A062158 a(n) = n^3 - n^2 + n - 1 = (n-1) * (n^2 + 1).

-1, 0, 5, 20, 51, 104, 185, 300, 455, 656, 909, 1220, 1595, 2040, 2561, 3164, 3855, 4640, 5525, 6516, 7619, 8840, 10185, 11660, 13271, 15024, 16925, 18980, 21195, 23576, 26129, 28860, 31775, 34880, 38181, 41684, 45395, 49320, 53465, 57836, 62439, 67280, 72365, 77700, 83291, 89144, 95265, 101660

Offset

0,3

Comments

Number of walks of length 4 between any two distinct vertices of the complete graph K_{n+1} (n >= 1). Example: a(2) = 5 because in the complete graph ABC we have the following walks of length 4 between A and B: ABACB, ABCAB, ACACB, ACBAB and ACBCB. - Emeric Deutsch, Apr 01 2004

1/a(n) for n >= 2, is in base n given by 0.repeat(0,0,1,1), due to (1/n^3 + 1/n^4)*(1/(1-1/n^4)) = 1/((n-1)*(n^2+1)). - Wolfdieter Lang, Jun 20 2014

For n>3, a(n) is 1220 in base n-1. - Bruno Berselli, Jan 26 2016

For odd n, a(n) * (n+1) / 2 + 1 also represents the first integer in a sum of n^4 consecutive integers that equals n^8. - 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 (4,-6,4,-1).

Formula

a(n) = round(n^4/(n+1)) for n >= 2.

a(n) = A062160(n, 4), for n > 2.

G.f.: (4*x-1)*(1+x^2)/(1-x)^4 (for the signed sequence). - Emeric Deutsch, Apr 01 2004

a(n) = floor(n^5/(n^2+n)) for n > 0. - Gary Detlefs, May 27 2010

a(n) = -A053698(-n). - Bruno Berselli, Jan 26 2016

Sum_{n>=2} 1/a(n) = A268086. - Amiram Eldar, Nov 18 2020

E.g.f.: exp(x)*(x^3 + 2*x^2 + x - 1). - Stefano Spezia, Apr 22 2023

Example

a(4) = 4^3 - 4^2 + 4 - 1 = 64 - 16 + 4 - 1 = 51.

Maple

[seq(n^3-n^2+n-1, n=0..49)]; # Zerinvary Lajos, Jun 29 2006
a:=n->sum(1+sum(n, k=1..n), k=2..n):seq(a(n), n=0...43); # Zerinvary Lajos, Aug 24 2008

Mathematica

Table[n^3 - n^2 + n - 1, {n, 0, 49}] (* Alonso del Arte, Apr 30 2014 *)

Prog

(PARI) { for (n=0, 1000, write("b062158.txt", n, " ", n*(n*(n - 1) + 1) - 1) ) } \\ Harry J. Smith, Aug 02 2009
(Magma) [n^3 - n^2 + n - 1 : n in [0..50]]; // Wesley Ivan Hurt, Dec 26 2016

Crossrefs

Cf. A002061, A023443, A053698, A060884, A060888, A062159, A062160, A268086.

Sequence in context: A190094 A134481 A358632 * A034133 A006504 A007045

Adjacent sequences: A062155 A062156 A062157 * A062159 A062160 A062161

Keyword

sign,easy

Author

Henry Bottomley, Jun 08 2001

Extensions

More terms from Emeric Deutsch, Apr 01 2004

Status

approved