

A028552


a(n) = n*(n+3).


53



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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OFFSET

0,2


COMMENTS

n(n3), for n >= 3, is the number of [body] diagonals of an ngonal 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 3set Y and a 3set Z, having two element in common, are subsets of an nset X then a(n4) is the number of 3subsets 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(n1) + a(m3)*b(n2), n>=2, b(0)=0, b(1)=1, have closed form solutions involving only powers of m and 3m 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(n1). (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


LINKS

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 rightmost corner?
F. P. Muga II, Extending the Golden Ratio and the Binetde Moivre Formula, March 2014; Preprint on ResearchGate.
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

a(n) = 2*A000096(n).
a(n) = a(n1) + 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))/ (xy).  R. J. Mathar, Mar 22 2011
G.f.: 2*x*(2  x)/(1  x)^3.  Arkadiusz Wesolowski, Dec 31 2011
a(n) = 3*a(n1)3*a(n2)+a(n3), 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


EXAMPLE

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 + ...


MAPLE

A028552 := proc(n) n*(n+3); end proc: # R. J. Mathar, Jan 29 2011


MATHEMATICA

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 *)


PROG

(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 */


CROSSREFS

Cf. A000096, A002522, A028387, A062145.
Sequence in context: A009876 A161958 A013921 * A217748 A009877 A009880
Adjacent sequences: A028549 A028550 A028551 * A028553 A028554 A028555


KEYWORD

nonn,easy,nice


AUTHOR

Patrick De Geest


STATUS

approved



