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

0,3

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).

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

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 *)

Prog

(PARI) j=[]; for(n=0, 300, if((floor(sqrt(4!*(n+1) + 1))) == ceil(sqrt(4!*(n+1) + 1)), j=concat(j, n))); j
(Magma) [(-3+2*(-1)^n*n+3*(-1)^n+6*n^2+18*n)/16: n in [0..60]]; // Vincenzo Librandi, Jul 16 2016

Crossrefs

Cf. A007310, A032766, A038179, A075889.

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

Sequence in context: A153357 A310591 A002732 * A343098 A036831 A084263

Adjacent sequences: A144062 A144063 A144064 * A144066 A144067 A144068

Keyword

nonn,easy

Author

Alexander R. Povolotsky, Sep 09 2008

Status

approved