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A048888 a(n) = Sum{T(m,n+1-m): m=1,2,...,n}, array T as in A048887. 16
0, 1, 2, 4, 7, 13, 23, 42, 76, 139, 255, 471, 873, 1627, 3044, 5718, 10779, 20387, 38673, 73561, 140267, 268065, 513349, 984910, 1892874, 3643569, 7023561, 13557019, 26200181, 50691977, 98182665, 190353369, 369393465, 717457655 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

From Marc LeBrun, Dec 12 2001: (Start)

Define a "numbral arithmetic" by replacing addition with binary bitwise inclusive-OR (so that [3] + [5] = [7] etc.) and multiplication becomes shift-&-OR instead of shift-&-add (so that [3] * [3] = [7] etc.). [d] divides [n] means there exists an [e] with [d] * [e] = [n]. For example the six divisors of [14] are [1], [2], [3], [6], [7] and [14]. Then it appears that this sequence gives the number of proper divisors of [2^n-1]. Conjecture confirmed by Richard Schroeppel, Dec 14 2001. (End)

The number of "prime endofunctions" on n points, meaning the cardinality of the subset of the A001372(n) mappings (or mapping patterns) up to isomorphism from n (unlabeled) points to themselves (endofunctions) which are neither the sum of prime endofunctions (i.e., whose disjoint connected components are prime endofunctions) nor the categorical product of prime endofunctions. The n for which a(n) is prime (n such that the number of prime endofunctions on n points is itself prime) are 2, 4, 5, 6, 9, 13, 19, ... - Jonathan Vos Post, Nov 19 2006

For n>=1, compositions p(1)+p(2)+...+p(m)=n such that p(k)<=p(1)+1, see example. - Joerg Arndt, Dec 28 2012

LINKS

Table of n, a(n) for n=0..33.

A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.

FORMULA

G.f.: Sum_{k>0} x^k*(1-x^k)/(1-2*x+x^(k+1)). - Vladeta Jovovic, Feb 25 2003

a(m) = Sum_{ n=2..m+1 } Fn(m) where Fn is a Fibonacci n-step number (Fibonacci, tetranacci, etc.) indexed as in A000045, A000073, A000078. - Gerald McGarvey, Sep 25 2004

EXAMPLE

From Joerg Arndt, Dec 28 2012: (Start)

There are a(6)=23 compositions p(1)+p(2)+...+p(m)=6 such that p(k)<=p(1)+1:

[ 1]  [ 1 1 1 1 1 1 ]

[ 2]  [ 1 1 1 1 2 ]

[ 3]  [ 1 1 1 2 1 ]

[ 4]  [ 1 1 2 1 1 ]

[ 5]  [ 1 1 2 2 ]

[ 6]  [ 1 2 1 1 1 ]

[ 7]  [ 1 2 1 2 ]

[ 8]  [ 1 2 2 1 ]

[ 9]  [ 2 1 1 1 1 ]

[10]  [ 2 1 1 2 ]

[11]  [ 2 1 2 1 ]

[12]  [ 2 1 3 ]

[13]  [ 2 2 1 1 ]

[14]  [ 2 2 2 ]

[15]  [ 2 3 1 ]

[16]  [ 3 1 1 1 ]

[17]  [ 3 1 2 ]

[18]  [ 3 2 1 ]

[19]  [ 3 3 ]

[20]  [ 4 1 1 ]

[21]  [ 4 2 ]

[22]  [ 5 1 ]

[23]  [ 6 ]

(End)

PROG

(PARI)

N = 66;  x = 'x + O('x^N);

gf = sum(n=0, N,  (1-x^n)*x^n/(1-2*x+x^(n+1)) ) + 'c0;

v = Vec(gf);  v[1]-='c0;  v

/* Joerg Arndt, Apr 14 2013 */

CROSSREFS

Cf. A007059.

Cf. A000312, A001372, A002861, A006961, A001373, A054050, A054745, A125024.

Sequence in context: A165648 A078038 A190502 * A280027 A026724 A054163

Adjacent sequences:  A048885 A048886 A048887 * A048889 A048890 A048891

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified March 26 10:42 EDT 2017. Contains 284111 sequences.