This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A082601 Tribonacci array: to get next row, right-adjust 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. LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened 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". 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=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 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; 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, 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 ++  -- 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 13 15:41 EST 2019. Contains 329106 sequences. (Running on oeis4.)