A116940 - OEIS

%I #61 Sep 16 2022 03:57:50

%S 0,3,6,11,16,23,30,39,48,59,70,83,96,111,126,143,160,179,198,219,240,

%T 263,286,311,336,363,390,419,448,479,510,543,576,611,646,683,720,759,

%U 798,839,880,923,966,1011,1056,1103,1150,1199,1248,1299,1350,1403,1456

%N Greatest m such that A116939(m) = n.

%C From _Andrew Rupinski_, Nov 30 2009: (Start)

%C 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.

%C 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)

%H Reinhard Zumkeller, <a href="/A116940/b116940.txt">Table of n, a(n) for n = 0..1000</a>

%H John Tyler Rascoe, <a href="/A116940/a116940.jpg">Illustration of the first 10 terms</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

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

%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - _Joerg Arndt_, Apr 02 2011

%F G.f.: x*(3 - x^2)/((1 + x)*(1 - x)^3). - _Arkadiusz Wesolowski_, Jan 01 2012

%F a(n) = 2n + ceiling(n^2/2). - _Wesley Ivan Hurt_, Jun 14 2013

%F a(n) = (2*n*(n + 4) - (-1)^n + 1)/4. - _Bruno Berselli_, Jun 14 2013

%F a(n) = A081352(n) - A236283(n + 1). - _Miko Labalan_, Dec 04 2016

%F From _Klaus Purath_, Jan 26 2020: (Start)

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

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

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

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

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

%F E.g.f.: (1/2)*(x*(5 + x)*cosh(x) + (1 + 5*x + x^2)*sinh(x)). - _Stefano Spezia_, Jan 26 2020

%F a(n) = A000217(2*n) - 2*A001859(n-1) for n>0. - _John Tyler Rascoe_, Jul 31 2022

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

%e a(n) = A000982(n) + A005843(n).

%e From _Andrew Rupinski_, Nov 30 2009: (Start)

%e a(1) = 3 and binomial(6,3)-binomial(6,1) = binomial(6,2)-binomial(6,0).

%e 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)

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

%t 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 *)

%t Table[(2*(n+2)^2 -(-1)^n -7)/4, {n,0,55}] (* _G. C. Greubel_, Jan 26 2020 *)

%o (Haskell)

%o import Data.List (elemIndices)

%o a116940 n = last $ elemIndices n $ takeWhile (<= n + 1) a116939_list

%o -- _Reinhard Zumkeller_, Jun 28 2013

%o (PARI) vector(56, n, (2*(n+1)^2 +(-1)^n -7)/4) \\ _G. C. Greubel_, Jan 26 2020

%o (Magma) [(2*n*(n+4) -(-1)^n +1)/4: n in [0..55]]; // _G. C. Greubel_, Jan 26 2020

%o (Sage) [(2*n*(n+4) -(-1)^n +1)/4 for n in (0..55)] # _G. C. Greubel_, Jan 26 2020

%Y Cf. A116939.

%Y Cf. A081352, A236283.

%Y Cf. A028884, A099392.

%Y Cf. A000217, A001859.

%K nonn,easy

%O 0,2

%A _Reinhard Zumkeller_, Feb 27 2006