A033818 Convolution of natural numbers n >= 1 with Lucas numbers L(k) for k >= -2.
3, 5, 9, 14, 22, 34, 53, 83, 131, 208, 332, 532, 855, 1377, 2221, 3586, 5794, 9366, 15145, 24495, 39623, 64100, 103704, 167784, 271467, 439229, 710673, 1149878, 1860526, 3010378, 4870877, 7881227, 12752075, 20633272, 33385316, 54018556, 87403839, 141422361, 228826165, 370248490
Offset: 1
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).
Programs
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GAP
List([1..50], n-> Lucas(1,-1,n+1)[2] +n-1) # G. C. Greubel, Jun 01 2019
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Magma
[Lucas(n+1) +n-1: n in [1..50]]; // G. C. Greubel, Jun 01 2019
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Mathematica
LinearRecurrence[{3,-2,-1,1}, {3,5,9,14}, 50] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2011, modified by G. C. Greubel, Jun 01 2019 *) Table[LucasL[n+1] +n-1, {n,1,50}] (* G. C. Greubel, Jun 01 2019 *) -
PARI
{a(n) = fibonacci(n+2) + fibonacci(n) + n-1}; \\ G. C. Greubel, Jun 01 2019 -
Sage
[lucas_number2(n+1,1,-1) +n-1 for n in (1..50)] # G. C. Greubel, Jun 01 2019
Formula
a(n) = L(1)*(F(n+1) - 1) + L(0)*F(n) - L(-1)*n, F(n): Fibonacci (A000045), L(n): Lucas (A000032) with L(-n) = (-1)^n*L(n).
G.f.: x*(3-4*x)/((1-x-x^2)*(1-x)^2).
a(n) = Lucas(n+1) + n - 1. - G. C. Greubel, Jun 01 2019
Extensions
Terms a(31) onward added by G. C. Greubel, Jun 01 2019