A062159 a(n) = n^5 - n^4 + n^3 - n^2 + n - 1.

-1, 0, 21, 182, 819, 2604, 6665, 14706, 29127, 53144, 90909, 147630, 229691, 344772, 501969, 711914, 986895, 1340976, 1790117, 2352294, 3047619, 3898460, 4929561, 6168162, 7644119, 9390024, 11441325, 13836446, 16616907, 19827444, 23516129, 27734490, 32537631, 37984352, 44137269

Offset: 0

Author

Henry Bottomley, Jun 08 2001

Keywords

• easy
• sign

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

Examples

			a(4) = 4^5 - 4^4 + 4^3 - 4^2 + 4 - 1 = 1024 - 256 + 64 - 16 + 4 - 1 = 819.
		

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).

Crossrefs

Cf. A023443, A002061, A062158, A060884, A060888.

Cf. A002061.

Programs

Maple

A062159:=n->n^5-n^4+n^3-n^2+n-1; seq(A062159(k), k=0..100); # Wesley Ivan Hurt, Nov 06 2013

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 *)

PARI

a(n) = { n*(n*(n*(n*(n - 1) + 1) - 1) + 1) - 1 } \\ Harry J. Smith, Aug 02 2009

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

Extensions

More terms from Emeric Deutsch, Apr 01 2004