A000579 - OEIS (original) (raw)

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

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) 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].

Leo Moser, Quicky 87, Mathematics Magazine, 26 (March 1953), p. 226.

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)^(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

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.

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[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

(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):

for i in range(6):

CROSSREFS

Cf. A053135, A053128, A000580 (partial sums), A000581, A000582, A000217, A000292, A000332, A000389 (first differences), A104712 (fifth column, k=6).

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