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 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. 15
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compositions of n where successive parts differ by at most 1, see example. [Joerg Arndt, Dec 10 2012] 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. FORMULA a(n) ~ c * d^n, where d = 1.668202067018461116361070469945501401879811945303435230637248..., c = 0.762436680050402638439806786781869262562176911054246754543346... . - Vaclav Kotesovec, Sep 02 2014 EXAMPLE 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) 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 *) PROG (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 xrange(-1, 2)]) def a(n): return sum([b(n, k) for k in xrange(n + 1)]) print map(a, xrange(51)) # Indranil Ghosh, Aug 14 2017, after Maple code CROSSREFS Cf. A003116, A034296. Column k=1 of A214246, A214248. Row sums of A309939. Sequence in context: A115315 A004698 A014217 * A326495 A026636 A026658 Adjacent sequences:  A034294 A034295 A034296 * A034298 A034299 A034300 KEYWORD nonn AUTHOR EXTENSIONS More terms from David Wasserman, Feb 02 2006 a(0)=1 prepended by Alois P. Heinz, Aug 14 2017 STATUS approved

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Last modified October 20 10:45 EDT 2019. Contains 328257 sequences. (Running on oeis4.)