A257639 a(n) is the minimal position at which the maximal value of row n appears in row n of triangle A008289.

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 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, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset

1,5

Comments

Except rows 3, 4, 10, 11, 21 of triangle A008289, all the other rows up to row number 10^6 contain a single maximal value.

Conjecture: for n >= 22 there is a unique maximal value in row n of triangle A008289.

Links

Gheorghe Coserea, Table of n, a(n) for n = 1..50000

Paul Erdős, On some asymptotic formulas in the theory of partitions, Bull. Amer. Math. Soc. 52 (1946), no. 2, 185--188.

B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97.

G. Szekeres, Some asymptotic formulas in the theory of partitions (II), Quart. J. Math. Oxford (2), 4(1953), 96-111.

Formula

a(n) = min argmax(k->Q(n,k), k=1..m), that is a(n) = min{k, Q(n,k) = max{Q(n,p), p=1..m}}, where m = A003056(n) and Q(n,k) is defined by A008289.

a(n) ~ K*sqrt(n) + O(1), where K = 2*sqrt(3)*log(2)/Pi = 0.76430413884... (A131266).

Example

For n=9, a(9)=2 because A003056(9)=3 and max{Q(9,p), p=1..3}=4 and Q(9,2)=4.

Prog

(PARI)
Q(N) = {
  my(q = vector(N)); q[1] = [1, 0, 0, 0];
  for (n = 2, N,
    my(m = (sqrtint(8*n+1) - 1)\2);
    q[n] = vector((1 + (m>>2)) << 2); q[n][1] = 1;
    for (k = 2, m, q[n][k] = q[n-k][k] + q[n-k][k-1]));
  return(q);
};
seq(N) = {
  my(a = vector(N), q = Q(N), vmx = apply(vecmax, q));
  for (n = 1, N, a[n] = vecmin(select(v->v==vmx[n], q[n], 1)));
  a;
};
seq(86) \\ updated by Gheorghe Coserea, Jun 02 2018

Crossrefs

Cf. A003056, A008289, A030699, A131266.

Sequence in context: A372406 A329195 A204164 * A180447 A295866 A115338

Adjacent sequences: A257636 A257637 A257638 * A257640 A257641 A257642

Keyword

nonn

Author

Gheorghe Coserea, Nov 04 2015

Status

approved