A293497 Triangular array read by rows: row n >= 1 is the list of integers from 0 to 2n-1.

0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8

Offset

0,5

Comments

a(n) = the least nonnegative n - 2 * T, where T is a triangular number.

a(n) = the least nonnegative n - k * (k + 1), where k is a nonnegative integer.

This sequence shares several properties with A053186 (square excess of n):

- same recursion formula a(n) = f(n,1) with f(n,m) = if n < m then n, otherwise f(n-m,m+2);

- same formula pattern a(n) = n - g(floor(f(n))), with f and g each other's inverse function: f(x)=sqrt(x) and g(x)=x^2 in the case of A053186, f(x)=(sqrt(1+4x)-1)/2 and g(x)=x(x+1) in the case of this sequence;

- similar graphic representation (arithmetically increasing sawtooth shape);

- both sequences appear to intertwine into A288969.

Odd-indexed rows of A002262. - Omar E. Pol, Oct 10 2017

Links

Table of n, a(n) for n=0..80.

Formula

a(n) = n - g(floor(f(n))), with f(x) = (sqrt(1+4x)-1)/2 and g(x) = x(x+1).

a(n) = f(n,1) with f(n,m) = if n < m then n, otherwise f(n-m,m+2).

a(n) = t - t^2 + n, where t = floor(sqrt(n+1) + 1/2). - Ridouane Oudra, May 03 2019

Example

Triangle begins:
0, 1;
0, 1, 2, 3;
0, 1, 2, 3, 4, 5;
0, 1, 2, 3, 4, 5, 6, 7;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
...

Mathematica

FOdd[x_] := x*(x + 1)
InvFOdd[x_] := (Sqrt[1 + 4 x] - 1)/2
GOdd[n_] := n - FOdd[Floor[InvFOdd[n]]]
Table[GOdd[n], {n, 0, 80}]

Crossrefs

Cf. A002262, A053186, A288969.

Sequence in context: A242305 A049263 A014588 * A278164 A328480 A279681

Adjacent sequences: A293494 A293495 A293496 * A293498 A293499 A293500

Keyword

nonn,tabf

Author

Luc Rousseau, Oct 10 2017

Status

approved