A144065 - OEIS

%I #45 Feb 23 2024 13:25:53

%S 0,1,4,6,11,14,21,25,34,39,50,56,69,76,91,99,116,125,144,154,175,186,

%T 209,221,246,259,286,300,329,344,375,391,424,441,476,494,531,550,589,

%U 609,650,671,714,736,781,804,851,875,924,949,1000,1026,1079,1106,1161,1189,1246,1275,1334,1364,1425

%N Values of k such that the expression sqrt(4!*(k+1) + 1) yields an integer.

%C Integers of form m*(m+5)/6 (nonnegative values of m are listed in A032766). - _Bruno Berselli_, Jul 18 2016

%H G. C. Greubel, <a href="/A144065/b144065.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6). - _Jaume Oliver Lafont_, Jan 21 2009

%F a(n) = (-3 + 2*(-1)^n*n + 3*(-1)^n + 6*n^2 + 18*n)/16. - _Alexander R. Povolotsky_, Jan 27 2009

%F a(n) = A001318(n+1) - 1. - _Peter Bala_, Mar 22 2009

%F G.f.: x*(1 + 3*x - x^3)/((1 + x)^2*(1 - x)^3). - _Jaume Oliver Lafont_, Aug 31 2009

%F a(n) = Sum_{i=1..n+3} numerator(i/2) - denominator(i/2). - _Wesley Ivan Hurt_, Feb 26 2017

%F Sum_{n>=1} 1/a(n) = (93+10*sqrt(3)*Pi)/75. - _Amiram Eldar_, Sep 22 2022

%p seq(seq(((24*a+b)^2-25)/24, b=[5,7,11,13,17,19,23,25]),a=0..10); # _Robert Israel_, Jul 15 2016

%t LinearRecurrence[{0,3,0,-3,0,1}, {0, 1, 4, 6, 11, 14}, 50] (* _G. C. Greubel_, Jul 15 2016 *)

%t Select[Range[0,1500],IntegerQ[Sqrt[4!(#+1)+1]]&] (* _Harvey P. Dale_, Sep 20 2019 *)

%o (PARI) j=[];for(n=0, 300,if((floor(sqrt(4!*(n+1) + 1))) == ceil(sqrt(4!*(n+1) + 1)), j=concat(j, n))); j

%o (Magma) [(-3+2*(-1)^n*n+3*(-1)^n+6*n^2+18*n)/16: n in [0..60]]; // _Vincenzo Librandi_, Jul 16 2016

%Y Cf. A007310, A032766, A038179, A075889.

%Y Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [_Bruno Berselli_, Jul 25 2016]

%K nonn,easy

%O 0,3

%A _Alexander R. Povolotsky_, Sep 09 2008