A103685 -id:A103685 - cp's OEIS frontend

A103684 Triangle read by rows, based on the morphism f: 1->{1,2}, 2->{1,3}, 3->{1}. First row is 1. If current row is a,b,c,..., then the next row is a,b,c,...,f(a),f(b),f(c),...

1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1

Offset: 1

Roger L. Bagula, Mar 26 2005

			[1], [1,1,2], [1,1,2,1,2,1,2,1,3], [1,1,2,1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,3,1,2,1,3,1,2,1], ...

G. C. Greubel, Table of n, a(n) for the first 8 rows, flattened

Cf. A073058, A103685, A103682.

NestList[ Flatten[ Join[ #, # /. {1 -> {1, 2}, 2 -> {1, 3}, 3->{1}}]] &, {1}, 4] // Flatten (* Robert G. Wilson v, Jul 09 2006 - corrected by G. C. Greubel, Oct 26 2017 *)

{a(n)=local(m,v,w); v=w=[1]; while(length(w)Michael Somos, Apr 16 2005 */

Image of {3} in the definition corrected by R. J. Mathar, Nov 18 2010

A115390 Binomial transform of tribonacci sequence A000073.

Original entry on oeis.org

0, 0, 1, 4, 12, 34, 96, 272, 772, 2192, 6224, 17672, 50176, 142464, 404496, 1148480, 3260864, 9258528, 26287616, 74638080, 211918912, 601698560, 1708394752, 4850622592, 13772308480, 39103533056, 111026143488, 315235058688, 895042726912, 2541282959872

Offset: 0

Jonathan Vos Post, Mar 08 2006

See also A117189 Binomial transform of the tribonacci sequence A000073.

			1*0 = 0.
1*0 + 1*0 = 0.
1*0 + 2*0 + 1*1 = 1.
1*0 + 3*0 + 3*1 + 1* 1 = 4.
1*0 + 4*0 + 6*1 + 4*1 + 1*2 = 12.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
J. Pan, Multiple Binomial Transforms and Families of Integer Sequences, J. Int. Seq. 13 (2010), 10.4.2
J. Pan, Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences, J. Int. Seq. 14 (2011) # 11.3.4, remark 14.
Eric Weisstein's World of Mathematics, Binomial Transform.
Index entries for linear recurrences with constant coefficients, signature (4,-4,2).

Cf. A000073, A117189. Trisection of A103685.

a115390 n = a115390_list !! n
a115390_list = 0 : 0 : 1 : map (* 2) (zipWith (-) a115390_list
   (tail $ map (* 2) $ zipWith (-) a115390_list (tail a115390_list)))
-- Reinhard Zumkeller, Oct 21 2011

b[0]=b[1]=0;b[2]=1;b[n_]:=b[n]=b[n-1]+b[n-2]+b[n-3]; a[n_]:=Sum[n!/(k!*(n-k)!)*b[k],{k,0,n}];Table[a[n],{n,0,27}] (* Farideh Firoozbakht, Mar 11 2006 *)

sum(sum(binomial(j-1,k-1)*2^(j-k)*binomial(n-j+k-1,2*k-1),j,k,n-k),k,1,n); /* Vladimir Kruchinin, Aug 18 2010 */

a(n) = Sum_{k=0..n} C(n,k)*A000073(k).
O.g.f.: -x^2/(-1+4*x-4*x^2+2*x^3). - R. J. Mathar, Apr 02 2008
a(n) = sum(sum(binomial(j-1,k-1)*2^(j-k)*binomial(n-j+k-1,2*k-1),j,k,n-k),k,1,n). - Vladimir Kruchinin, Aug 18 2010

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

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A103684 Triangle read by rows, based on the morphism f: 1->{1,2}, 2->{1,3}, 3->{1}. First row is 1. If current row is a,b,c,..., then the next row is a,b,c,...,f(a),f(b),f(c),...

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A115390 Binomial transform of tribonacci sequence A000073.

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A073357 Binomial transform of tribonacci numbers.

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