OFFSET
0,8
COMMENTS
Starting with "1", equals the eigensequence of a triangle with A000579 = binomial(n,6) = (1, 7, 28, 84, 210, ...) as the left column and the rest 1's. - Gary W. Adamson, Jul 24 2010
Number of binary strings of length n with at least six 0's. Equivalently, number of subsets of an n-element set with at least six elements. - Enrique Navarrete, Nov 15 2025
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
J. Eckhoff, Der Satz von Radon in konvexen Produktstrukturen II, Monat. f. Math., 73 (1969), 7-30.
Index entries for linear recurrences with constant coefficients, signature (8,-27,50,-55,36,-13,2).
FORMULA
From Paul Barry, Aug 23 2004: (Start)
G.f.: x^6/((1 - 2*x)*(1 - x)^6).
a(n) = Sum_{k=0..n} binomial(n, k+6) = Sum_{k=6..n} binomial(n, k).
a(n) = 2*a(n-1) + binomial(n-1, 5). (End)
From Enrique Navarrete, Nov 15 2025: (Start)
a(n) = 8*a(n-1) - 27*a(n-2) + 50*a(n-3) - 55*a(n-4) + 36*a(n-5) - 13*a(n-6) + 2*a(n-7).
E.g.f.: exp(x)*(exp(x) - 1 - x - x^2/2 - x^3/6 - x^4/24 - x^5/120). (End)
MAPLE
a:= n-> (Matrix(7, (i, j)-> if (i=j-1) then 1 elif j=1 then [8, -27, 50, -55, 36, -13, 2][i] else 0 fi)^(n))[1, 7]:
seq(a(n), n=0..30); # Alois P. Heinz, Aug 05 2008
MATHEMATICA
Table[Sum[Binomial[n, k+6], {k, 0, n}], {n, 0, 30}] (* Zerinvary Lajos, Jul 08 2009 *)
Table[2^n-Total[Binomial[n, Range[0, 5]]], {n, 0, 40}] (* Harvey P. Dale, Oct 24 2017 *)
PROG
(Haskell)
a035038 n = a035038_list !! n
a035038_list = map (sum . drop 6) a007318_tabl
-- Reinhard Zumkeller, Jun 20 2015
(Magma) [n le 5 select 0 else (&+[Binomial(n, j): j in [6..n]]): n in [0..50]]; // G. C. Greubel, Mar 20 2023
(SageMath) [sum(binomial(n, j) for j in range(6, n+1)) for n in range(51)] # G. C. Greubel, Mar 20 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved