A106627 Product L(n)*L_4(n), where L(n) are Lucas numbers and L_4(n) are Lucas 4-step numbers.

8, 1, 9, 28, 105, 286, 918, 2871, 8977, 27892, 87084, 271635, 847182, 2641991, 8240325, 25700488, 80156033, 249994997, 779700654, 2431777739, 7584375260

Offset

0,1

Comments

a(n) is semiprime iff n is an element of A001606 (an index of a prime Lucas number) and an element of A104577 (an index of a prime Lucas 4-step number). The only known such are n = 2, 8, 16, 19, 71, (through 145858).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,4,5,9,3,-2,1,-1).

Formula

a(n) = A000032(n) * A073817(n).

a(n) = +a(n-1) +4*a(n-2) +5*a(n-3) +9*a(n-4) +3*a(n-5) -2*a(n-6) +a(n-7) -a(n-8). - R. J. Mathar, Dec 22 2010

G.f.: (8 -7*x -24*x^2 -25*x^3 -36*x^4 -9*x^5 +4*x^6 -x^7) / (1 -x -4*x^2 -5*x^3 -9*x^4 -3*x^5 +2*x^6 -x^7 +x^8). - Colin Barker, Jun 17 2012

Example

a(0) = 8 because L(0) * L_4(0) = 2 * 4.
a(1) = 1 because L(1) * L_4(1) = 1 * 1.
a(2) = 9 because L(2) * L_4(2) = 3 * 3.
a(3) = 28 because L(3) * L_4(3) = 4 * 7.
a(4) = 105 because L(4) * L_4(4) = 7 * 15.
a(5) = 286 because L(5) * L_4(5) = 11 * 26.
a(6) = 918 because L(6) * L_4(6) = 18 * 51.

Mathematica

LinearRecurrence[{1, 4, 5, 9, 3, -2, 1, -1}, {8, 1, 9, 28, 105, 286, 918, 2871}, 40] (* G. C. Greubel, Feb 19 2019 *)

Prog

(PARI) my(x='x+O('x^40)); Vec((8-7*x-24*x^2-25*x^3-36*x^4-9*x^5+4*x^6 -x^7)/(1-x-4*x^2-5*x^3-9*x^4-3*x^5+2*x^6-x^7+x^8)) \\ G. C. Greubel, Feb 19 2019
(Magma) I:=[8, 1, 9, 28, 105, 286, 918, 2871]; [n le 8 select I[n] else Self(n-1)+4*Self(n-2)+5*Self(n-3)+9*Self(n-4)+3*Self(n-5)-2*Self(n-6) + Self(n-7)-Self(n-8): n in [1..30]]; // G. C. Greubel, Feb 19 2019
(Sage) ((8-7*x-24*x^2-25*x^3-36*x^4-9*x^5+4*x^6-x^7)/(1-x-4*x^2-5*x^3 -9*x^4-3*x^5+2*x^6-x^7+x^8)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 19 2019
(GAP) a:=[8, 1, 9, 28, 105, 286, 918, 2871];; for n in [9..30] do a[n]:=a[n-1] +4*a[n-2]+5*a[n-3]+9*a[n-4]+3*a[n-5]-2*a[n-6]+a[n-7]-a[n-8]; od; a; # G. C. Greubel, Feb 19 2019

Crossrefs

Cf. A000032, A000040, A001358, A001606, A073446, A073817, A104576.

Sequence in context: A298519 A238271 A217430 * A302152 A160925 A345059

Adjacent sequences: A106624 A106625 A106626 * A106628 A106629 A106630

Keyword

easy,nonn

Author

Jonathan Vos Post, May 11 2005

Status

approved