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A257639 a(n) is the minimal position at which the maximal value of row n appears in row n of triangle A008289. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A134914 A329195 A204164 * A180447 A295866 A115338

Adjacent sequences:  A257636 A257637 A257638 * A257640 A257641 A257642

KEYWORD

nonn

AUTHOR

Gheorghe Coserea, Nov 04 2015

STATUS

approved

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Last modified June 1 07:40 EDT 2020. Contains 334759 sequences. (Running on oeis4.)