A116940 Greatest m such that A116939(m) = n.

0, 3, 6, 11, 16, 23, 30, 39, 48, 59, 70, 83, 96, 111, 126, 143, 160, 179, 198, 219, 240, 263, 286, 311, 336, 363, 390, 419, 448, 479, 510, 543, 576, 611, 646, 683, 720, 759, 798, 839, 880, 923, 966, 1011, 1056, 1103, 1150, 1199, 1248, 1299, 1350, 1403, 1456

Offset: 0

Author

Reinhard Zumkeller, Feb 27 2006

Keywords

• nonn
• easy

Comments

From Andrew Rupinski, Nov 30 2009: (Start)

For n > 0, a(n) appears to be the set such that binomial(2*a(n),r) - binomial(2*a(n),r-2) = binomial(2*a(n),s) - binomial(2*a(n),s-2) for some r != s.

As a consequence of the Weyl Dimension Formula and the above comment, a(n) also appears to be the indices k such that the Symplectic Group Sp(k) has two fundamental irreducible representations of the same dimension. (End)

Examples

			a(n) = A000982(n) + A005843(n).
From _Andrew Rupinski_, Nov 30 2009: (Start)
a(1) = 3 and binomial(6,3)-binomial(6,1) = binomial(6,2)-binomial(6,0).
a(1) = 3 and the fundamental representations of Sp(3) are of dimensions 6, 14 and 14. a(2) = 6 and the fundamental representations of Sp(6) are of dimensions 12, 65, 208, 429, 572, and 429. (End)
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• John Tyler Rascoe, Illustration of the first 10 terms.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A116939.

Cf. A081352, A236283.

Cf. A028884, A099392.

Cf. A000217, A001859.

Programs

Haskell

import Data.List (elemIndices)
a116940 n = last $ elemIndices n $ takeWhile (<= n + 1) a116939_list
-- Reinhard Zumkeller, Jun 28 2013

Magma

[(2*n*(n+4) -(-1)^n +1)/4: n in [0..55]]; // G. C. Greubel, Jan 26 2020

Maple

seq( (2*(n+2)^2 -(-1)^n -7)/4, n=0..55); # G. C. Greubel, Jan 26 2020

Mathematica

a = {0}; Do[AppendTo[a, If[Count[a, #-1] > #-1, #+1, #-1]] &@ a[[n]], {n, 1500}]; Most@ Values@ Map[Last, PositionIndex@ a] - 1 (* Michael De Vlieger, Dec 07 2016, Version 10 *)
Table[(2*(n+2)^2 -(-1)^n -7)/4, {n,0,55}] (* G. C. Greubel, Jan 26 2020 *)

PARI

vector(56, n, (2*(n+1)^2 +(-1)^n -7)/4) \\ G. C. Greubel, Jan 26 2020

Sage

[(2*n*(n+4) -(-1)^n +1)/4 for n in (0..55)] # G. C. Greubel, Jan 26 2020

Formula

a(0) = 0, a(n+1) = a(n) + 2*floor(n/2) + 3.

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Joerg Arndt, Apr 02 2011

G.f.: x*(3 - x^2)/((1 + x)*(1 - x)^3). - Arkadiusz Wesolowski, Jan 01 2012

a(n) = 2n + ceiling(n^2/2). - Wesley Ivan Hurt, Jun 14 2013

a(n) = (2*n*(n + 4) - (-1)^n + 1)/4. - Bruno Berselli, Jun 14 2013

a(n) = A081352(n) - A236283(n + 1). - Miko Labalan, Dec 04 2016

From Klaus Purath, Jan 26 2020: (Start)

a(n) = binomial(n+2, 2) + floor((n-1)/2).

a(n) = floor(A028884(n)/2) - n.

a(n) = (n+1)^2 - A099392(n+1).

a(2*n)^2 - a(2*n-1)*a(2*n+1) = 3, n > 0.

a(2*n+1)^2 - a(2*n)*a(2*n+2) = (2*n+3)^2. (End)

E.g.f.: (1/2)*(x*(5 + x)*cosh(x) + (1 + 5*x + x^2)*sinh(x)). - Stefano Spezia, Jan 26 2020

a(n) = A000217(2*n) - 2*A001859(n-1) for n>0. - John Tyler Rascoe, Jul 31 2022

Sum_{n>=1} 1/a(n) = 11/8 + tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)). - Amiram Eldar, Sep 16 2022