A129589 a(n) = 2*A000129(n) + A000129(n-1) - n.

1, 3, 9, 25, 65, 163, 401, 977, 2369, 5731, 13849, 33449, 80769, 195011, 470817, 1136673, 2744193, 6625091, 15994409, 38613945, 93222337, 225058659, 543339697, 1311738097, 3166815937, 7645370019, 18457556025, 44560482121, 107578520321, 259717522819

Offset

1,2

Comments

a(n)/a(n-1) tends to 1 + sqrt(2).

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Formula

a(n) = 2*A000129(n) + A000129(n-1) - n; where A000129 = the Pell sequence. a(n) = A000129(n+1) - n.

From Colin Barker, Sep 02 2019: (Start)

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

a(n) = (-(1 - sqrt(2))^(1+n) + (1+sqrt(2))^(1+n)) / (2*sqrt(2)) - n.

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4) for n>4.

(End)

Example

a(5) = 65 = 2*A000129(5) + A000129(4) - 5 = 2*29 + 12 - 5.
a(5) = 65 = A000129(6) - 5 = 70 - 5.

Prog

(PARI) Vec(x*(1 - x + x^2 + x^3) / ((1 - x)^2*(1 - 2*x - x^2)) + O(x^35)) \\ Colin Barker, Sep 02 2019

Crossrefs

Cf. A000129.

Sequence in context: A195417 A295142 A002064 * A335472 A096322 A058396

Adjacent sequences: A129586 A129587 A129588 * A129590 A129591 A129592

Keyword

nonn,easy

Author

Gary W. Adamson, Sep 19 2007

Status

approved