A307707 Lexicographically earliest sequence of nonnegative integers in which, for all k >= 0, there are exactly k pairs of consecutive terms whose sum is k.

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Apr 23 2019

Keywords

• nonn
• look

Comments

The old definition was "Lexicographically earliest sequence starting with a(1) = 0 such that a(n) is the number of pairs of contiguous terms whose sum is a(n)".

From Paul Curtz, Apr 27 2019: This can be written as a triangle:

0

1 1

1 2 1

2 2 2 2

2 3 2 3 2

3 3 3 3 3 3

3 4 3 4 3 4 3

...

Links

• Jean-Marc Falcoz, Table of n, a(n) for n = 1..11326

Crossrefs

Cf. A002024.

Cf. also A007590, A057353, A106466 and A238410.

For other versions see A307720 and A378117.

Programs

Mathematica

m = 107; a[1]=0;
a24[n_] := Ceiling[(Sqrt[8n+1]-1)/2];
Array[a, m] /. Solve[Table[a[n] + a[n+1] == a24[n], {n, 1, m-1}]][[1]] (* Jean-François Alcover, Jun 02 2019, after Rémy Sigrist's formula *)

PARI

v=0; rem=wanted=1; for (n=1, 107, print1 (v", "); v=wanted-v; if (rem--==0, rem=wanted++)) \\ Rémy Sigrist, Apr 23 2019

Formula

a(n) + a(n+1) = A002024(n). - Rémy Sigrist, Apr 24 2019

Let t_m = m*(m+1)/2. Write n = t_m - i with m >= 1 and 0 <= i < m. Then a(n) = m/2 if m is even, or if m is odd, a(n) = (m-1)/2 + (i-1 mod 2). - N. J. A. Sloane, Nov 16 2024

Extensions

Definition clarified by Rémy Sigrist and N. J. A. Sloane, Nov 17 2024