A034297 Number of ordered positive integer solutions (m_1, m_2, ..., m_k) (for some k) to Sum_{i=1..k} m_i=n with |m_i-m_{i-1}| <= 1 for i = 2 ... k.

1, 1, 2, 4, 6, 11, 17, 29, 47, 78, 130, 215, 357, 595, 990, 1651, 2748, 4584, 7643, 12744, 21256, 35451, 59133, 98636, 164531, 274463, 457837, 763746, 1274060, 2125356, 3545491, 5914545, 9866602, 16459421, 27457549, 45804648, 76411272, 127469285, 212644336

Offset: 0

Author

Erich Friedman

Keywords

• nonn

Comments

Compositions of n where successive parts differ by at most 1, see example. [Joerg Arndt, Dec 10 2012]

Examples

			From _Joerg Arndt_, Dec 10 2012: (Start)
The a(6) = 17 such compositions of 6 are
[ #]     composition
[ 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]    [ 1 2 3 ]
[10]    [ 2 1 1 1 1 ]
[11]    [ 2 1 1 2 ]
[12]    [ 2 1 2 1 ]
[13]    [ 2 2 1 1 ]
[14]    [ 2 2 2 ]
[15]    [ 3 2 1 ]
[16]    [ 3 3 ]
[17]    [ 6 ]
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4500
• Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018.

Crossrefs

Cf. A003116, A034296.

Column k=1 of A214246, A214248.

Row sums of A309939.

Programs

Maple

b:= proc(n, i) option remember;
      `if`(n=i, 1, `if`(n<0 or i<1, 0, add(b(n-i, i+j), j=-1..1)))
    end:
a:= n-> add(b(n, k), k=0..n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 06 2012

Mathematica

b[n_, i_] := b[n, i] = If[n == i, 1, If[n<0 || i<1, 0, Sum[b[n-i, i+j], {j, -1, 1}] ]]; a[n_] := Sum[b[n, k], {k, 1, n}]; Array[a, 50] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

PARI

N=70;  nil=-1;
T = matrix(N, N, i, j, nil);
doIt(last, left) = my(c); c = T[last, left]; if (c == nil, c = 0; for (i = max(1, last - 1), last + 1, c += b(i, left - i)); T[last, left] = c); c;
b(last, left) = if (left == 0, return(1)); if (left < 0, return(0)); doIt(last, left);
a(n) = sum (i = 1, n, b(i, n - i));
vector(N,n,a(n))  \\ David Wasserman, Feb 02 2006

Python

from sympy.core.cache import cacheit
@cacheit
def b(n, i): return 1 if n==i else 0 if n<0 or i<1 else sum(b(n - i, i + j) for j in range(-1, 2))
def a(n): return sum(b(n, k) for k in range(n + 1))
print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 14 2017, after Maple code

Formula

a(n) ~ c * d^n, where d = 1.668202067018461116361070469945501401879811945303435230637248..., c = 0.762436680050402638439806786781869262562176911054246754543346... . - Vaclav Kotesovec, Sep 02 2014

Extensions

More terms from David Wasserman, Feb 02 2006

a(0)=1 prepended by Alois P. Heinz, Aug 14 2017