1: for n from 4 to 35 do t[n]:=t[n-1]+t[n-2]+t[n-3]+t[n-4] od: seq(add(binomial(n,k)*t[k],k=0..n),n=0..30); # end of first Maple program
G:=x^3/(1-5*x+8*x^2-6*x^3+x^4): Gser:=series(G,x=0,33): seq(coeff(Gser,x,n),n=0..30); # Emeric Deutsch, Apr 09 2006

LinearRecurrence[{5,-8,6,-1}, {0,0,0,1}, 25] (* G. C. Greubel, Nov 03 2016 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,6,-8,5]^n*[0;0;0;1])[1,1] \\ Charles R Greathouse IV, Jun 28 2017

a(n) = Sum_{k=0..n} C(n,k) * A000078(k).
G.f.: x^3/(1-5*x+8*x^2-6*x^3+x^4). - Emeric Deutsch, Apr 09 2006
a(n) = 5*a(n-1) - 8*a(n-2) + 6*a(n-3) - a(n-4). - G. C. Greubel, Nov 03 2016

Definition corrected by Franklin T. Adams-Watters, Mar 13 2006

More terms from Emeric Deutsch, Apr 09 2006

cp's OEIS Frontend

A117189 Binomial transform of the tribonacci sequence A000073 (shifted left twice).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A073357 Binomial transform of tribonacci numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A103685 Consider the morphism 1->{1,2}, 2->{1,3}, 3->{1}; a(n) is the total number of '3' after n substitutions.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A192801 Constant term in the reduction of the polynomial (x+2)^n by x^3->x^2+x+1. See Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A116521 Binomial transform of tetranacci sequence A000078.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions