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%I #168 Feb 12 2024 02:28:04
%S 0,4,10,18,28,40,54,70,88,108,130,154,180,208,238,270,304,340,378,418,
%T 460,504,550,598,648,700,754,810,868,928,990,1054,1120,1188,1258,1330,
%U 1404,1480,1558,1638,1720,1804,1890,1978,2068,2160,2254,2350,2448,2548,2650
%N a(n) = n*(n+3).
%C n*(n-3), for n >= 3, is the number of [body] diagonals of an n-gonal prism. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
%C 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
%C 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
%C 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
%C Starting with offset 1 = binomial transform of [4, 6, 2, 0, 0, 0, ...]. - _Gary W. Adamson_, Jan 09 2009
%C The sequence provides all nonnegative integers m such that 4*m + 9 is a square. - _Vincenzo Librandi_, Mar 03 2013
%C 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
%C 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
%C Partial sums of the even composites (which are A005843 without the 2). - _R. J. Mathar_, Sep 09 2015
%C 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:
%C __ __ __ __ __ __ __ __ __ __
%C |__| |__| | |__| | | |__| | | |
%C |_____| |__ __| | |__ __| | |
%C |__ __ __| |__ __ __| |
%C |__ __ __ __|
%C 4 10 18 28 - _Stefano Spezia_, May 29 2023
%H Charles R Greathouse IV, <a href="/A028552/b028552.txt">Table of n, a(n) for n = 0..10000</a>
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/consemor.htm">Palindromic Quasipronics of the form n(n+x)</a>.
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/996452/expected-number-of-turns-for-a-rook-to-move-to-top-right-most-corner">Expected number of turns for a rook to move to top right-most corner?</a>.
%H Felix P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, Preprint on ResearchGate, March 2014.
%H Aleksandar Petojević, <a href="http://dx.doi.org/10.5937/MatMor0801037P">A Note about the Pochhammer Symbol</a>, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 2*A000096(n).
%F a(A002522(n)) = A156798(n). - _Reinhard Zumkeller_, Feb 16 2009
%F a(n) = a(n-1) + 2*(n+1) for n>0, with a(0)=0. - _Vincenzo Librandi_, Aug 05 2010
%F 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
%F G.f.: 2*x*(2 - x)/(1 - x)^3. - _Arkadiusz Wesolowski_, Dec 31 2011
%F 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
%F a(n) = 4*C(n+1,2) - 2*C(n,2) for n>=0. - _Felix P. Muga II_, Mar 11 2014
%F a(-3 - n) = a(n) for all n in Z. - _Michael Somos_, Mar 18 2014
%F E.g.f.: (x^3 + 4*x)*exp(x). - _G. C. Greubel_, Jul 20 2017
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/3 - 5/18. - _Amiram Eldar_, Jan 15 2021
%F From _Amiram Eldar_, Feb 12 2024: (Start)
%F Product_{n>=1} (1 - 1/a(n)) = 2*cos(sqrt(13)*Pi/2)/Pi.
%F Product_{n>=1} (1 + 1/a(n)) = -6*cos(sqrt(5)*Pi/2)/Pi. (End)
%e 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 + ...
%p A028552 := proc(n) n*(n+3); end proc: # _R. J. Mathar_, Jan 29 2011
%t LinearRecurrence[{3,-3,1},{0,4,10},50] (* _Harvey P. Dale_, Feb 05 2012 *)
%t Table[ChineseRemainder[{n, n + 1}, {n + 2, n + 3}], {n, -1, 80}] (* _Zak Seidov_, Oct 25 2014 *)
%o (Magma) [n*(n+3): n in [0..150]]; // _Vincenzo Librandi_, Apr 21 2011
%o (PARI) a(n)=n*(n+3) \\ _Charles R Greathouse IV_, Mar 16 2012
%o (Maxima) makelist(n*(n+3),n,0,20); /* _Martin Ettl_, Jan 22 2013 */
%Y Cf. A000096, A002522, A005843, A028387, A062145, A156798.
%K nonn,easy,nice
%O 0,2
%A _Patrick De Geest_
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