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A014106 a(n) = n*(2*n + 3). 56
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
REFERENCES
Jolley, Summation of Series, Dover (1961).
LINKS
Jeff Bowermaster, Loose Rectangular Spiral
Sergio Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234.
Eric Weisstein's World of Mathematics, Star Polygon
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)
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
Cf. A091823. See A110325 for another version.
Sequence in context: A185233 A065351 A002503 * A110325 A331775 A140342
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified July 15 17:56 EDT 2024. Contains 374333 sequences. (Running on oeis4.)