A154139 Indices k such that 4 plus the k-th triangular number is a perfect square.

0, 6, 9, 39, 56, 230, 329, 1343, 1920, 7830, 11193, 45639, 65240, 266006, 380249, 1550399, 2216256, 9036390, 12917289, 52667943, 75287480, 306971270, 438807593, 1789159679, 2557558080, 10427986806, 14906540889, 60778761159, 86881687256

Offset: 1

Author

R. J. Mathar, Oct 18 2009

Keywords

• nonn
• easy

Comments

Also numbers n such that (ceiling(sqrt(n*(n+1)/2)))^2 - n*(n+1)/2 = 4. - Ctibor O. Zizka, Nov 10 2009

Examples

			0*(0+1)/2+4 = 2^2. 6*(6+1)/2+4 = 5^2. 9*(9+1)/2+4 = 7^2. 39*(39+1)/2+4 = 28^2.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• F. T. Adams-Watters, SeqFan Discussion, Oct 2009
• Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-1,1).

Crossrefs

Cf. A000217, A000290, A006451.

Programs

Magma

I:=[0, 6, 9, 39,56]; [n le 5 select I[n] else Self(n-1)+6*Self(n-2)-6*Self(n-3)-Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 11 2012

Maple

a := proc (n) if type(sqrt(4+(1/2)*n*(n+1)), integer) = true then n else end if end proc: seq(a(n), n = 0 .. 10^7); # Emeric Deutsch, Oct 31 2009

Mathematica

LinearRecurrence[{1, 6, -6, -1, 1}, {0, 6, 9, 39, 56}, 40] (* Vincenzo Librandi, Dec 11 2012 *)
Join[{0}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 4 &] ] (* G. C. Greubel, Sep 03 2016 *)
Join[{0},Position[Accumulate[Range[66000]]+4,?(IntegerQ[Sqrt[#]]&)]//Flatten] (* The program generates the first 13 terms of the sequence. *) (* _Harvey P. Dale, Feb 18 2023 *)

Formula

a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - a(n-4) + a(n-5).

G.f.: x^2*(6 +3*x -6*x^2 -x^3)/((1-x)*(x^2-2*x-1)*(x^2+2*x-1)) = 1 + 1/2*(4+11*x)/(x^2-2*x-1) + 1/2/(x-1) + 1/2*(-3+2*x)/(x^2+2*x-1).

For n>4, a(n) = 6*a(n-2) - a(n-4) + 2. - Ctibor O. Zizka, Nov 10 2009

Extensions

a(17)-a(18) from Emeric Deutsch, Oct 31 2009

a(19)-a(25) from Donovan Johnson, Nov 01 2010

More terms from Max Alekseyev, Jan 24 2012