A033817 Convolution of natural numbers n >= 1 with Lucas numbers L(k) for k >= -4.

7, 10, 16, 21, 28, 36, 47, 62, 84, 117, 168, 248, 375, 578, 904, 1429, 2276, 3644, 5855, 9430, 15212, 24565, 39696, 64176, 103783, 167866, 271552, 439317, 710764, 1149972, 1860623, 3010478, 4870980, 7881333, 12752184, 20633384, 33385431, 54018674, 87403960, 141422485

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, A033814.

Programs

GAP

List([1..40], n-> Lucas(1, -1, n-1)[2] +4*n+1 ) # G. C. Greubel, Jun 01 2019

Magma

[Lucas(n-1) + 4*n + 1 : n in [1..40]]; // G. C. Greubel, Jun 01 2019

Mathematica

Table[LucasL[n-1] +4*n+1, {n,1,40}] (* G. C. Greubel, Jun 01 2019 *)

PARI

vector(40, n, fibonacci(n) + fibonacci(n-2) +4*n+1) \\ G. C. Greubel, Jun 01 2019

Sage

[lucas_number2(n-1,1,-1) +4*n+1 for n in (1..40)] # G. C. Greubel, Jun 01 2019

Formula

a(n) = L(-1)*(F(n+1)-1) + L(-2)*F(n) - L(-3)*n, F(n): Fibonacci (A000045), L(n): Lucas (A000032) with L(-n)=(-1)^n*L(n)

G.f.: x*(7-11*x)/((1-x-x^2)*(1-x)^2).

a(n) = Lucas(n-1) + 4*n + 1. - G. C. Greubel, Jun 01 2019

Extensions

Terms a(31) onward added by G. C. Greubel, Jun 01 2019