A088210 -id:A088210 - cp's OEIS frontend

A209819 Triangle of coefficients of polynomials u(n,x) jointly generated with A209820; see the Formula section.

1, 1, 3, 1, 5, 7, 1, 5, 17, 17, 1, 5, 21, 53, 41, 1, 5, 21, 81, 157, 99, 1, 5, 21, 89, 289, 449, 239, 1, 5, 21, 89, 361, 973, 1253, 577, 1, 5, 21, 89, 377, 1389, 3133, 3433, 1393, 1, 5, 21, 89, 377, 1565, 5085, 9745, 9273, 3363, 1, 5, 21, 89, 377, 1597, 6285

Offset: 1

Clark Kimberling, Mar 23 2012

Let T(n,k) be the general term.
T(n,n): A001333
T(n,n-1): A088210
Row sums: A003561
Alternating row sums: 1,-2,3,-4,5,-6,7,-8,...
Limiting row: F(2), F(5),F(8),...where F=A000045 (Fibonacci numbers)
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...3
1...5...7
1...5...17...17
1...5...21...53...41
First three polynomials u(n,x): 1, 1 + 3x, 1 + 5x + 7x^2.

Cf. A209820, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209819 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209820 *)

u(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A088211 Denominators of convergents of the continued fraction with the n+1 partial quotients: [2;2,2,...(n 2's)...,2,n+1], starting with [1], [2;2], [2;2,3], [2;2,2,4], ...

Original entry on oeis.org

1, 2, 7, 22, 65, 186, 519, 1422, 3841, 10258, 27143, 71270, 185921, 482314, 1245191, 3201182, 8199169, 20931234, 53276679, 135246390, 342508097, 865501658, 2182728199, 5494630702, 13808551681, 34648530866, 86815769095, 217237177222

Offset: 0

Paul D. Hanna, Sep 23 2003

Numerators are A088210.

			A088210(3)/a(3) = [2;2,2,4] = 53/22.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Cf. A088210, A000129.

LinearRecurrence[{4, -2, -4, -1}, {1, 2, 7, 22}, 30] (* Paolo Xausa, Feb 08 2024 *)

G.f.: (1-2*x+x^2+2*x^3)/(1-2*x-x^2)^2.

a(n) = A000129(n+1) + (n-1)*A000129(n), where A000129 are the Pell numbers. [Corrected by Paolo Xausa, Feb 08 2024]

cp's OEIS Frontend

A209819 Triangle of coefficients of polynomials u(n,x) jointly generated with A209820; see the Formula section.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula