|
%I #35 Nov 09 2021 18:39:46
%S 1,42,1379,47124,1599205,54335358,1845747527,62701403688,
%T 2130000094537,72357312787410,2458018570699691,83500274463891516,
%U 2836551311028252973,96359244313163973414,3273377755262716618895,111198484435049515150416,3777475093033912744231057
%N A fourth-order linear divisibility sequence related to the Pell numbers.
%C 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
%H Michael De Vlieger, <a href="/A238537/b238537.txt">Table of n, a(n) for n = 1..654</a>
%H Peter Bala, <a href="/A238536/a238536.pdf">A family of linear divisibility sequences of order four</a>
%H E. L. Roettger and H. C. Williams, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Roettger/roettger12.html">Appearance of Primes in Fourth-Order Odd Divisibility Sequences</a>, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Divisibility_sequence">Divisibility sequence</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pell_number">Pell number</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (28,202,28,-1)
%F a(n) = (1/5)*A000129(3*n)*A001333(n).
%F a(n) = (1/(20*sqrt(2)))*((1 + sqrt(2))^(3*n) - (1 - sqrt(2))^(3*n))*( (1 + sqrt(2))^n + (1 - sqrt(2))^n ).
%F O.g.f.: x*(1 + 14*x + x^2)/( (1 + 6*x + x^2)*(1 - 34*x + x^2) ).
%F Recurrence equation: a(n) = 28*a(n-1) + 202*a(n-2) + 28*a(n-3) - a(n-4).
%F a(n) = (1/10) * (Pell(4n) + (-1)^n*Pell(2n)), with Pell(n) = A000129(n). - _Ralf Stephan_, Mar 01 2014
%t LinearRecurrence[{28, 202, 28, -1}, {1, 42, 1379, 47124}, 17] (* _Jean-François Alcover_, Nov 02 2019 *)
%Y Cf. A000032, A000045, A215466, A238536, A238538, A238539, A238540, A238541.
%K nonn,easy
%O 1,2
%A _Peter Bala_, Feb 28 2014
|
