A144065 Values of k such that the expression sqrt(4!*(k+1) + 1) yields an integer.

0, 1, 4, 6, 11, 14, 21, 25, 34, 39, 50, 56, 69, 76, 91, 99, 116, 125, 144, 154, 175, 186, 209, 221, 246, 259, 286, 300, 329, 344, 375, 391, 424, 441, 476, 494, 531, 550, 589, 609, 650, 671, 714, 736, 781, 804, 851, 875, 924, 949, 1000, 1026, 1079, 1106, 1161, 1189, 1246, 1275, 1334, 1364, 1425

Offset: 0

Author

Alexander R. Povolotsky, Sep 09 2008

Keywords

• nonn
• easy

Comments

Integers of form m*(m+5)/6 (nonnegative values of m are listed in A032766). - Bruno Berselli, Jul 18 2016

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A007310, A032766, A038179, A075889.

Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [Bruno Berselli, Jul 25 2016]

Programs

Magma

[(-3+2*(-1)^n*n+3*(-1)^n+6*n^2+18*n)/16: n in [0..60]]; // Vincenzo Librandi, Jul 16 2016

Maple

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

Mathematica

LinearRecurrence[{0,3,0,-3,0,1}, {0, 1, 4, 6, 11, 14}, 50] (* G. C. Greubel, Jul 15 2016 *)
Select[Range[0,1500],IntegerQ[Sqrt[4!(#+1)+1]]&] (* Harvey P. Dale, Sep 20 2019 *)

PARI

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

Formula

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6). - Jaume Oliver Lafont, Jan 21 2009

a(n) = (-3 + 2*(-1)^n*n + 3*(-1)^n + 6*n^2 + 18*n)/16. - Alexander R. Povolotsky, Jan 27 2009

a(n) = A001318(n+1) - 1. - Peter Bala, Mar 22 2009

G.f.: x*(1 + 3*x - x^3)/((1 + x)^2*(1 - x)^3). - Jaume Oliver Lafont, Aug 31 2009

a(n) = Sum_{i=1..n+3} numerator(i/2) - denominator(i/2). - Wesley Ivan Hurt, Feb 26 2017

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