A033814 Convolution of positive integers n with Lucas numbers L(k)(A000032) for k >= 4.

7, 25, 61, 126, 238, 426, 737, 1247, 2079, 3432, 5628, 9188, 14955, 24293, 39409, 63874, 103466, 167534, 271205, 438955, 710387, 1149580, 1860216, 3010056, 4870543, 7880881, 12751717, 20632902, 33384934, 54018162, 87403433, 141421943, 228825735, 370248048

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A000032, A000045, A023548, A023537.

Programs

GAP

List([1..40], n-> Lucas(1, -1, n+7)[2] -11*n-29 ) # G. C. Greubel, Jun 01 2019

Magma

[Lucas(n+7) - 11*n - 29 : n in [1..40]]; // G. C. Greubel, Jun 01 2019

Mathematica

Table[LucasL[n+7] -11*n-29, {n,1,40}] (* G. C. Greubel, Jun 01 2019 *)

PARI

vector(40, n, fibonacci(n+8) + fibonacci(n+6) -11*n-29) \\ G. C. Greubel, Jun 01 2019

Python

from sympy import lucas
def a(n): return lucas(n+7) - 11*n - 29
print([a(n) for n in range(1, 35)]) # Michael S. Branicky, Jul 25 2021

Sage

[lucas_number2(n+7,1,-1) -11*n-29 for n in (1..40)] # G. C. Greubel, Jun 01 2019

Formula

a(n) = L(7)*(F(n+1)-1) + L(6)*F(n) - L(5)*n, F(n): Fibonacci (A000045) and L(n): Lucas (A000032).

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

a(n) = A000032(n+7) - 11*n - 29. - G. C. Greubel, Jun 01 2019