A027976 n-th diagonal sum of right justified array T given by A027960.

1, 1, 4, 6, 10, 18, 29, 47, 78, 126, 204, 332, 537, 869, 1408, 2278, 3686, 5966, 9653, 15619, 25274, 40894, 66168, 107064, 173233, 280297, 453532, 733830, 1187362, 1921194, 3108557, 5029751, 8138310, 13168062, 21306372, 34474436, 55780809, 90255245, 146036056, 236291302, 382327358

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).

Crossrefs

Cf. A000032, A000045, A004695, A027960, A102283.

Programs

GAP

a:=[1,1,4,6,10];; for n in [6..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]-a[n-4]-a[n-5]; od; a; # G. C. Greubel, Sep 26 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1 + 2*x^2)/((1-x^3)*(1-x-x^2)) )); // G. C. Greubel, Sep 26 2019

Maple

seq(coeff(series((1 + 2*x^2)/((1-x^3)*(1-x-x^2)), x, n+1), x, n), n = 0..40); # G. C. Greubel, Sep 26 2019

Mathematica

LinearRecurrence[{1,1,1,-1,-1}, {1,1,4,6,10}, 41] (* or *) Table[ (Fibonacci[n+1] +LucasL[n+2] -2*Sin[2*Pi*n/3]/Sqrt[3] -2)/2, {n,0,40}] (* G. C. Greubel, Sep 26 2019 *)

PARI

my(x='x+O('x^40)); Vec((1 + 2*x^2)/((1-x^3)*(1-x-x^2))) \\ G. C. Greubel, Sep 26 2019

Sage

def A027976_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1 + 2*x^2)/((1-x^3)*(1-x-x^2))).list()
A027976_list(40) # G. C. Greubel, Sep 26 2019

Formula

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

From G. C. Greubel, Sep 26 2019: (Start)

a(n) = (Fibonacci(n) + 4*Fibonacci(n+1) - A102283(n) - 2)/2.

a(n) = (Fibonacci(n+1) + Lucas(n+2) - 2*sin(2*Pi*n/3)/sqrt(3) - 2)/2. (End)

Extensions

Terms a(28) onward added by G. C. Greubel, Sep 26 2019