OFFSET
0,2
COMMENTS
If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-8) is the number of 8-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
7-dimensional square numbers, sixth partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = Sum_{i=0..n} C(n+6,i+6)*b(i), where b(i) = [1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
2*a(n) is number of ways to place 6 queens on an (n+6) X (n+6) chessboard so that they diagonally attack each other exactly 15 times. The maximal possible attack number, p=binomial(k,2)=15 for k=6 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form a corresponding complete graph. - Antal Pinter, Dec 27 2015
Coefficients in the terminating series identity 1 - 9*n/(n + 8) + 44*n*(n - 1)/((n + 8)*(n + 9)) - 156*n*(n - 1)*(n - 2)/((n + 8)*(n + 9)*(n + 10)) + ... = 0 for n = 1,2,3,.... Cf. A005585 and A053347. - Peter Bala, Feb 18 2019
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
FORMULA
a(n) = (-1)^n*A053120(2*n+7, 7)/64 (1/64 of eighth unsigned column of Chebyshev T-triangle, zeros omitted).
G.f.: (1+x)/(1-x)^8.
a(n) = 2*C(n+7, 7)-C(n+6, 6). - Paul Barry, Mar 04 2003
a(n) = C(n+6,6)+2*C(n+6,7). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
a(n) = (-1)^n*A084930(n+3, 3)/64. Compare with the first line above. - Wolfdieter Lang, Aug 04 2014
a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8) for n>7. - Wesley Ivan Hurt, Jan 01 2016
From Amiram Eldar, Jan 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 24871/25 - 7168*log(2)/5.
Sum_{n>=0} (-1)^n/a(n) = 1792*Pi/5 - 28126/25. (End)
MAPLE
MATHEMATICA
s1=s2=s3=s4=s5=0; lst={}; Do[s1+=n^2; s2+=s1; s3+=s2; s4+=s3; s5+=s4; AppendTo[lst, s5], {n, 0, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
CoefficientList[Series[(1 + x) / (1 - x)^8, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 )
Table[SeriesCoefficient[(1 + x)/(1 - x)^8, {x, 0, n}], {n, 0, 28}] (* or *)
Table[Binomial[n + 6, 6] (2 n + 7)/7, {n, 0, 30}] (* Michael De Vlieger, Dec 31 2015 *)
PROG
(Magma) [Binomial(n+6, 6) + 2*Binomial(n+6, 7): n in [0..35]]; // Vincenzo Librandi, Jun 09 2013
(PARI) a(n)=binomial(n+6, 6)*(2*n+7)/7 \\ Charles R Greathouse IV, Sep 24 2015
(Python)
A050486_list, m = [], [2]+[1]*7
for _ in range(10**2):
A050486_list.append(m[-1])
for i in range(7):
m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016
CROSSREFS
Fourth column (s=3, without leading zeros) of A111125. - Wolfdieter Lang, Oct 18 2012
Cf. A084960 (unsigned fourth column divided by 64). - Wolfdieter Lang, Aug 04 2014
KEYWORD
nonn,easy
AUTHOR
Barry E. Williams, Dec 26 1999
STATUS
approved