A308663 Partial sums of A097805.

1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 8, 9, 12, 15, 16, 16, 17, 21, 27, 31, 32, 32, 33, 38, 48, 58, 63, 64, 64, 65, 71, 86, 106, 121, 127, 128, 128, 129, 136, 157, 192, 227, 248, 255, 256, 256, 257, 265, 293, 349, 419, 475, 503, 511, 512

Offset

0,3

Comments

Curtz (1965), page 15, from right to left, gives (F1):

1/2;

1/4, 3/4;

1/8, 4/8, 7/8;

1/16, 5/16, 11/16, 15/16;

... .

Numerators + Denominators = (C) =

3;

5, 7;

9, 12, 15;

17, 21, 27, 31;

... .

This is the current sequence without powers of 2.

The triangle (P) for a(n) is

1;

1, 2;

2, 3, 4;

4, 5, 7, 8;

8, 9, 12, 15, 16;

... .

(C) is the core of (P).

Extension of (F1). (F2) =

0/1;

0/1, 1/1;

0/2, 1/2, 2/2;

0/4, 1/4, 3/4, 4/4;

0/8, 1/8, 4/8, 7/8, 8/8;

... .

(Mentioned, without 0's, op. cit., page 16.)

a(n) = Numerators + Denominators.

Row sums of triangle (P): A084858(n).

From right to left, with alternating signs: 1, 1, 3, 2, 12, 8, 48, 32, ..., see A098646.

For triangle (C), row sums give A167667(n+1).

From right to left, with alternating signs: A098646(n).

Rank of A016116(n): 0 together with A117142.

Links

Table of n, a(n) for n=0..54.

Paul Curtz, Accélération de la convergence de certaines séries alternées à l'aide des fonctions de sommation, Thèse de 3ème Cycle d'Analyse Numérique, Faculté des Sciences de l'Université de Paris, 4 mai 1965.

Formula

T(n,k) = ceiling(2^(n-1)) + Sum_{j=0..k-1} binomial(n-1,j). - Alois P. Heinz, Jun 15 2019

a(n+1) = a(n) + A097805(n+1) for n >= 0.

Crossrefs

Cf. A097805.

Sequence in context: A153155 A225085 A134310 * A305713 A259201 A342518

Adjacent sequences: A308660 A308661 A308662 * A308664 A308665 A308666

Keyword

nonn,tabl

Author

Paul Curtz, Jun 15 2019

Extensions

Edited by N. J. A. Sloane, Sep 15 2019

Status

approved