A057944 Largest triangular number less than or equal to n; write m-th triangular number m+1 times.

0, 1, 1, 3, 3, 3, 6, 6, 6, 6, 10, 10, 10, 10, 10, 15, 15, 15, 15, 15, 15, 21, 21, 21, 21, 21, 21, 21, 28, 28, 28, 28, 28, 28, 28, 28, 36, 36, 36, 36, 36, 36, 36, 36, 36, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 66, 66, 66, 66, 66, 66

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
(Python)
from math import comb, isqrt
def A057944(n): return comb((m:=isqrt(k:=n+1<<1))+(k>m*(m+1)), 2) # Chai Wah Wu, Nov 09 2024

Crossrefs

Cf. A000217, A003056, A056944, A057945, A127739.

Sequence in context: A108581 A073080 A171601 * A281258 A080607 A013322

Adjacent sequences: A057941 A057942 A057943 * A057945 A057946 A057947

Keyword

easy,nonn,tabl,changed

Author

Henry Bottomley, Oct 05 2000

Extensions

Keyword tabl added by Reinhard Zumkeller, Feb 03 2012

Status

approved