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A154139 Indices k such that 4 plus the k-th triangular number is a perfect square. 4
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 (list; graph; refs; listen; history; text; internal format)
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 *)

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 * A187998 A177181 A126110

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

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Last modified March 26 10:42 EDT 2017. Contains 284111 sequences.