OFFSET
0,3
COMMENTS
Number of walks of length 6 between any two distinct nodes of the complete graph K_{n+1} (n>=1). - Emeric Deutsch, Apr 01 2004
For odd n, a(n) * (n+1) / 2 + 1 also represents the first integer in a sum of n^6 consecutive integers that equals n^12. - Patrick J. McNab, Dec 26 2016
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
FORMULA
a(n) = round(n^6/(n+1)) for n>2 = A062160(n,6).
G.f.: (76x^3 + 6x^2 + 27x^4 + 6x^5 + 6x - 1)/(1-x)^6 (for the signed sequence). - Emeric Deutsch, Apr 01 2004
a(n) = (n^6 - 1)/(n+1). a(n) = (n-1)(n^2 - n + 1)(n^2 + n + 1) = (n-1)*A002061(n)*A002061(n+1). - Alexander Adamchuk, Apr 12 2006
a(0)=-1, a(1)=0, a(2)=21, a(3)=182, a(4)=819, a(5)=2604, a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Harvey P. Dale, Dec 20 2015
E.g.f.: exp(x)*(x^5 + 9*x^4 + 20*x^3 + 10*x^2 + x - 1). - Stefano Spezia, Apr 22 2023
EXAMPLE
a(4) = 4^5 - 4^4 + 4^3 - 4^2 + 4 - 1 = 1024 - 256 + 64 - 16 + 4 - 1 = 819.
MAPLE
MATHEMATICA
Table[n^5-n^4+n^3-n^2+n-1, {n, 0, 100}] (* Wesley Ivan Hurt, Nov 06 2013 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {-1, 0, 21, 182, 819, 2604}, 40] (* Harvey P. Dale, Dec 20 2015 *)
PROG
(PARI) { for (n=0, 1000, write("b062159.txt", n, " ", n*(n*(n*(n*(n - 1) + 1) - 1) + 1) - 1) ) } \\ Harry J. Smith, Aug 02 2009
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Henry Bottomley, Jun 08 2001
EXTENSIONS
More terms from Emeric Deutsch, Apr 01 2004
STATUS
approved