A238537 A fourth-order linear divisibility sequence related to the Pell numbers.

1, 42, 1379, 47124, 1599205, 54335358, 1845747527, 62701403688, 2130000094537, 72357312787410, 2458018570699691, 83500274463891516, 2836551311028252973, 96359244313163973414, 3273377755262716618895, 111198484435049515150416, 3777475093033912744231057

Offset

1,2

Comments

Let P and Q be integers. The Lucas sequences U(n) and V(n) (which depend on P and Q) are a pair of integer sequences that satisfy the recurrence equation a(n) = P*a(n-1) - Q*a(n-2) with the initial conditions U(0) = 0, U(1) = 1 and V(0) = 2, V(1) = P, respectively. The sequence {U(n)}n>=1 is a divisibility sequence, i.e., U(n) divides U(m) whenever n divides m and U(n) <> 0. In general the sequence V(n) is not a divisibility sequence. However, it can be shown that if p >= 3 is an odd integer then the sequence {U(p*n)*V(n)}n>=1 is a divisibility sequence satisfying a linear recurrence of order 4. For a proof and a generalization of this result see the Bala link. Here we take p = 3 with P = 2 and Q = -1, for which U(n) is the sequence of Pell numbers A000129, and consider the normalized divisibility sequence with initial term equal to 1. For other sequences of this type see A238536 and A238538

Links

Michael De Vlieger, Table of n, a(n) for n = 1..654

Peter Bala, A family of linear divisibility sequences of order four

E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.

Wikipedia, Divisibility sequence

Wikipedia, Pell number

Index entries for linear recurrences with constant coefficients, signature (28,202,28,-1)

Formula

a(n) = (1/5)*A000129(3*n)*A001333(n).

a(n) = (1/(20*sqrt(2)))*((1 + sqrt(2))^(3*n) - (1 - sqrt(2))^(3*n))*( (1 + sqrt(2))^n + (1 - sqrt(2))^n ).

O.g.f.: x*(1 + 14*x + x^2)/( (1 + 6*x + x^2)*(1 - 34*x + x^2) ).

Recurrence equation: a(n) = 28*a(n-1) + 202*a(n-2) + 28*a(n-3) - a(n-4).

a(n) = (1/10) * (Pell(4n) + (-1)^n*Pell(2n)), with Pell(n) = A000129(n). - Ralf Stephan, Mar 01 2014

Mathematica

LinearRecurrence[{28, 202, 28, -1}, {1, 42, 1379, 47124}, 17] (* Jean-François Alcover, Nov 02 2019 *)

Crossrefs

Cf. A000032, A000045, A215466, A238536, A238538, A238539, A238540, A238541.

Sequence in context: A331858 A331805 A348775 * A077123 A121974 A096048

Adjacent sequences: A238534 A238535 A238536 * A238538 A238539 A238540

Keyword

nonn,easy

Author

Peter Bala, Feb 28 2014

Status

approved