A085293 Product of Lucas (A000204) and a Pell Companion series (A002203).

2, 18, 56, 238, 902, 3564, 13862, 54238, 211736, 827298, 3231362, 12623044, 49308482, 192613698, 752401496, 2939092798, 11480914982, 44847668844, 175187526662, 684331472398, 2673190054136, 10442227799538, 40790261396162, 159338166024964, 622419427368002

Offset: 1

Author

Gary W. Adamson, Jun 24 2003

Keywords

• nonn
• easy

Comments

Convergent a(n+1)/a(n) = ((1+sqrt(5))/2)*(1+sqrt(2)) = (1.618...)*(2.414213...) = 3.9062796... = (1 + sqrt(2) + sqrt(5) + sqrt(10))/2.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,7,2,-1).

Crossrefs

Cf. A000204, A002203, A085292.

Programs

Magma

I:=[2,18,56,238]; [n le 4 select I[n] else 2*Self(n-1) + 7*Self(n-2) + 2*Self(n-3) - Self(n-4):n in [1..30]]; // Marius A. Burtea, Aug 25 2019

Formula

a(n) = A000204(n) * A002203(n), n > 0.

a(n) = 2*A085292(n).

a(n) = (((1+sqrt(5))/2)^n + ((1-sqrt(5))/2)^n) * ((1+sqrt(2))^n + (1-sqrt(2))^n).

From Colin Barker, Oct 15 2013: (Start)

a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).

G.f.: -2*x*(2*x^3 - 3*x^2 - 7*x - 1) / (x^4 - 2*x^3 - 7*x^2 - 2*x + 1). (End)

E.g.f.: 4*(exp(x/2)*(cosh(x/sqrt(2))*cosh(sqrt(5/2)*x)*cosh(sqrt(5)*x/2)+sinh(x/sqrt(2))*sinh(sqrt(5/2)*x)*sinh(sqrt(5)*x/2))-1). - Stefano Spezia, Aug 25 2019

Extensions

More terms from David Wasserman, Jan 31 2005

More terms from Colin Barker, Oct 16 2013