login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A154141 Indices k such that 8 plus the k-th triangular number is a perfect square. 4
1, 7, 16, 46, 97, 271, 568, 1582, 3313, 9223, 19312, 53758, 112561, 313327, 656056, 1826206, 3823777, 10643911, 22286608, 62037262, 129895873, 361579663, 757088632, 2107440718, 4412635921, 12283064647, 25718726896, 71590947166, 149899725457, 417262618351 (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 = 8. - Ctibor O. Zizka, Nov 10 2009

LINKS

Table of n, a(n) for n=1..30.

F. T. Adams-Watters, SeqFan Discussion, Oct 2009.

FORMULA

{k: 8+k*(k+1)/2 in A000290}

Conjectures: (Start)

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

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

G.f.: ( 4 + 1/(x-1) - 3/(x^2+2*x-1) + (6+15*x)/(x^2-2*x-1) )/2. (End)

a(1..4) = (1,7,16,46); a(n) = 6*a(n-2) - a(n-4) + 2, for n>4. - Ctibor O. Zizka, Nov 10 2009

EXAMPLE

1*(1+1)/2+8 = 3^2. 7*(7+1)/2+8 = 6^2. 16*(16+1)/2+8 = 12^2. 46*(46+1)/2+8 = 33^2.

MATHEMATICA

Join[{1}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 8 &]] (* G. C. Greubel, Sep 03 2016 *)

Select[Range[0, 2 10^7], IntegerQ[Sqrt[8 + # (# + 1) / 2]] &] (* Vincenzo Librandi, Sep 03 2016 *)

PROG

(PARI) isok(n) = issquare(8 + n*(n+1)/2); \\ Michel Marcus, Sep 03 2016

(Magma) [1] cat [n: n in [0..2*10^7] | (Ceiling(Sqrt(n*(n+ 1)/2)))^2-n*(n+1)/2 eq 8]; /* or */ [n: n in [0..2*10^7] | IsSquare(8+n*(n+1)/2)]; // Vincenzo Librandi, Sep 03 2016

CROSSREFS

Cf. A000217, A000290, A006451.

Sequence in context: A066009 A304013 A037241 * A173661 A152530 A065099

Adjacent sequences: A154138 A154139 A154140 * A154142 A154143 A154144

KEYWORD

nonn

AUTHOR

R. J. Mathar, Oct 18 2009

EXTENSIONS

a(17)-a(24) from Donovan Johnson, Nov 01 2010

a(25)-a(30) from Lars Blomberg, Jul 07 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 05:41 EST 2022. Contains 358649 sequences. (Running on oeis4.)