A273626 A fourth-order divisibility sequence: a(n) = (1/14)*(Pell(4*n) + Pell(2*n)).
1, 30, 995, 33660, 1142629, 38810970, 1318402631, 44786716920, 1521429030985, 51683794848150, 1755727563817259, 59643053188493940, 2026108079758297261, 68828031652259981010, 2338126968060165944975, 79427488882178225107440, 2698196495024745460575889
Offset: 1
Links
- Colin Barker, Table of n, a(n) for n = 1..600
- P. Bala, Lucas sequences and divisibility sequences
- Wikipedia, Lucas Sequence
- Index entries for linear recurrences with constant coefficients, signature (40,-206,40,-1).
Programs
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Magma
I:=[1,30,995,33660]; [n le 4 select I[n] else 40*Self(n-1)-206*Self(n-2)+40*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 04 2016
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Maple
#A273626 A000129 := proc (n) option remember; if n <= 1 then n else 2*A000129(n-1) + A000129(n-2) end if end proc: seq(1/14*(A000129(4*n) + A000129(2*n)), n = 1..20);
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Mathematica
LinearRecurrence[{40,-206,40,-1},{1,30,995,33660},100] (* G. C. Greubel, Jun 02 2016 *)
Formula
a(n) = sqrt(2)/56*( (sqrt(2) + 1)^(4*n) - (sqrt(2) - 1)^(4*n) + (sqrt(2) + 1)^(2*n) - (sqrt(2) - 1)^(2*n) ).
a(n) = 1/14*Pell(2*n)*A081555(n).
a(n) = -a(-n).
a(n) = 40*a(n-1) - 206*a(n-2) + 40*a(n-3) - a(n-4) for n>4.
O.g.f.: x*(x^2 - 10*x + 1)/((x^2 - 6*x + 1)*(x^2 - 34*x + 1)).
Comments