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%I #101 Jul 21 2023 20:05:19
%S 0,5,14,27,44,65,90,119,152,189,230,275,324,377,434,495,560,629,702,
%T 779,860,945,1034,1127,1224,1325,1430,1539,1652,1769,1890,2015,2144,
%U 2277,2414,2555,2700,2849,3002,3159,3320,3485,3654,3827,4004,4185,4370
%N a(n) = n*(2*n + 3).
%C If Y is a 2-subset of a 2n-set X then, for n >= 1, a(n-1) is the number of (2n-2)-subsets of X intersecting Y. - _Milan Janjic_, Nov 18 2007
%C This sequence can also be derived from 1*(2+3)=5, 2*(3+4)=14, 3*(4+5)=27, and so forth. - _J. M. Bergot_, May 30 2011
%C Consider the partitions of 2n into exactly two parts. Then a(n) is the sum of all the parts in the partitions of 2n + the number of partitions of 2n + the total number of partition parts of 2n. - _Wesley Ivan Hurt_, Jul 02 2013
%C a(n) is the number of self-intersecting points of star polygon {(2*n+3)/(n+1)}. - _Bui Quang Tuan_, Mar 25 2015
%C Bisection of A000096. - _Omar E. Pol_, Dec 16 2016
%C a(n+1) is the number of function calls required to compute Ackermann's function ack(2,n). - _Olivier Gérard_, May 11 2018
%C a(n-1) is the least denominator d > n of the best rational approximation of sqrt(n^2-2) by x/d (see example and PARI code). - _Hugo Pfoertner_, Apr 30 2019
%C The number of cells in a loose n X n+1 rectangular spiral where n is even. See loose rectangular spiral image. - _Jeff Bowermaster_, Aug 05 2019
%D Jolley, Summation of Series, Dover (1961).
%H Vincenzo Librandi, <a href="/A014106/b014106.txt">Table of n, a(n) for n = 0..920</a>
%H Jeff Bowermaster, <a href="/A014106/a014106.png">Loose Rectangular Spiral</a>
%H Sergio Falcon, <a href="http://dx.doi.org/10.4236/am.2014.515216">Relationships between Some k-Fibonacci Sequences</a>, Applied Mathematics, 2014, 5, 2226-2234.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Leo Tavares, <a href="/A014106/a014106.jpg">Illustration: Hex-tangles</a>
%H Leo Tavares, <a href="/A014106/a014106_1.jpg">Illustration: Second Hex-tangles</a>
%H Leo Tavares, <a href="/A014106/a014106_2.jpg">Illustration: Ob-tangles</a>
%H Leo Tavares, <a href="/A014106/a014106_3.jpg">Illustration: Trap-tangles</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StarPolygon.html">Star Polygon</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) - 1 = A091823(n). - _Howard A. Landman_, Mar 28 2004
%F A014107(-n) = a(n), A000384(n+1) = a(n)+1. - _Michael Somos_, Nov 06 2005
%F G.f.: x*(5 - x)/(1 - x)^3. - _Paul Barry_, Feb 27 2003
%F E.g.f: x*(5 + 2*x)*exp(x). - _Michael Somos_, Nov 06 2005
%F a(n) = a(n-1) + 4*n + 1, n > 0. - _Vincenzo Librandi_, Nov 19 2010
%F a(n) = 4*A000217(n) + n. - _Bruno Berselli_, Feb 11 2011
%F Sum_{n>=1} 1/a(n) = 8/9 -2*log(2)/3 = 0.4267907685155920.. [Jolley eq. 265]
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 + log(2)/3 - Pi/6. - _Amiram Eldar_, Jul 03 2020
%F From _Leo Tavares_, Jan 27 2022: (Start)
%F a(n) = A000384(n+1) - 1. See Hex-tangles illustration.
%F a(n) = A014105(n) + n*2. See Second Hex-tangles illustration.
%F a(n) = 2*A002378(n) + n. See Ob-tangles illustration.
%F a(n) = A005563(n) + 2*A000217(n). See Trap-tangles illustration. (End)
%e a(5-1) = 44: The best approximation of sqrt(5^2-2) = sqrt(23) by x/d with d <= k is 24/5 for all k < 44, but sqrt(23) ~= 211/44 is the first improvement. - _Hugo Pfoertner_, Apr 30 2019
%p A014106 := proc(n) n*(2*n+3) ; end proc: # _R. J. Mathar_, Feb 13 2011
%p seq(k*(2*k+3), k=1..100); # _Wesley Ivan Hurt_, Jul 02 2013
%t Table[n (2 n + 3), {n, 0, 120}] (* _Michael De Vlieger_, Apr 02 2015 *)
%t LinearRecurrence[{3,-3,1},{0,5,14},50] (* _Harvey P. Dale_, Jul 21 2023 *)
%o (PARI) a(n)=2*n^2+3*n
%o (PARI) \\ least denominator > n in best rational approximation of sqrt(n^2-2)
%o for(n=2,47,for(k=n,oo,my(m=denominator(bestappr(sqrt(n^2-2),k)));if(m>n,print1(k,", ");break(1)))) \\ _Hugo Pfoertner_, Apr 30 2019
%o (Magma) [n*(2*n+3): n in [0..50]]; // _Vincenzo Librandi_, Apr 25 2011
%Y Cf. A091823. See A110325 for another version.
%Y Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A028895, A045944, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734, A139273.
%Y Cf. A000384, A014105, A002378, A005563, A000217.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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