

A210881


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


4



1, 1, 3, 3, 4, 4, 4, 7, 5, 7, 5, 9, 10, 9, 11, 6, 11, 13, 17, 14, 18, 7, 13, 16, 22, 27, 23, 29, 8, 15, 19, 27, 35, 44, 37, 47, 9, 17, 22, 32, 43, 57, 71, 60, 76, 10, 19, 25, 37, 51, 70, 92, 115, 97, 123, 11, 21, 28, 42, 59, 83, 113, 149, 186, 157, 199, 12, 23, 31
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OFFSET

1,3


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+1. 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(k1), F(k+2), 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>2.


EXAMPLE

First six rows:
1
1...3
3...4....4
4...7....5....7
5...9....10...9....11
6...11...13...17...14...18
First three polynomials u(n,x): 1, 1 + 3x, 3 + 4x + 4x^2.


MATHEMATICA

u[1, x_] := 1; u[2, x_] := 3 x + 1; 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[%] (* A210881 *)


CROSSREFS

Cf. A208510, A210874, A210875.
Sequence in context: A243348 A136546 A278765 * A058729 A021303 A303821
Adjacent sequences: A210878 A210879 A210880 * A210882 A210883 A210884


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Mar 30 2012


STATUS

approved



