OFFSET
0,2
COMMENTS
a(n) is the maximal number of regions in 6-space formed by n-1 5-dimensional hypercubes. - Christian Schroeder, Jan 04 2016
a(n) is the number of binary words of length n matching the regular expression 0*1*0*1*0*1*0*. A000124, A000125, A000127, A006261 count binary words of the form 0*1*0*, 1*0*1*0*, 0*1*0*1*0*, and 1*0*1*0*1*0*, respectively. - Manfred Scheucher, Jun 22 2023
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
FORMULA
a(n) = Sum_{k=0..3} binomial(n+1, 2*k). - Len Smiley, Oct 20 2001
O.g.f.: (1 - 5*x + 11*x^2 - 13*x^3 + 9*x^4 - 3*x^5 + x^6)/(1-x)^7. - R. J. Mathar, Apr 02 2008
a(n) = a(n-1) + A006261(n-1). - Christian Schroeder, Jan 04 2016
a(n) = (n^6 - 9*n^5 + 55*n^4 - 75*n^3 + 304*n^2 + 444*n + 720)/720. - Gerry Martens , May 04 2016
E.g.f.: (720 + 720*x + 360*x^2 + 120*x^3 + 30*x^4 + 6*x^5 + x^6)*exp(x)/6!. - Ilya Gutkovskiy, May 04 2016
MAPLE
MATHEMATICA
Table[Sum[Binomial[n, k], {k, 0, 6}], {n, 0, 40}] (* Harvey P. Dale, Jan 16 2012 *)
PROG
(Haskell)
a008859 = sum . take 7 . a007318_row -- Reinhard Zumkeller, Nov 24 2012
(PARI) a(n)=sum(k=0, 6, binomial(n, k)) \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [(&+[Binomial(n, k): k in [0..6]]): n in [0..40]]; // G. C. Greubel, Sep 13 2019
(Sage) [sum(binomial(n, k) for k in (0..6)) for n in (0..40)] # G. C. Greubel, Sep 13 2019
(GAP) List([0..40], n-> Sum([0..6], k-> Binomial(n, k)) ); # G. C. Greubel, Sep 13 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved