A033550 a(n) = A005248(n) - n.

2, 2, 5, 15, 43, 118, 316, 836, 2199, 5769, 15117, 39592, 103670, 271430, 710633, 1860483, 4870831, 12752026, 33385264, 87403784, 228826107, 599074557, 1568397585, 4106118220, 10749957098, 28143753098, 73681302221, 192900153591

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also distinct compositions of the wheel graph W_n. - Ralf Stephan, Jan 02 2003

Links

• A. Knopfmacher and M. E. Mays, Graph Compositions. I: Basic Enumeration, Integers 1(2001), #A04.
• Index entries for linear recurrences with constant coefficients, signature (5,-8,5,-1).

Crossrefs

Cf. A000032, A000045, A005248.

Programs

GAP

List([0..50], n-> Lucas(1,-1, 2*n)[2] - n ); # G. C. Greubel, Oct 12 2019

Magma

[Lucas(2*n) - n: n in [0..50]]; // G. C. Greubel, Oct 12 2019

Maple

with(combinat); seq(fibonacci(2*n+1)+fibonacci(2*n-1)-n, n=0..50); # G. C. Greubel, Oct 12 2019

Mathematica

Table[LucasL[2*n]-n, {n,0,50}] (* G. C. Greubel, Oct 12 2019 *)

PARI

a(n)=fibonacci(2*n+1)+fibonacci(2*n-1)-n

Sage

[lucas_number2(2*n, 1,-1) - n for n in range(50)] # G. C. Greubel, Oct 12 2019

Formula

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

G.f.: (2 - 8*x + 11*x^2 - 4*x^3)/((1-3*x+x^2)*(1-x)^2).

a(n) = Lucas(2*n) - n. - G. C. Greubel, Oct 12 2019

E.g.f.: 2*exp(3*x/2)*cosh(sqrt(5)*x/2) - exp(x)*x. - Stefano Spezia, Oct 14 2024