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A028552
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a(n) = n*(n+3).
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56
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0, 4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180, 208, 238, 270, 304, 340, 378, 418, 460, 504, 550, 598, 648, 700, 754, 810, 868, 928, 990, 1054, 1120, 1188, 1258, 1330, 1404, 1480, 1558, 1638, 1720, 1804, 1890, 1978, 2068
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refs;
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history;
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internal format)
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OFFSET
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0,2
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COMMENTS
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n(n-3), for n >= 3, is the number of [body] diagonals of an n-gonal prism. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
a(n) = A028387(n)-1. Half of the difference between n(n+1)(n+2)(n+3) and the largest square less than it. Calling this difference "SquareMod": a(n) = (1/2)*SquareMod(n(n+1)(n+2)(n+3)). - Rainer Rosenthal, Sep 04 2004
n != -2 such that x^4 + x^3 - n*x^2 + x + 1 is reducible over the integers. Starting at 10: n such that x^4 + x^3 - n*x^2 + x + 1 is a product of irreducible quadratic factors over the integers. - James R. Buddenhagen, Apr 19 2005
If a 3-set Y and a 3-set Z, having two element in common, are subsets of an n-set X then a(n-4) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007
Starting with offset 1 = binomial transform of [4, 6, 2, 0, 0, 0, ...]. - Gary W. Adamson, Jan 09 2009
a(A002522(n)) = A156798(n). - Reinhard Zumkeller, Feb 16 2009
The sequence provides all nonnegative integers m such that 4*m + 9 is a square. - Vincenzo Librandi, Mar 03 2013
The second order linear recurrence relations b(n)=3*b(n-1) + a(m-3)*b(n-2), n>=2, b(0)=0, b(1)=1, have closed form solutions involving only powers of m and 3-m where m>=4 is a positive integer; and lim_{n->inf} b(n+1)/b(n) = 4. - Felix P. Muga II, Mar 18 2014
If a rook is placed at a corner of an n X n chessboard, the expected number of moves for it to reach the opposite corner is a(n-1). (See Math StackExchange link.) - Eric M. Schmidt, Oct 29 2014
Partial sums of the even composites (which are A005843 without the 2). - R. J. Mathar, Sep 09 2015
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LINKS
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Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
P. De Geest, Palindromic Quasipronics of the form n(n+x)
Milan Janjic, Two Enumerative Functions
Math StackExchange, Expected number of turns for a rook to move to top right-most corner?
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = 2*A000096(n).
a(n) = a(n-1) + 2*(n+1) for n>0, with a(0)=0. - Vincenzo Librandi, Aug 05 2010
Sum_{n>=1} 1/a(n) = 11/18 via Sum_{n>=0} 1/((n+x)*(n+y)) = (psi(x)-psi(y))/ (x-y). - R. J. Mathar, Mar 22 2011
G.f.: 2*x*(2 - x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 31 2011
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3), with a(0)=0, a(1)=4, a(2)=10. - Harvey P. Dale, Feb 05 2012
a(n) = 4*C(n+1,2)-2*C(n,2) for n>=0. - Felix P. Muga II, Mar 11 2014
a(-3 - n) = a(n) for all n in Z. - Michael Somos, Mar 18 2014
E.g.f.: (x^3 + 4*x)*exp(x). - G. C. Greubel, Jul 20 2017
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EXAMPLE
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G.f. = 4*x + 10*x^2 + 18*x^3 + 28*x^4 + 40*x^5 + 54*x^6 + 70*x^7 + 88*x^8 + ...
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MAPLE
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A028552 := proc(n) n*(n+3); end proc: # R. J. Mathar, Jan 29 2011
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {0, 4, 10}, 50] (* Harvey P. Dale, Feb 05 2012 *)
Table[ChineseRemainder[{n, n + 1}, {n + 2, n + 3}], {n, -1, 80}] (* Zak Seidov, Oct 25 2014 *)
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PROG
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(MAGMA) [n*(n+3): n in [0..150]]; // Vincenzo Librandi, Apr 21 2011
(PARI) a(n)=n*(n+3) \\ Charles R Greathouse IV, Mar 16 2012
(Maxima) makelist(n*(n+3), n, 0, 20); /* Martin Ettl, Jan 22 2013 */
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CROSSREFS
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Cf. A000096, A002522, A028387, A062145.
Sequence in context: A009876 A161958 A013921 * A217748 A009877 A009880
Adjacent sequences: A028549 A028550 A028551 * A028553 A028554 A028555
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Patrick De Geest
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STATUS
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approved
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