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A014106
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a(n) = n*(2*n + 3).
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55
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0, 5, 14, 27, 44, 65, 90, 119, 152, 189, 230, 275, 324, 377, 434, 495, 560, 629, 702, 779, 860, 945, 1034, 1127, 1224, 1325, 1430, 1539, 1652, 1769, 1890, 2015, 2144, 2277, 2414, 2555, 2700, 2849, 3002, 3159, 3320, 3485, 3654, 3827, 4004, 4185, 4370
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OFFSET
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0,2
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COMMENTS
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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
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
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
a(n) is the number of self-intersecting points of star polygon {(2*n+3)/(n+1)}. - Bui Quang Tuan, Mar 25 2015
Bisection of A000096. - Omar E. Pol, Dec 16 2016
a(n+1) is the number of function calls required to compute Ackermann's function ack(2,n). - Olivier Gérard, May 11 2018
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
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
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REFERENCES
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Jolley, Summation of Series, Dover (1961).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..920
Jeff Bowermaster, Loose Rectangular Spiral
Sergio Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234.
Milan Janjic, Two Enumerative Functions
Leo Tavares, Illustration: Hex-tangles
Leo Tavares, Illustration: Second Hex-tangles
Leo Tavares, Illustration: Ob-tangles
Leo Tavares, Illustration: Trap-tangles
Eric Weisstein's World of Mathematics, Star Polygon
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) - 1 = A091823(n). - Howard A. Landman, Mar 28 2004
A014107(-n) = a(n), A000384(n+1) = a(n)+1. - Michael Somos, Nov 06 2005
G.f.: x*(5 - x)/(1 - x)^3. - Paul Barry, Feb 27 2003
E.g.f: x*(5 + 2*x)*exp(x). - Michael Somos, Nov 06 2005
a(n) = a(n-1) + 4*n + 1, n > 0. - Vincenzo Librandi, Nov 19 2010
a(n) = 4*A000217(n) + n. - Bruno Berselli, Feb 11 2011
Sum_{n>=1} 1/a(n) = 8/9 -2*log(2)/3 = 0.4267907685155920.. [Jolley eq. 265]
Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 + log(2)/3 - Pi/6. - Amiram Eldar, Jul 03 2020
From Leo Tavares, Jan 27 2022: (Start)
a(n) = A000384(n+1) - 1. See Hex-tangles illustration.
a(n) = A014105(n) + n*2. See Second Hex-tangles illustration.
a(n) = 2*A002378(n) + n. See Ob-tangles illustration.
a(n) = A005563(n) + 2*A000217(n). See Trap-tangles illustration. (End)
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EXAMPLE
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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
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MAPLE
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A014106 := proc(n) n*(2*n+3) ; end proc: # R. J. Mathar, Feb 13 2011
seq(k*(2*k+3), k=1..100); # Wesley Ivan Hurt, Jul 02 2013
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MATHEMATICA
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Table[n (2 n + 3), {n, 0, 120}] (* Michael De Vlieger, Apr 02 2015 *)
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PROG
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(PARI) a(n)=2*n^2+3*n
(PARI) \\ least denominator > n in best rational approximation of sqrt(n^2-2)
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
(Magma) [n*(2*n+3): n in [0..50]]; // Vincenzo Librandi, Apr 25 2011
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CROSSREFS
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Cf. A091823. See A110325 for another version.
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.
Cf. A000384, A014105, A002378, A005563, A000217.
Sequence in context: A185233 A065351 A002503 * A110325 A331775 A140342
Adjacent sequences: A014103 A014104 A014105 * A014107 A014108 A014109
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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