login
Product L(n)*L_4(n), where L(n) are Lucas numbers and L_4(n) are Lucas 4-step numbers.
1

%I #19 Sep 08 2022 08:45:18

%S 8,1,9,28,105,286,918,2871,8977,27892,87084,271635,847182,2641991,

%T 8240325,25700488,80156033,249994997,779700654,2431777739,7584375260

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

%C 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).

%H G. C. Greubel, <a href="/A106627/b106627.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,5,9,3,-2,1,-1).

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

%F 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

%F 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

%e a(0) = 8 because L(0) * L_4(0) = 2 * 4.

%e a(1) = 1 because L(1) * L_4(1) = 1 * 1.

%e a(2) = 9 because L(2) * L_4(2) = 3 * 3.

%e a(3) = 28 because L(3) * L_4(3) = 4 * 7.

%e a(4) = 105 because L(4) * L_4(4) = 7 * 15.

%e a(5) = 286 because L(5) * L_4(5) = 11 * 26.

%e a(6) = 918 because L(6) * L_4(6) = 18 * 51.

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

%o (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

%o (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

%o (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

%o (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

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

%K easy,nonn

%O 0,1

%A _Jonathan Vos Post_, May 11 2005