A107066 Expansion of 1/(1-2*x+x^5).

1, 2, 4, 8, 16, 31, 60, 116, 224, 432, 833, 1606, 3096, 5968, 11504, 22175, 42744, 82392, 158816, 306128, 590081, 1137418, 2192444, 4226072, 8146016, 15701951, 30266484, 58340524, 112454976, 216763936, 417825921, 805385358, 1552430192, 2992405408, 5768046880

Offset: 0

Author

Paul Barry, May 10 2005

Keywords

• easy
• nonn

Comments

Row sums of number triangle A107065.

Same as A018922 plus first 3 additional terms. - Vladimir Joseph Stephan Orlovsky, Jul 08 2011

a(n) is the number of binary words of length n containing no subword 01011. - Alois P. Heinz, Mar 14 2012

Examples

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 31*x^5 + 60*x^6 + 116*x^7 + 224*x^8 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Otto Dunkel, Solutions of a probability difference equation, Amer. Math. Monthly, 32 (1925), 354-370; see p. 356.
• Jia Huang, A coin flip game and generalizations of Fibonacci numbers, arXiv:2501.07463 [math.CO], 2025. See p. 7.
• Thomas Langley, Jeffrey Liese, and Jeffrey Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011), Article #11.4.2.
• Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,-1).

Crossrefs

Cf. A018922, A119407 (partial sums), A000078 (first differences).

Cf. A209888. - Alois P. Heinz, Mar 14 2012

Column k = 1 of array A140996 (with a different offset) and second main diagonal of A140995.

Column k = 4 of A172119 (with a different offset).

Programs

GAP

a:=[1,2,4,8,16];; for n in [6..40] do a[n]:=2*a[n-1]-a[n-5]; od; a; # G. C. Greubel, Jun 12 2019

Magma

I:=[1,2,4,8,16]; [n le 5 select I[n] else 2*Self(n-1) - Self(n-5): n in [1..40]]; // G. C. Greubel, Jun 12 2019

Mathematica

CoefficientList[Series[1/(1 - 2*z + z^5), {z, 0, 40}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)
LinearRecurrence[{2,0,0,0,-1}, {1,2,4,8,16}, 40] (* G. C. Greubel, Jun 12 2019 *)

PARI

{a(n) = if( n<0, n = -n; polcoeff( -x^5 / (1 - 2*x^4 + x^5) + x * O(x^n), n), polcoeff( 1 / (1 - 2*x + x^5) + x * O(x^n), n))} /* Michael Somos, Dec 28 2012 */

Sage

(1/(1-2*x+x^5)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 12 2019

Formula

a(n) = 2*a(n-1) - a(n-5).

a(n) = Sum_{k=0..floor(n/5)} C(n-4*k, k) * 2^(n-2*k) *(-1)^k.

a(n) = A018922(n-3) for n >= 3. - R. J. Mathar, Mar 09 2007

First difference of A119407. - Michael Somos, Dec 28 2012

From Petros Hadjicostas, Jun 12 2019: (Start)

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

Setting k = 1 in the double recurrence for array A140996, we get that a(n+5) = 1 + a(n+1) + a(n+2) + a(n+3) + a(n+4) for n >= 0, which of course we can prove using other methods as well. See also Dunkel (1925).

(End)

a(n) = Sum_{k=0..n+3} A000078(k). - Greg Dresden, Jan 01 2021