A122068 Expansion of x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3).

1, 3, 10, 35, 126, 462, 1715, 6419, 24157, 91238, 345401, 1309574, 4970070, 18874261, 71705865, 272491891, 1035680954, 3936821259, 14965658694, 56893879910, 216295686467, 822315097387, 3126323230541, 11885921055638

Offset: 1

Author

Gary W. Adamson, Oct 15 2006

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.
• R. Witula, P. Lorenc, M. Rozanski, and M. Szweda, Sums of the rational powers of roots of cubic polynomials, Zeszyty Naukowe Politechniki Slaskiej, Seria: Matematyka Stosowana z. 4, Nr. kol. 1920, 2014.
• Index entries for linear recurrences with constant coefficients, signature (7,-14,7).

Crossrefs

Cf. A087946, A081567.

Cf. A215007, A215008. - Roman Witula, May 16 2014

Programs

GAP

a:=[1,3,10];; for n in [4..30] do a[n]:=7*(a[n-1]-2*a[n-2]+a[n-3]); od; a; # G. C. Greubel, Oct 03 2019

Magma

I:=[1,3,10]; [n le 3 select I[n] else 7*(Self(n-1) -2*Self(n-2) + Self(n-3)): n in [1..30]]; // G. C. Greubel, Oct 03 2019

Maple

seq(coeff(series(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3), x, n+1), x, n), n =1..30); # G. C. Greubel, Oct 03 2019

Mathematica

M = {{2,1,0,0,0,0}, {1,2,1,0,0,0}, {0,1,2,1,0,0}, {0,0,1,2,1,0}, {0,0,0, 1,2,1}, {0,0,0,0,1,2}}; v[1] = {1,1,1,1,1,1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n,30}]
Rest@CoefficientList[Series[x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3), {x, 0, 30}], x] (* G. C. Greubel, Oct 03 2019 *)
LinearRecurrence[{7,-14,7},{1,3,10},30] (* Harvey P. Dale, Mar 08 2020 *)

PARI

Vec(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3)+O(x^30)) \\ Charles R Greathouse IV, Sep 27 2012

Sage

def A122068_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3)).list()
a=A122068_list(30); a[1:] # G. C. Greubel, Oct 03 2019

Formula

From Roman Witula, May 16 2014: (Start)

a(n) = (1/2)*Sum_{k=0..2}(1 - 1/sqrt(7)*cot(2^k * alpha))* (2*sin(2^k * alpha))^(2n), where alpha := 2*Pi/7.

a(n) = (A215007(n) + A215008(n+1) - 2*A215008(n))/2. (End)

a(n) = binomial(2*n-1, n-1) + Sum_{k=1..n} (-1)^k*binomial(2*n, n+7*k). - Greg Dresden, Jan 28 2023