A028552 a(n) = n*(n+3).

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

Offset: 0

Author

Patrick De Geest

Keywords

• nonn
• easy
• nice

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

Examples

			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 + ...
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
• Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
• Milan Janjic, Two Enumerative Functions.
• Mathematics Stack Exchange, Expected number of turns for a rook to move to top right-most corner?.
• 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.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000096, A002522, A005843, A028387, A062145, A156798.

Programs

Magma

[n*(n+3): n in [0..150]]; // Vincenzo Librandi, Apr 21 2011

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 *)
Table[ChineseRemainder@@TakeDrop[Range[n,n+3],2],{n,-1,50}] (* Harvey P. Dale, Mar 14 2025 *)

Maxima

makelist(n*(n+3),n,0,20); /* Martin Ettl, Jan 22 2013 */

PARI

a(n)=n*(n+3) \\ Charles R Greathouse IV, Mar 16 2012

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)