A027181 a(n) = Lucas(n+4) - 2*(n+3).

1, 3, 8, 17, 33, 60, 105, 179, 300, 497, 817, 1336, 2177, 3539, 5744, 9313, 15089, 24436, 39561, 64035, 103636, 167713, 271393, 439152, 710593, 1149795, 1860440, 3010289, 4870785, 7881132, 12751977, 20633171, 33385212, 54018449, 87403729, 141422248

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Let F be a homogeneous polynomial in n + 4 variables f0, f1, f2, g0, g1, ..., gn, defined as the determinant of a Sylvester matrix of polynomials f2*x^2 + f1*x + f0 and gn*x^n + ... + g1*x + g0. It appears that a(n) is equal to the l1-norm of F, i.e., the sum of absolute values of coefficients of F. - Anton Mosunov, Apr 13 2019

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Crossrefs

Cf. A000032, A000045, A027170, A192978.

Programs

GAP

List([0..40], n-> Lucas(1,-1,n+4)[2] -2*(n+3)); # G. C. Greubel, Jul 24 2019

Magma

[Lucas(n+4) - (2*n+6): n in [0..40]]; // Vincenzo Librandi, Apr 16 2011

Mathematica

LinearRecurrence[{3,-2,-1,1},{1,3,8,17},40] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2012 *)
Table[LucasL[n+4]-2*(n+3), {n,0,40}] (* G. C. Greubel, Jul 24 2019 *)

PARI

Vec((1 + x^2)/((1 - x)^2*(1 - x - x^2)) + O(x^40)) \\ Colin Barker, Mar 10 2017

PARI

vector(40, n, n--; f=fibonacci; f(n+5)+f(n+3)-2*(n+3)) \\ G. C. Greubel, Jul 24 2019

Sage

[lucas_number2(n+4,1,-1) -2*(n+3) for n in range(40)] # G. C. Greubel, Apr 14 2019

Formula

a(n) = Sum_{k=0..floor(n/2)} A027170(n-k, k).

G.f.: (1 + x^2)/((1 - x)^2*(1 - x - x^2)).

From Colin Barker, Mar 10 2017: (Start)

a(n) = -4 + (2^(-1-n)*((1-sqrt(5))^n*(-15+7*sqrt(5)) + (1+sqrt(5))^n*(15+7*sqrt(5))))/sqrt(5) - 2*(1+n).

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

(End)