

A057944


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


17



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
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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*(t1), 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

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


AUTHOR

Henry Bottomley, Oct 05 2000


EXTENSIONS

Keyword tabl added by Reinhard Zumkeller, Feb 03 2012


STATUS

approved



