

A062158


a(n) = n^3  n^2 + n  1.


10



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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/(11/n^4)) = 1/((n1)*(n^2+1)).  Wolfdieter Lang, Jun 20 2014
For n>3, a(n) is 1220 in base n1.  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


FORMULA

a(n) = round[n^4/(n+1)] for n > 2 = A062160(n, 4).
G.f.: (4*x1)*(1+x^2)/(1x)^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]


EXAMPLE

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


MAPLE

[seq(n^3n^2+n1, 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.
Sequence in context: A297569 A190094 A134481 * 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



