A023550 Convolution of natural numbers >= 2 and (F(2), F(3), F(4), ...).
2, 7, 16, 32, 59, 104, 178, 299, 496, 816, 1335, 2176, 3538, 5743, 9312, 15088, 24435, 39560, 64034, 103635, 167712, 271392, 439151, 710592, 1149794, 1860439, 3010288, 4870784, 7881131, 12751976, 20633170, 33385211, 54018448, 87403728, 141422247, 228826048
Offset: 1
Links
- Vincenzo Librandi, 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..40], n-> Lucas(1, -1, n+4)[2] -2*n-7 ); # G. C. Greubel, Jun 01 2019
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Magma
[Lucas(n+4) - 2*n - 7 : n in [1..40]]; // G. C. Greubel, Jun 01 2019
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Mathematica
LinearRecurrence[{3,-2,-1,1}, {2,7,16,32}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *) -
PARI
Vec(x*(2-x)*(1+x)/((1-x)^2*(1-x-x^2)) + O(x^40)) \\ Colin Barker, Mar 11 2017
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PARI
vector(40, n, fibonacci(n+5) + fibonacci(n+3) -2*n-7) \\ G. C. Greubel, Jun 01 2019
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Sage
[lucas_number2(n+4,1,-1) -2*n-7 for n in (1..40)] # G. C. Greubel, Jun 01 2019
Formula
a(n) = A023537(n) + 2*n.
From Colin Barker, Mar 11 2017: (Start)
G.f.: x*(2-x)*(1+x) / ((1-x)^2*(1-x-x^2)).
a(n) = (2^(-1-n)*((1-sqrt(5))^n*(-15+7*sqrt(5)) + (1+sqrt(5))^n*(15+7*sqrt(5)))) / sqrt(5) - 2*n - 7.
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4.
(End)
a(n) = Lucas(n+4) - 2*n - 7. - G. C. Greubel, Jun 01 2019