# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a014106 Showing 1-1 of 1 %I A014106 #101 Jul 21 2023 20:05:19 %S A014106 0,5,14,27,44,65,90,119,152,189,230,275,324,377,434,495,560,629,702, %T A014106 779,860,945,1034,1127,1224,1325,1430,1539,1652,1769,1890,2015,2144, %U A014106 2277,2414,2555,2700,2849,3002,3159,3320,3485,3654,3827,4004,4185,4370 %N A014106 a(n) = n*(2*n + 3). %C A014106 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 A014106 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 A014106 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 A014106 a(n) is the number of self-intersecting points of star polygon {(2*n+3)/(n+1)}. - _Bui Quang Tuan_, Mar 25 2015 %C A014106 Bisection of A000096. - _Omar E. Pol_, Dec 16 2016 %C A014106 a(n+1) is the number of function calls required to compute Ackermann's function ack(2,n). - _Olivier Gérard_, May 11 2018 %C A014106 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 A014106 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 A014106 Jolley, Summation of Series, Dover (1961). %H A014106 Vincenzo Librandi, Table of n, a(n) for n = 0..920 %H A014106 Jeff Bowermaster, Loose Rectangular Spiral %H A014106 Sergio Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234. %H A014106 Milan Janjic, Two Enumerative Functions %H A014106 Leo Tavares, Illustration: Hex-tangles %H A014106 Leo Tavares, Illustration: Second Hex-tangles %H A014106 Leo Tavares, Illustration: Ob-tangles %H A014106 Leo Tavares, Illustration: Trap-tangles %H A014106 Eric Weisstein's World of Mathematics, Star Polygon %H A014106 Index entries for linear recurrences with constant coefficients, signature (3,-3,1). %F A014106 a(n) - 1 = A091823(n). - _Howard A. Landman_, Mar 28 2004 %F A014106 A014107(-n) = a(n), A000384(n+1) = a(n)+1. - _Michael Somos_, Nov 06 2005 %F A014106 G.f.: x*(5 - x)/(1 - x)^3. - _Paul Barry_, Feb 27 2003 %F A014106 E.g.f: x*(5 + 2*x)*exp(x). - _Michael Somos_, Nov 06 2005 %F A014106 a(n) = a(n-1) + 4*n + 1, n > 0. - _Vincenzo Librandi_, Nov 19 2010 %F A014106 a(n) = 4*A000217(n) + n. - _Bruno Berselli_, Feb 11 2011 %F A014106 Sum_{n>=1} 1/a(n) = 8/9 -2*log(2)/3 = 0.4267907685155920.. [Jolley eq. 265] %F A014106 Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 + log(2)/3 - Pi/6. - _Amiram Eldar_, Jul 03 2020 %F A014106 From _Leo Tavares_, Jan 27 2022: (Start) %F A014106 a(n) = A000384(n+1) - 1. See Hex-tangles illustration. %F A014106 a(n) = A014105(n) + n*2. See Second Hex-tangles illustration. %F A014106 a(n) = 2*A002378(n) + n. See Ob-tangles illustration. %F A014106 a(n) = A005563(n) + 2*A000217(n). See Trap-tangles illustration. (End) %e A014106 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 A014106 := proc(n) n*(2*n+3) ; end proc: # _R. J. Mathar_, Feb 13 2011 %p A014106 seq(k*(2*k+3), k=1..100); # _Wesley Ivan Hurt_, Jul 02 2013 %t A014106 Table[n (2 n + 3), {n, 0, 120}] (* _Michael De Vlieger_, Apr 02 2015 *) %t A014106 LinearRecurrence[{3,-3,1},{0,5,14},50] (* _Harvey P. Dale_, Jul 21 2023 *) %o A014106 (PARI) a(n)=2*n^2+3*n %o A014106 (PARI) \\ least denominator > n in best rational approximation of sqrt(n^2-2) %o A014106 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 A014106 (Magma) [n*(2*n+3): n in [0..50]]; // _Vincenzo Librandi_, Apr 25 2011 %Y A014106 Cf. A091823. See A110325 for another version. %Y A014106 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 A014106 Cf. A000384, A014105, A002378, A005563, A000217. %K A014106 nonn,easy %O A014106 0,2 %A A014106 _N. J. A. Sloane_ # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE