A014106 a(n) = n*(2*n + 3).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

a(n-1) is the dimension of the second cohomology group of 2n+1-dimensional Heisenberg Lie algebra h_{2n+1}. - Rafik Khalfi, Jan 27 2025

Examples

			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
		

References

• Jolley, Summation of Series, Dover (1961).

Links

• 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
• L. J. Santharoubane, Cohomology of Heisenberg Lie algebras, Proc. Amer. Math. Soc. 87 (1983), 23-28.
• 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).

Crossrefs

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.

Programs

Magma

[n*(2*n+3): n in [0..50]]; // Vincenzo Librandi, Apr 25 2011

Maple

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

Mathematica

Table[n (2 n + 3), {n, 0, 120}] (* Michael De Vlieger, Apr 02 2015 *)
LinearRecurrence[{3,-3,1},{0,5,14},50] (* Harvey P. Dale, Jul 21 2023 *)

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

Formula

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)