A220104 k appears k*(k+1) times.

1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset

1,3

Comments

The current sequence is, loosely, the inverse function of the 2*binomial(n,3) sequence A007290.

A007290 has alternative formulas, thus yielding alternative formulas for the current sequence.

The formula below was inspired by Sum_{i = 0..n-2} (i*(i+1)) = n*(n-1)*(n-2)/3 given in A007290.

By definition: A002378 = run lengths. - Reinhard Zumkeller, Jan 01 2013

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

For c(n) = floor((3*n)^(1/3)), e(n) = n - (c(n)*(c(n)+1)*(c(n)+2))/3, explicit formula is a(n) = c(n) + sgn(abs(e(n)) + e(n)).

a(n) = floor(t + 1/(3*t)), where t = (3*(n - 1))^(1/3), for n > 1. - Ridouane Oudra, Oct 30 2023

Example

For n = 21 the solution is found as the following: c(21) = 3, e(21) = 1, and finally a(21) = 4.

Mathematica

Flatten[Array[Table[#, #(#+1)]&, 6]] (* Paolo Xausa, Dec 10 2023 *)

Prog

(Haskell)
a220104 n = a220104_list !! (n-1)
a220104_list = concatMap (\x -> take (a002378 x) $ repeat x) [1..]
-- Reinhard Zumkeller, Jan 01 2013

Crossrefs

Cf. A002378, A007290.

Sequence in context: A123119 A099396 A126235 * A191228 A340763 A286103

Adjacent sequences: A220101 A220102 A220103 * A220105 A220106 A220107

Keyword

nonn

Author

Can Atilgan and Murat Erşen Berberler (muratersenberberler(AT)gmail.com), Dec 05 2012

Status

approved