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A062158
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a(n) = n^3 - n^2 + n - 1 = (n-1) * (n^2 + 1).
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13
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-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
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OFFSET
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0,3
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COMMENTS
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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 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
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LINKS
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FORMULA
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a(n) = round(n^4/(n+1)) 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
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EXAMPLE
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a(4) = 4^3 - 4^2 + 4 - 1 = 64 - 16 + 4 - 1 = 51.
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MAPLE
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a:=n->sum(1+sum(n, k=1..n), k=2..n):seq(a(n), n=0...43); # Zerinvary Lajos, Aug 24 2008
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MATHEMATICA
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PROG
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(PARI) { for (n=0, 1000, write("b062158.txt", n, " ", n*(n*(n - 1) + 1) - 1) ) } \\ Harry J. Smith, Aug 02 2009
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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