A008859 a(n) = Sum_{k=0..6} binomial(n,k).

1, 2, 4, 8, 16, 32, 64, 127, 247, 466, 848, 1486, 2510, 4096, 6476, 9949, 14893, 21778, 31180, 43796, 60460, 82160, 110056, 145499, 190051, 245506, 313912, 397594, 499178, 621616, 768212, 942649, 1149017, 1391842, 1676116, 2007328

Offset: 0

Author

N. J. A. Sloane and R. K. Guy

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A008860, A008861, A008862, A008863, A006261, A000127, A007318, A219531.

Programs

GAP

List([0..40], n-> Sum([0..6], k-> Binomial(n,k)) ); # G. C. Greubel, Sep 13 2019

Haskell

a008859 = sum . take 7 . a007318_row  -- Reinhard Zumkeller, Nov 24 2012

Magma

[(&+[Binomial(n,k): k in [0..6]]): n in [0..40]]; // G. C. Greubel, Sep 13 2019

Maple

A008859 := proc(n)
    add(binomial(n,k),k=0..6) ;
end proc: # R. J. Mathar, Oct 30 2015

Mathematica

Table[Sum[Binomial[n,k],{k,0,6}],{n,0,40}] (* Harvey P. Dale, Jan 16 2012 *)

PARI

a(n)=sum(k=0,6,binomial(n,k)) \\ Charles R Greathouse IV, Sep 24 2015

Sage

[sum(binomial(n,k) for k in (0..6)) for n in (0..40)] # G. C. Greubel, Sep 13 2019

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