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A082601 Tribonacci array: to get the next row, right-adjust the previous 3 rows and add them, then append a final 0. 5
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 0, 0, 1, 4, 6, 2, 0, 0, 1, 5, 10, 7, 1, 0, 0, 1, 6, 15, 16, 6, 0, 0, 0, 1, 7, 21, 30, 19, 3, 0, 0, 0, 1, 8, 28, 50, 45, 16, 1, 0, 0, 0, 1, 9, 36, 77, 90, 51, 10, 0, 0, 0, 0, 1, 10, 45, 112, 161, 126, 45, 4, 0, 0, 0, 0, 1, 11, 55, 156, 266, 266, 141, 30, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Coefficients of tribonacci polynomials: t_0 = 1, t_1 = x, t_2 = x^2 + x, t_n = x*(t_{n-1} + t_{n-2} + t_{n-3}).

Row sums are tribonacci numbers.

From Petros Hadjicostas, Jun 10 2020: (Start)

To prove a Swamy inequality for the above tribonacci polynomials, we use Guilfoyle's (1967) technique. We write t_n as the determinant of an n X n matrix and then apply Hadamard's inequality.

Since x*t_{n-3} + x*t_{n-2} + x*t_{n-1} - t_n = 0 (with the above initial conditions), we may prove that for n >= 3, t_n = det(A_n), where A_n is the n X n matrix A_n = [[x,-1,0,0,0,...,0,0,0,0,0], [x,x,-1,0,0,...,0,0,0,0,0], [x,x,x,-1,0,...,0,0,0,0,0], [0,x,x,x,-1,...,0,0,0,0,0], ..., [0,0,0,0,0,...,x,x,x,-1,0], [0,0,0,0,0,...,0,x,x,x,-1], [0,0,0,0,0,...,0,0,x,x,x]]).

Using Hadamard's inequality, we obtain t_n^2 <= 3*x^2*(2*x^2 + 1)*(x^2 + 1)*(3*x^2 + 1)^(n-3) for all integers n >= 3 and all real x. (Of course, it is not true for n = 0, 1, 2.)

Guilfoyle's technique can be applied for Werner Schulte's polynomial sequence below, i.e., for p^2*U(n) + p*q*U(n+1) + q^2*U(n+2) - U(n+3) = 0. The first three rows and first three columns of the matrix A_n depend on the initial conditions. We omit the details. (End)

REFERENCES

Thomas Koshy, Fibonacci and Lucas numbers with Applications, Vol. 2, Wiley, 2019; see p. 33. [He gives Swamy inequalities for the Fibonacci and the Lucas polynomials. Vol. 1 was published in 2001. - Petros Hadjicostas, Jun 10 2020]

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Richard Guilfoyle, Comment to the solution of Problem E1846, Amer. Math. Monthly, 74(5), 1967, 593.

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, 2001; Chapter 47: Tribonacci Polynomials: ("In 1973, V.E. Hoggatt, Jr. and M. Bicknell generalized Fibonacci polynomials to Tribonacci polynomials tx(x)"); Table 47.1, page 534: "Tribonacci Array".

M. N. S. Swamy and R. E. Giudici, Solution to Problem E1846, Amer. Math. Monthly, 74(5), 1967, 592-593.

Wikipedia, Hadamard's inequality.

FORMULA

G.f.: x/(1 - x - x^2*y - x^3*y^2). - Vladeta Jovovic, May 30 2003

From Werner Schulte, Feb 22 2017: (Start)

T(n,k) = Sum_{j=0..floor(k/2)} binomial(k-j,j)*binomial(n-k,k-j) for 0 <= k and k <= floor(2*n/3) with binomial(i,j) = 0 for i<j (see Dennis P. Walsh at A078802).

Based on two integers p and q define the integer sequence U(n) by U(0) = 0 and U(1) = 0 and U(n+2) = Sum_{k=0..floor(2*n/3)} T(n,k)*p^k*q^(2*n-3*k) for n >= 0. That yields the g.f. f(p,q,x) = x^2/(1 - q^2*x - p*q*x^2 - p^2*x^3) and the recurrence U(n+3) = q^2*U(n+2) + p*q*U(n+1) + p^2*U(n) for n >= 0 with initial values U(0) = U(1) = 0 and U(2) = 1. For p = q = +/-1, you'll get tribonacci numbers A000073. For p = -1 and q = 1, you'll get A021913. (End)

EXAMPLE

Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:

  1;

  1,  0;

  1,  1,  0;

  1,  2,  1,  0;

  1,  3,  3,  0,  0;

  1,  4,  6,  2,  0,  0;

  1,  5, 10,  7,  1,  0,  0;

  ...

From Petros Hadjicostas, Jun 10 2020: (Start)

The n-th tribonacci polynomial is t_n = Sum_{k=0..n} T(n,k)*x^(n-k), so, for example:

t_4 = x^4 + 3*x^3 + 3*x^2;

t_5 = x^5 + 4*x^4 + 6*x^3 + 2*x^2;

t_6 = x^6 + 5*x^5 + 10*x^4 + 7*x^3 + x^2;

t_7 = x^7 + 6*x^6 + 15*x^5 + 16*x^4 + 6*x^3.

We have

t_4 = det([[x,-1,0,0]; [x,x,-1,0]; [x,x,x,-1]; [0,x,x,x]]);

t_5 = det([[x,-1,0,0,0]; [x,x,-1,0,0]; [x,x,x,-1,0]; [0,x,x,x,-1]; [0,0,x,x,x]]);

t_6 = det([[x,-1,0,0,0,0]; [x,x,-1,0,0,0]; [x,x,x,-1,0,0]; [0,x,x,x,-1,0]; [0,0,x,x,x,-1]; [0,0,0,x,x,x]]);

t_7 = det([[x,-1,0,0,0,0,0]; [x,x,-1,0,0,0,0]; [x,x,x,-1,0,0,0]; [0,x,x,x,-1,0,0]; [0,0,x,x,x,-1,0]; [0,0,0,x,x,x,-1]; [0,0,0,0,x,x,x]]). (End)

MAPLE

G:=x*y/(1-x-x^2*y-x^3*y^2): Gs:=simplify(series(G, x=0, 18)): for n from 1 to 16 do P[n]:=sort(coeff(Gs, x^n)) od: seq(seq(coeff(P[i], y^j), j=1..i), i=1..16);

MATHEMATICA

Table[SeriesCoefficient[x/(1 - x - x^2*y - x^3*y^2), {x, 0, n}, {y, 0, k}], {n, 13}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Feb 22 2017 *)

PROG

(Haskell)

a082601 n k = a082601_tabl !! n !! k

a082601_row n = a082601_tabl !! n

a082601_tabl = [1] : [1, 0] : [1, 1, 0] : f [0, 0, 1] [0, 1, 0] [1, 1, 0]

   where f us vs ws = ys : f (0:vs) (0:ws) ys where

                      ys = zipWith3 (((+) .) . (+)) us vs ws ++ [0]

-- Reinhard Zumkeller, Apr 13 2014

CROSSREFS

Closely related to A078802. A better version of A082870. Cf. A000073.

Cf. A002426 (central terms).

Sequence in context: A173438 A103493 A121480 * A286509 A213887 A279589

Adjacent sequences:  A082598 A082599 A082600 * A082602 A082603 A082604

KEYWORD

nonn,tabl,easy

AUTHOR

Gary W. Adamson, May 24 2003

EXTENSIONS

Edited by Anne Donovan and N. J. A. Sloane, May 27 2003

More terms from Emeric Deutsch, May 06 2004

STATUS

approved

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Last modified September 22 22:24 EDT 2020. Contains 337291 sequences. (Running on oeis4.)