OFFSET
0,4
LINKS
Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
FORMULA
a(n) = floor((sqrt(1+8*n)-1)/2)*floor((sqrt(1+8*n)+1)/2)/2 = (trinv(n)*(trinv(n)-1))/2 = A000217(A003056(n)) = n - A002262(n)
a(n) = (1/2)*t*(t-1), where t = floor(sqrt(2*n+1)+1/2) = A002024(n+1). - Ridouane Oudra, Oct 20 2019
Sum_{n>=1} 1/a(n)^2 = 2*Pi^2/3 - 4. - Amiram Eldar, Aug 14 2022
EXAMPLE
a(35) = 28 since 28 and 36 are successive triangular numbers and 28 <= 35 < 36.
MAPLE
A057944 := proc(n)
k := (-1+sqrt(1+8*n))/2 ;
k := floor(k) ;
k*(k+1)/2 ;
end proc; # R. J. Mathar, Nov 05 2011
MATHEMATICA
f[n_] := Block[{a = Floor@ Sqrt[1 + 8 n]}, Floor[(a - 1)/2]*Floor[(a + 1)/2]/2]; Array[f, 72, 0]
t0=0; t1=1; k=1; Table[If[n < t1, t0, k++; t0=t1; t1=t1+k; t0], {n, 0, 72}]
With[{nn=15}, Table[#[[1]], #[[2]]+1]&/@Thread[{Accumulate[Range[ 0, nn]], Range[ 0, nn]}]]//Flatten (* Harvey P. Dale, Mar 01 2020 *)
PROG
(Haskell)
a057944 n = a057944_list !! n -- common flat access
a057944_list = concat a057944_tabl
a057944' n k = a057944_tabl !! n !! k -- access when seen as a triangle
a057944_row n = a057944_tabl !! n
a057944_tabl = zipWith ($) (map replicate [1..]) a000217_list
-- Reinhard Zumkeller, Feb 03 2012
(PARI) a(n)=my(t=(sqrtint(8*n+7)-1)\2); t*(t+1)/2 \\ Charles R Greathouse IV, Jan 26 2013
CROSSREFS
KEYWORD
AUTHOR
Henry Bottomley, Oct 05 2000
EXTENSIONS
Keyword tabl added by Reinhard Zumkeller, Feb 03 2012
STATUS
approved