A097177 a(n) = Sum_{k=0..n} binomial(floor((n+1)/2), floor((k+1)/2)) * 10^k.

1, 11, 111, 1221, 11221, 133331, 1133331, 14466441, 114466441, 1561110551, 11561110551, 167672165661, 1167672165661, 17934888731771, 117934888731771, 1911423761908881, 11911423761908881, 203053799952796991

Offset

0,2

Comments

a(n) = (11/10)*{1, 101, 101, 10201, 10201, 1030301, ...} - 100*{0, 1, 0, 100, 0, 10000, ...} - (1/10)*{1, 1, 1, 1, 1, 1, ...}.

Partial sums of A097178.

Links

G. C. Greubel, Table of n, a(n) for n = 0..980

Index entries for linear recurrences with constant coefficients, signature (1,201,-201,-10100,10100)

Formula

G.f.: (1+10*x-101*x^2-900*x^3)/((1-x)*(1-100*x^2)*(1-101*x^2)).

a(n) = (11/20)*( (1-sqrt(101))*(-sqrt(101))^n + (1+sqrt(101))*(sqrt(101))^n ) - 5*(10^n - (-10)^n) - 1/10.

a(n) = a(n-1) + 201*a(n-2) - 201*a(n-3) - 10100*a(n-4) + 10100*a(n-5).

Maple

seq(coeff(series((1+10*x-101*x^2-900*x^3)/((1-x)*(1-100*x^2)*(1-101*x^2)), x, n+1), x, n), n = 0 ..20); # G. C. Greubel, Sep 17 2019

Mathematica

Accumulate[LinearRecurrence[{0, 201, 0, -10100}, {1, 10, 100, 1110}, 20]] (* or *) LinearRecurrence[{1, 201, -201, -10100, 10100}, {1, 11, 111, 1221, 11221}, 20] (* Harvey P. Dale, Dec 08 2018 *)

Prog

(PARI) my(x='x+O('x^20)); Vec((1+10*x-101*x^2-900*x^3)/((1-x)*(1-100*x^2)*(1-101*x^2))) \\ G. C. Greubel, Sep 17 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+10*x-101*x^2-900*x^3)/((1-x)*(1-100*x^2)*(1-101*x^2)) )); // G. C. Greubel, Sep 17 2019
(Sage)
def A097177_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P((1+10*x-101*x^2-900*x^3)/((1-x)*(1-100*x^2)*(1-101*x^2))).list()
A097177_list(20) # G. C. Greubel, Sep 17 2019
(GAP) a:=[1, 11, 111, 1221, 11221];; for n in [6..20] do a[n]:=a[n-1] + 201*a[n-2] - 201*a[n-3] - 10100*a[n-4] + 10100*a[n-5]; od; a; # G. C. Greubel, Sep 17 2019

Crossrefs

Cf. A097175, A097176, A097178.

Sequence in context: A240367 A338679 A199764 * A136982 A083441 A261269

Adjacent sequences: A097174 A097175 A097176 * A097178 A097179 A097180

Keyword

easy,nonn

Author

Paul Barry, Jul 30 2004

Status

approved