A023552 Convolution of natural numbers >= 3 and Fibonacci numbers.

3, 7, 15, 28, 50, 86, 145, 241, 397, 650, 1060, 1724, 2799, 4539, 7355, 11912, 19286, 31218, 50525, 81765, 132313, 214102, 346440, 560568, 907035, 1467631, 2374695, 3842356, 6217082, 10059470, 16276585, 26336089, 42612709, 68948834, 111561580, 180510452

Offset

1,1

Links

Colin Barker, Table of n, a(n) for n = 1..1000

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

Formula

G.f.: x*(3-2*x)/((1-x-x^2)*(1-x)^2). - Ralf Stephan, Apr 28 2004

From Colin Barker, Mar 11 2017: (Start)

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4.

a(n) = -5 + (2^(-1-n)*((1-sqrt(5))^n*(-13+5*sqrt(5)) + (1+sqrt(5))^n*(13+5*sqrt(5)))) / sqrt(5) - n. (End)

a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (n+5). - G. C. Greubel, Jul 08 2019

Mathematica

LinearRecurrence[{3, -2, -1, 1}, {3, 7, 15, 28}, 40] (* or *) Rest[ CoefficientList[Series[(x(3-2x))/((1-x-x^2)(1-x)^2), {x, 0, 40}], x]]  (* Harvey P. Dale, Apr 24 2011 *)
With[{F=Fibonacci}, Table[F[n+4]+2*F[n+2]-n-5, {n, 40}]] (* G. C. Greubel, Jul 08 2019 *)

Prog

(PARI) Vec(x*(3-2*x)/((1-x-x^2)*(1-x)^2) + O(x^40)) \\ Colin Barker, Mar 11 2017
(PARI) vector(40, n, f=fibonacci; f(n+4)+2*f(n+2)-n-5) \\ G. C. Greubel, Jul 08 2019
(Magma) F:=Fibonacci; [F(n+4)+2*F(n+2)-n-5: n in [1..40]]; // G. C. Greubel, Jul 08 2019
(Sage) f=fibonacci; [f(n+4)+2*f(n+2)-n-5 for n in (1..40)] # G. C. Greubel, Jul 08 2019
(GAP) F:=Fibonacci;; List([1..40], n-> F(n+4)+2*F(n+2)-n-5) # G. C. Greubel, Jul 08 2019

Crossrefs

Cf. A000032, A000045, A213579.

Sequence in context: A101498 A027965 A130145 * A293316 A364293 A009859

Adjacent sequences: A023549 A023550 A023551 * A023553 A023554 A023555

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved