%I #44 Jan 29 2023 02:27:31

%S 1,1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,

%T 231,253,276,300,325,351,378,406,435,465,496,528,561,595,630,666,703,

%U 741,780,820,861,903,946,990,1035,1081,1128,1176,1225,1275,1326,1378,1431

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

%H Gennady Eremin, <a href="/A179865/b179865.txt">Table of n, a(n) for n = 1..500</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

%F G.f.: x*(1 + x/(1-x)^3). - _Gennady Eremin_, Feb 23 2021

%F For n > 1, a(n+1) = a(n) + n. - _Gennady Eremin_, Mar 12 2021

%F E.g.f.: x*(2 + x*exp(x))/2. - _Stefano Spezia_, Jan 29 2023

%e G.f. = x + x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 15*x^6 + 21*x^7 + 28*x^8 + ...

%e For n = 4, the 6 numbers are 1000, 1001, 1011, 1100, 1101, 1110.

%o (Python)

%o def A179865(n):

%o if n==1: return 1

%o return n*(n-1)//2 # _Gennady Eremin_, Mar 14 2021

%Y Cf. A000217.

%K nonn,base,easy

%O 1,3

%A _Vladimir Shevelev_, Jul 30 2010, Aug 03 2010

%E Edited by _N. J. A. Sloane_, Aug 08 2010

%E More terms from _Michel Marcus_, Feb 23 2021