A024310 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 2), t = (Lucas numbers).

2, 6, 17, 26, 59, 97, 191, 308, 565, 915, 1606, 2598, 4436, 7178, 12037, 19476, 32273, 52219, 85845, 138900, 227133, 367509, 598828, 968924, 1575046, 2548478, 4136169, 6692462, 10850455, 17556405, 28444379, 46023972, 74532629, 120596327, 195238738, 315902914, 511328632, 827347106

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

Formula

G.f.: (2 +4*x +5*x^2 -5*x^3 -4*x^4 +2*x^5 -2*x^6 +x^7)/((1-x-x^2)*(1-x^2-x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

From G. C. Greubel, Feb 21 2022: (Start)

a(n) = Sum_{j=2..floor((n+3)/2)} j*Lucas(n-j+2).

a(n) = 5*Fibonacci(n+3) - (m+1)*Lucas(n-m+5) + m*Lucas(n-m+4), where m = floor((n+3)/2).

a(2*n) = 5*Fibonacci(2*n+3) - (n+5)*Fibonacci(n+3) - n*Fibonacci(n+1).

a(2*n+1) = 5*Fibonacci(2*n+2) - (n+5)*Fibonacci(n+2) - n*Fibonacci(n). (End)

Mathematica

With[{m=Floor[(n+3)/2]}, Table[5*Fibonacci[n+3] -(m+1)*LucasL[n-m+5] + m*LucasL[n -m+4], {n, 40}]] (* G. C. Greubel, Feb 21 2022 *)

Prog

(Sage) [sum(j*lucas_number2(n+2-j, 1, -1) for j in (2..floor((n+3)/2))) for n in (1..50)] # G. C. Greubel, Feb 21 2022
(Magma) [(&+[j*Lucas(n+2-j): j in [2..Floor((n+3)/2)]]) : n in [1..40]]; // G. C. Greubel, Feb 21 2022

Crossrefs

Cf. A000032, A000045.

Sequence in context: A069014 A105146 A076660 * A064516 A001441 A204450

Adjacent sequences: A024307 A024308 A024309 * A024311 A024312 A024313

Keyword

nonn

Author

Clark Kimberling

Extensions

Terms a(29) onward added by G. C. Greubel, Feb 21 2022

Status

approved