login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000579 Figurate numbers or binomial coefficients C(n,6).
(Formerly M4390 N1847)
117
0, 0, 0, 0, 0, 0, 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, 736281, 906192, 1107568, 1344904, 1623160, 1947792, 2324784, 2760681, 3262623 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
Number of triangles (all of whose vertices lie inside the circle) formed when n points in general position on a circle are joined by straight lines - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 25 2000
Figurate numbers based on 6-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 13 of these numbers. - Jonathan Vos Post, Nov 28 2004
a(n) = A110555(n+1,6). - Reinhard Zumkeller, Jul 27 2005
a(n) is the number of terms in the expansion of (a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7)^n. - Sergio Falcon, Feb 12 2007
Only prime in this sequence is 7. - Artur Jasinski, Dec 02 2007
6-dimensional triangular numbers, sixth partial sums of binomial transform of [1, 0, 0, 0, ...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009, R. J. Mathar, Jul 07 2009
The number of n-digit numbers the binary expansion of which contains 3 runs of 0's. Generally, the number of n-digit numbers with k runs of 0's is Sum_{i = k..n-k} binomial(i-1, k-1)*binomial(n-i, k) = C(n,2*k) = A034839(n,k) - Vladimir Shevelev, Jul 30 2010
The dimension of the space spanned by a 6-form that couples to M5-brane worldsheets wrapping 6-cycles inside tori (ref. Green,Miller,Vanhove eq. 3.10). - Stephen Crowley, Jan 09 2012
For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 5 elements, which is 3*C(n+1,6) (for n>=5), hence a(n) = 3*C(n+1,6) = 3*A000579(n+1). - Serhat Bulut, Oktay Erkan Temizkan, Mar 13 2015
a(n) = fallfac(n, 6)/6! is also the number of independent components of an antisymmetric tensor of rank 6 and dimension n >= 1. Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 645120. - Philippe A.J.G. Chevalier, Dec 28 2015
Coordination sequence for 6-dimensional cyclotomic lattice Z[zeta_7].
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Charles W. Trigg: Mathematical Quickies. New York: Dover Publications, Inc., 1985, p. 11, #32
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, Preprint, 2012.
Philippe A. J. G. Chevalier, On a Mathematical Method for Discovering Relations Between Physical Quantities: a Photonics Case Study, Slides from a talk presented at ICOL2014.
Philippe A. J. G. Chevalier, A "table of Mendeleev" for physical quantities?, Slides from a talk, May 14 2014, Leuven, Belgium.
Philippe A. J. G. Chevalier, Dimensional exploration techniques for photonics, Slides of a talk, 2016.
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
Michael B. Green, Stephen D. Miller, and Pierre Vanhove, Small representations, string instantons, and Fourier modes of Eisenstein series, arXiv:1111.2983 [hep-th], 2011-2013.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Leo Moser, Quicky 87, Mathematics Magazine, 26 (March 1953), p. 226.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Hermann Stamm-Wilbrandt, Sum of Pascal's triangle reciprocals [Cached copy from the Wayback Machine]
Eric Weisstein's World of Mathematics, Composition
FORMULA
G.f.: x^6/(1-x)^7.
E.g.f.: exp(x)*x^6/720.
a(n) = (n^6 - 15*n^5 + 85*n^4 - 225*n^3 + 274*n^2 - 120*n)/720.
Conjecture: a(n+3) = Sum_{0 <= k, L, m <= n; k + L + m <= n} k*L*m. - Ralf Stephan, May 06 2005
Convolution of the nonnegative numbers (A001477) with the hexagonal numbers (A000389). Also convolution of the triangular numbers (A000217) with the tetrahedral numbers (A000292). - Sergio Falcon, Feb 12 2007
a(n) = n*(n - 1)*(n - 2)*(n - 3)*(n - 4)*(n - 5)/720. - Artur Jasinski, Dec 02 2007, R. J. Mathar, Jul 07 2009
Equals binomial transform of [1, 6, 15, 20, 15, 6, 1, 0, 0, 0, ...]. - Gary W. Adamson, Aug 02 2008
a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 0, a(5) = 0, a(6) = 1, a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Harvey P. Dale, Dec 30 2012
Sum_{n >= 0} a(n)/n! = e/720. Sum_{n >= 5} a(n)/(n-5)! = 4051*e/720. See A067653 regarding the second ratio. - Richard R. Forberg, Dec 26 2013
Sum_{n >= 6} 1/a(n) = 6/5. - Hermann Stamm-Wilbrandt, Jul 13 2014
Sum_{n >= 6} (-1)^(n + 1)/a(n) = 192*log(2) - 661/5 = 0.8842586675... Also see A242023. - Richard R. Forberg, Aug 11 2014
a(n) = a(5-n) for all n in Z. - Michael Somos, Oct 07 2014
0 = a(n)*(+a(n+1) +5*a(n+2)) + a(n+1)*(-7*a(n+1) +a(n+2)) for all n in Z. - Michael Somos, Oct 07 2014
a(n) = 3*C(n+1,6) = 3*A000579(n+1). - Serhat Bulut, Oktay Erkan Temizkan, Mar 13 2015
a(n) = A000292(n-5)*A000292(n-2)/20. - R. J. Mathar, Nov 29 2015
EXAMPLE
a(9) = 84 = (1, 3, 3, 1) dot (1, 6, 15, 20) = (1 + 18 + 45 + 20). - Gary W. Adamson, Aug 02 2008
G.f. = x^6 + 7*x^7 + 28*x^8 + 84*x^9 + 210*x^10 + 462*x^11 + 924*x^12 + ...
For A = {1,2,3,4,5,6} subsets with 5 elements are {1,2,3,4,5}, {1,2,3,4,6}, {1,2,3,5,6}, {1,2,4,5,6}, {1,3,4,5,6}, {2,3,4,5,6}. Sum of 2 smallest elements of each subset: a(6) = (1+2) + (1+2) + (1+2) + (1+2) + (1+3) + (2+3) = 21 = 3*C(6+1,6) = 3*A000579(6+1). - Serhat Bulut, Oktay Erkan Temizkan, Mar 13 2015
a(7) = 7 from the seven independent components of an antisymmetric tensor A of rank 6 and dimension 7: A(1,2,3,4,5,6), A(1,2,3,4,5,7), A(1,2,3,4,6,7), A(1,2,3,5,6,7) A(1,2,4,5,6,7), A(1,2,3,5,6,7) and A(2,3,4,5,6,7). See a Dec 10 2015 comment. - Wolfdieter Lang, Dec 10 2015
MAPLE
A000579 := n->binomial(n, 6);
ZL := [S, {S=Prod(B, B, B, B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=7..40); # Zerinvary Lajos, Mar 13 2007
A000579:=-1/(z-1)**7; # Simon Plouffe in his 1992 dissertation, referring to offset 0.
seq(binomial(n, 6), n=0..33); # Zerinvary Lajos, Jun 16 2008
G(x):=x^6*exp(x): f[0]:=G(x): for n from 1 to 39 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/6!, n=6..39); # Zerinvary Lajos, Apr 05 2009
MATHEMATICA
Table[Binomial[n, 6], {n, 6, 50}] (* Stefan Steinerberger, Apr 02 2006 *)
Table[n(n - 1)(n - 2)(n - 3)(n - 4)(n - 5)/720, {n, 0, 100}] (* Artur Jasinski, Dec 02 2007 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 0, 0, 0, 0, 1}, 50] (* Harvey P. Dale, Dec 30 2012 *)
CoefficientList[ Series[ -7x^6/(x-1)^7, {x, 0, 35}], x]/7 (* Robert G. Wilson v, Jan 29 2015 *)
PROG
(PARI) a(n)=binomial(n, 6) \\ Charles R Greathouse IV, Nov 20 2012
(Magma) [Binomial(n, 6) : n in [0..50]]; // Wesley Ivan Hurt, Jul 13 2014
(Python)
A000579_list, m = [], [1, -5, 10, -10, 5, -1, 0]
for _ in range(10**2):
A000579_list.append(m[-1])
for i in range(6):
m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016
CROSSREFS
Cf. A053135, A053128, A000580 (partial sums), A000581, A000582, A000217, A000292, A000332, A000389 (first differences), A104712 (fifth column, k=6).
Sequence in context: A023032 A341204 A278969 * A290994 A049017 A019501
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Some formulas that referred to other offsets corrected by R. J. Mathar, Jul 07 2009
I changed the offset to 0. This will require some further adjustments to the formulas. - N. J. A. Sloane, Aug 01 2010
Shevelev comment inserted and further adaptations to offset by R. J. Mathar, Aug 03 2010
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 3 20:15 EST 2024. Contains 370512 sequences. (Running on oeis4.)