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A238870 Number of compositions of n with c(1) = 1, c(i+1) - c(i) <= 1, and c(i+1) - c(i) != 0. 3
1, 1, 0, 1, 1, 0, 2, 2, 1, 4, 4, 4, 9, 10, 11, 21, 25, 30, 51, 62, 80, 125, 157, 208, 309, 399, 536, 772, 1013, 1373, 1938, 2574, 3503, 4882, 6540, 8918, 12329, 16611, 22672, 31183, 42182, 57588, 78952, 107092, 146202, 200037, 271831, 371057, 507053, 689885, 941558, 1285655, 1750672, 2388951, 3260459, 4442179, 6060948 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Number of fountains of n coins with at most two successive coins on the same level.

LINKS

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

a(n) ~ c / r^n, where r = 0.733216317061133379740342579187365700397652443391231594... and c = 0.172010618097928709454463097802313209201440229976513439... . - Vaclav Kotesovec, Feb 17 2017

EXAMPLE

The a(10) = 4 such compositions are:

:

:   1:  [ 1 2 1 2 1 2 1 ]  (composition)

:

:  o o o

: ooooooo   (rendering as composition)

:

:     O   O   O

:    O O O O O O O  (rendering as fountain of coins)

:

:

:   2:  [ 1 2 1 2 3 1 ]

:

:     o

:  o oo

: oooooo

:

:           O

:      O   O O

:     O O O O O O

:

:

:   3:  [ 1 2 3 1 2 1 ]

:

:   o

:  oo o

: oooooo

:

:       O

:      O O   O

:     O O O O O O

:

:

:   4:  [ 1 2 3 4 ]

:

:    o

:   oo

:  ooo

: oooo

:

:         O

:        O O

:       O O O

:      O O O O

:

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, add(

      `if`(i=j, 0, b(n-j, j)), j=1..min(n, i+1)))

    end:

a:= n-> b(n, 0):

seq(a(n), n=0..60);  # Alois P. Heinz, Mar 11 2014

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[If[i == j, 0, b[n-j, j]], {j, 1, Min[n, i+1]}]];

a[n_] := b[n, 0];

a /@ Range[0, 60] (* Jean-Fran├žois Alcover, Nov 07 2020, after Alois P. Heinz *)

PROG

(Sage) # translation of the Maple program by Alois P. Heinz

@CachedFunction

def F(n, i):

    if n == 0: return 1

    return sum( (i!=j) * F(n-j, j) for j in [1..min(n, i+1)] ) # A238870

#    return sum( F(n-j, j) for j in [1, min(n, i+1)] ) # A005169

def a(n): return F(n, 0)

print([a(n) for n in [0..50]])

# Joerg Arndt, Mar 20 2014

CROSSREFS

Cf. A005169 (fountains of coins), A001524 (weakly unimodal fountains of coins).

Cf. A186085 (1-dimensional sand piles), A227310 (rough sand piles).

Cf. A023361 (fountains of coins with all valleys at lowest level).

Sequence in context: A165038 A305191 A261357 * A213946 A145036 A341146

Adjacent sequences:  A238867 A238868 A238869 * A238871 A238872 A238873

KEYWORD

nonn

AUTHOR

Joerg Arndt, Mar 09 2014

STATUS

approved

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Last modified August 3 20:28 EDT 2021. Contains 346441 sequences. (Running on oeis4.)