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

1,2

Comments

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

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).

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000217, A000290, A006451.

Sequence in context: A038263 A004989 A147355 * A330983 A299914 A187998

Adjacent sequences: A154136 A154137 A154138 * A154140 A154141 A154142

Keyword

nonn,easy

Author

R. J. Mathar, Oct 18 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

Status

approved