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

%I #13 Aug 01 2024 09:58:57

%S 7,10,16,21,28,36,47,62,84,117,168,248,375,578,904,1429,2276,3644,

%T 5855,9430,15212,24565,39696,64176,103783,167866,271552,439317,710764,

%U 1149972,1860623,3010478,4870980,7881333,12752184,20633384,33385431,54018674,87403960,141422485

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

%H G. C. Greubel, <a href="/A033817/b033817.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3, -2, -1, 1).

%F 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)

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

%F a(n) = Lucas(n-1) + 4*n + 1. - _G. C. Greubel_, Jun 01 2019

%t Table[LucasL[n-1] +4*n+1, {n,1,40}] (* _G. C. Greubel_, Jun 01 2019 *)

%o (PARI) vector(40, n, fibonacci(n) + fibonacci(n-2) +4*n+1) \\ _G. C. Greubel_, Jun 01 2019

%o (Magma) [Lucas(n-1) + 4*n + 1 : n in [1..40]]; // _G. C. Greubel_, Jun 01 2019

%o (Sage) [lucas_number2(n-1,1,-1) +4*n+1 for n in (1..40)] # _G. C. Greubel_, Jun 01 2019

%o (GAP) List([1..40], n-> Lucas(1, -1, n-1)[2] +4*n+1 ) # _G. C. Greubel_, Jun 01 2019

%Y Cf. A000032, A000045, A023548, A023537, A033814.

%K easy,nonn

%O 1,1

%A _Wolfdieter Lang_

%E Terms a(31) onward added by _G. C. Greubel_, Jun 01 2019