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A069778
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q-factorial numbers 3!_q.
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8
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1, 6, 21, 52, 105, 186, 301, 456, 657, 910, 1221, 1596, 2041, 2562, 3165, 3856, 4641, 5526, 6517, 7620, 8841, 10186, 11661, 13272, 15025, 16926, 18981, 21196, 23577, 26130, 28861, 31776, 34881, 38182, 41685, 45396, 49321
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of proper n-colorings of the 4-cycle with one vertex color fixed (offset 2). - Michael Somos, Jul 19 2002
n such that x^3 + x^2 + x + n factors over the integers. - James Buddenhagen (jbuddenh(AT)gmail.com), Apr 19 2005
If Y is a 4-subset of an n-set X then, for n>=5, a(n-5) is the number of 5-subsets of X having at least two elements in common with Y. - Milan R. Janjic (agnus(AT)blic.net), Dec 08 2007
Equals row sums of the Connell (A001614) sequence as a triangle:/Q 1;/Q 2, 4;/Q 5, 7, 9;/Q 10, 12, 14, 16;/Q ... [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2008]
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REFERENCES
| T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
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LINKS
| C. P. Simoes, Teste de Desempenho Mental.
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FORMULA
| a(n) = (n + 1)*(n^2 + n + 1)
a(n)=(n+1)^3-2*T(n) where T(n) is the n-th triangular number: n*(n+1)/2 (A000217) 1^3-2x0=1; 2^3-2x1=6; 3^3-2x3=21; 4^3-2x6=52; 5^3-2*10=105 - Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 14 2006
a(n)= n^8 mod (n^3+n), with offset 1..a(1)=1 [From Gary Detlefs, May 02 2010]
a(0)=1, a(1)=6, a(2)=21, a(3)=52, a(n)=4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4) [From Harvey P. Dale, Jul 11 2011]
G.f.: (1+2*x+3*x^2)/(1-x)^4 [From Harvey P. Dale, Jul 11 2011]
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EXAMPLE
| For 2-colorings only 1212 is proper so a(2-2)=1. The proper 3-colorings are: 1212,1313,1213,1312,1232,1323 so a(3-2)=6.
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MAPLE
| n->sum(-2+sum(2-sum(2, j=1..n), j=1..n), j=1..n):seq(a(n)/(-2), n=1..40); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 06 2008]
seq(n^8 mod (n^3+n), n=1..37); [From Gary Detlefs, May 02 2010]
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MATHEMATICA
| a[n_]:=n^3-n^2+n^1; lst={}; Do[AppendTo[lst, a[n]], {n, 0, 5!}]; lst [From Vladimir Orlovsky, Dec 22 2008]
LinearRecurrence[{4, -6, 4, -1}, {1, 6, 21, 52}, 40] (* or *) Table[(n+1)(n^2+n+1), {n, 0, 40}] (* From Harvey P. Dale, Jul 11 2011 *)
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PROG
| (PARI) a(n)=(n+1)*(n^2+n+1)
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CROSSREFS
| Cf. A069777.
A001614 [From Gary W. Adamson, Sep 01 2008]
Sequence in context: A028345 A097124 A135454 * A015644 A067680 A115052
Adjacent sequences: A069775 A069776 A069777 * A069779 A069780 A069781
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KEYWORD
| easy,nonn
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AUTHOR
| Frank Adams-Watters (FrankTAW(AT)Netscape.net), Apr 07 2002
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