|
|
A154144
|
|
Indices k such that 13 plus the k-th triangular number is a perfect square.
|
|
3
|
|
|
2, 8, 23, 53, 138, 312, 807, 1821, 4706, 10616, 27431, 61877, 159882, 360648, 931863, 2102013, 5431298, 12251432, 31655927, 71406581, 184504266, 416188056, 1075369671, 2425721757, 6267713762, 14138142488, 36530912903, 82403133173, 212917763658, 480280656552
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
LINKS
|
|
|
FORMULA
|
Conjectures: (Start)
a(n) = +a(n-1) + 6*a(n-2) - 6*a(n-3) - a(n-4) + a(n-5).
G.f.: x*(2 +6*x +3*x^2 -6*x^3 -3*x^4)/((1-x) * (x^2-2*x-1) * (x^2+2*x-1))
G.f.: ( 6 + (-3-2*x)/(x^2+2*x-1) + 1/(x-1) + (8+19*x)/(x^2-2*x-1) )/2 . (End)
a(1..4) = (2,8,23,53); a(n) = 6*a(n-2) - a(n-4) + 2, for n>2. - Ctibor O. Zizka, Nov 10 2009
|
|
EXAMPLE
|
2*(2+1)/2+13 = 4^2. 8*(8+1)/2+13 = 7^2. 23*(23+1)/2+13 = 17^2. 53*(53+1)/2+13 = 38^2.
|
|
MATHEMATICA
|
With[{nn=25000}, Transpose[Select[Thread[{Range[nn], Accumulate[ Range[nn]]}], IntegerQ[Sqrt[#[[2]]+13]]&]][[1]]] (* Harvey P. Dale, Jan 13 2012 *)
Join[{2, 8}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 13 &]] (* G. C. Greubel, Sep 03 2016 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|