A179865 Number of n-bit binary numbers containing one run of 0's.

1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431

Offset

1,3

Links

Gennady Eremin, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

For n>=2, a(n) = A000217(n-1).

G.f.: x*(1 + x/(1-x)^3). - Gennady Eremin, Feb 23 2021

For n > 1, a(n+1) = a(n) + n. - Gennady Eremin, Mar 12 2021

E.g.f.: x*(2 + x*exp(x))/2. - Stefano Spezia, Jan 29 2023

Example

G.f. = x + x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 15*x^6 + 21*x^7 + 28*x^8 + ...
For n = 4, the 6 numbers are 1000, 1001, 1011, 1100, 1101, 1110.

Prog

(Python)
def A179865(n):
    if n==1: return 1
    return n*(n-1)//2  # Gennady Eremin, Mar 14 2021

Crossrefs

Cf. A000217.

Sequence in context: A253145 A161680 A000217 * A105340 A176659 A109811

Adjacent sequences: A179862 A179863 A179864 * A179866 A179867 A179868

Keyword

nonn,base,easy

Author

Vladimir Shevelev, Jul 30 2010, Aug 03 2010

Extensions

Edited by N. J. A. Sloane, Aug 08 2010

More terms from Michel Marcus, Feb 23 2021

Status

approved