

A210874


Triangular array U(n,k) of coefficients of polynomials defined in Comments.


4



1, 2, 3, 3, 5, 4, 4, 7, 7, 7, 5, 9, 10, 12, 11, 6, 11, 13, 17, 19, 18, 7, 13, 16, 22, 27, 31, 29, 8, 15, 19, 27, 35, 44, 50, 47, 9, 17, 22, 32, 43, 57, 71, 81, 76, 10, 19, 25, 37, 51, 70, 92, 115, 131, 123, 11, 21, 28, 42, 59, 83, 113, 149, 186, 212, 199, 12, 23, 31
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OFFSET

1,2


COMMENTS

Polynomials u(n,k) are defined by u(n,x)=x*u(n1,x)+(x^2)*u(n2,x)+n*(x+1), where u(1)=1 and u(2,x)=3x+2. The array (U(n,k)) is defined by rows:
u(n,x)=U(n,1)+U(n,2)*x+...+U(n,n1)*x^(n1).
In each column, the first number is a Lucas number and the difference between each two consecutive terms is a Fibonacci number (see the Formula section).
Alternating row sums: 1,2,3,5,8,13,21,... (signed Fibonacci numbers)


LINKS

Table of n, a(n) for n=1..69.


FORMULA

Column k consists of the partial sums of the following sequence: L(k), F(k+1), F(k+1), F(k+1), F(k+1),..., where L=A000032 (Lucas numbers) and F=000045 (Fibonacci numbers. That is, U(n+1,k)U(n,k)=F(k+1) for n>1.


EXAMPLE

First six rows:
1
2...3
3...5...4
4...7...7.....7
5...9...10....12...11
6...11...13...17...19...18
First three polynomials u(n,x): 1, 2 + 3x, 3 + 5x + 4x^2.


MATHEMATICA

u[1, x_] := 1; u[2, x_] := 3 x + 2; z = 14;
u[n_, x_] := x*u[n  1, x] + (x^2)*u[n  2, x] + n*(x + 1);
Table[Expand[u[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A210874 *)


CROSSREFS

Cf. A208510, A210881, A210875, A210880.
Sequence in context: A268087 A257004 A126571 * A244796 A080391 A273494
Adjacent sequences: A210871 A210872 A210873 * A210875 A210876 A210877


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Mar 30 2012


STATUS

approved



