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A028552 a(n) = n*(n+3). 62
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, 2160, 2254, 2350, 2448, 2548, 2650 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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
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->infinity} 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 Mathematics Stack Exchange 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
a(n) is the number of segments necessary to represent n squares of area 1, 4, ..., n^2 having the upper and left sides overlapped:
__ __ __ __ __ __ __ __ __ __
|__| |__| | |__| | | |__| | | |
|_____| |__ __| | |__ __| | |
|__ __ __| |__ __ __| |
|__ __ __ __|
4 10 18 28 - Stefano Spezia, May 29 2023
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
FORMULA
a(n) = 2*A000096(n).
a(A002522(n)) = A156798(n). - Reinhard Zumkeller, Feb 16 2009
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
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/3 - 5/18. - Amiram Eldar, Jan 15 2021
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = 2*cos(sqrt(13)*Pi/2)/Pi.
Product_{n>=1} (1 + 1/a(n)) = -6*cos(sqrt(5)*Pi/2)/Pi. (End)
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
Sequence in context: A161958 A013921 A217748 * A009877 A009880 A197057
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified April 15 23:50 EDT 2024. Contains 371696 sequences. (Running on oeis4.)