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Tribonacci array.
4

%I #18 Nov 27 2016 22:10:32

%S 1,1,1,1,1,2,1,1,3,3,1,4,6,2,1,5,10,7,1,1,6,15,16,6,1,7,21,30,19,3,1,

%T 8,28,50,45,16,1,1,9,36,77,90,51,10,1,10,45,112,161,126,45,4,1,11,55,

%U 156,266,266,141,30,1,1,12,66,210,414,504,357,126,15,1,13,78,275,615,882

%N Tribonacci array.

%C Row sums are tribonacci numbers.

%C From _Gary W. Adamson_, Nov 15 2016: (Start)

%C With an alternative format:

%C 1, 0, 0, 0, 0, 0, 0, ...

%C 1, 1, 1, 0, 0, 0, 0, ...

%C 1, 2, 3, 2, 1, 0, 0, ...

%C 1, 3, 6, 7, 6, 3, 1, ...

%C ... (where the k-th row is (1 + x + x^2)^k), let q(x) = (r(x) * r(x^3) * r(x^9) * r(x^27) * ...). Then q(x) is the binomial sequence beginning (1, k, ...). Example: (1, 3, 6, 10, ...) = q(x) with r(x) = (1, 3, 6, 7, 3, 1, 0, 0, 0). (End)

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

%H Reinhard Zumkeller, <a href="/A082870/b082870.txt">Rows n = 0..150 of triangle, flattened</a>

%F G.f.: x/(1 - x - x^2*y - x^3*y^2). - _Vladeta Jovovic_, May 30 2003

%e Triangle begins:

%e 1,

%e 1,

%e 1, 1,

%e 1, 2, 1,

%e 1, 3, 3,

%e 1, 4, 6, 2,

%e 1, 5, 10, 7, 1,

%e 1, 6, 15, 16, 6,

%o (Haskell)

%o a082870 n k = a082870_tabf !! n !! k

%o a082870_row n = a082870_tabf !! n

%o a082870_tabf = map (takeWhile (> 0)) a082601_tabl

%o -- _Reinhard Zumkeller_, Apr 13 2014

%Y A082601 is a better version. Cf. A000073, A078802.

%Y Cf. A004396 (row lengths).

%K nonn,tabf

%O 0,6

%A _Gary W. Adamson_, May 24 2003

%E More terms from _Vladeta Jovovic_, May 30 2003