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A186085 Number of 1-dimensional sand piles with n grains. 8
1, 1, 1, 1, 2, 3, 5, 8, 13, 22, 36, 60, 100, 166, 277, 461, 769, 1282, 2137, 3565, 5945, 9916, 16540, 27589, 46022, 76769, 128062, 213628, 356366, 594483, 991706, 1654352, 2759777, 4603843, 7680116, 12811951, 21372882, 35654237, 59478406, 99221923, 165522118, 276124217, 460630839 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Number of compositions of n where the first and the last parts are 1 and the absolute difference between consecutive parts is <=1 (smooth compositions).

Such a composition [c1,c2,c3,...] corresponds to a sand-pile with c1(=1) grains in the first positions, c2 in the second, and so on.  Assuming the critical slope is 1 (for the pile to be stable) we obtain the conditions on the compositions.

With the additional requirement of unimodality one gets A001522. [Joerg Arndt, Dec 09 2012]

Dropping the requirement that the first and last parts are 1 gives A034297. Restriction to weakly increasing (or decreasing) sums gives A034296. [Joerg Arndt, Jun 02 2013]

Also the number of compositions of n with first part 1, up-steps of at most 1, and no two consecutive up-steps.  The sand piles are recovered by shifting the rows above the bottom row to the left by one position relative to the next lower row. [Joerg Arndt, Mar 30 2014]

Also fountains of coins (cf. A005169) with no consecutive up-steps. Shift the top rows in the previous comment by half a position. [Joerg Arndt, Mar 30 2014]

LINKS

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

FORMULA

G.f.: 1 + x/(1-x - x^3*B(x)) where B(x) equals the g.f. of the antidiagonal sums of triangle A186084 [Paul D. Hanna].

G.f.: 1 + x/(1-x - x^3/(1-x^2 - x^5/(1-x^3 - x^7/(1-x^4 - x^9/(1 -...))))) (continued fraction).  [Paul D. Hanna].

G.f.: 1/(1 - x/(1-x^3/(1-x^2/(1 - x^3/(1-x^7/(1-x^4/(1 - x^5/(1-x^11/(1-x^6/(1 -...)))))))))) (continued fraction).  [Paul D. Hanna].

The g.f. T(x,y) of triangle A186084 satisfies: T(x,y) = 1/(1 - x*y - x^3*y^2*T(x,x*y)); therefore, the g.f. of this sequence is A(x) = 1 + x*T(x,1). [Paul D. Hanna]

a(n) ~ c/r^n where r = 0.5994477646147968266874606710272382... and c = 0.213259838728143595595398989847345... [Paul D. Hanna]

G.f.: 1 + 1/Q(0), where Q(k)= 1/x^(k+1) - 1 - 1/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 07 2013

G.f.: G(1), where G(k) = 1 + x^k/( 1 - x^k * G(k+1) ) (continued fraction). [Joerg Arndt, Jun 29 2013]

EXAMPLE

The a(7)=8 smooth compositions of 7 are:

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

:

: ooooooo  (rendering of sand pile)

:

:   2:      [ 1 1 1 1 2 1 ]

:

:     o

: oooooo

:

:   3:      [ 1 1 1 2 1 1 ]

:

:    o

: oooooo

:

:   4:      [ 1 1 2 1 1 1 ]

:

:   o

: oooooo

:

:   5:      [ 1 1 2 2 1 ]

:

:   oo

: ooooo

:

:   6:      [ 1 2 1 1 1 1 ]

:

:  o

: oooooo

:

:   7:      [ 1 2 1 2 1 ]

:

:  o o

: ooooo

:

:   8:      [ 1 2 2 1 1 ]

:

:  oo

: ooooo

MAPLE

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

      `if`(n<0 or i<1, 0, add(b(n-i, i+j), j=-1..1)))

    end:

a:= n-> `if`(n=0, 1, b(n-1, 1)):

seq(a(n), n=0..50);  # Alois P. Heinz, Jun 11 2013

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, If[i == 1, 1, 0], If[n<0 || i<1, 0, Sum[b[n-i, i+j], {j, -1, 1}]]]; a[n_] := If[n == 0, 1, b[n-1, 1]]; Table[a[n], {n, 0, 50}] (* Jean-Fran├žois Alcover, Feb 03 2014, after Alois P. Heinz *)

PROG

(PARI) {a(n)=local(Txy=1+x*y); for(i=1, n, Txy=1/(1-x*y-x^3*y^2*subst(Txy, y, x*y+x*O(x^n)))); polcoeff(subst(1+x*Txy, y, 1), n, x)} /* Paul D. Hanna */

(PARI) /* continued fraction for terms up to 460630839: */

Vec(1/ (1-x/ (1-x^3/ (1-x^2/ (1-x^3/ (1-x^7/ (1-x^4/ (1-x^5/ (1-x^11/ (1-x^6/(1-x*O(x^0) ))))))))))) /* Paul D. Hanna */

(PARI) N = 66; x = 'x + O('x^N);

Q(k) = if(k>N, 1, 1/x^(k+1) - 1 - 1/Q(k+1) );

gf = 1 + 1/Q(0);

Vec(gf) /* Joerg Arndt, May 07 2013 */

CROSSREFS

Cf. A186084 (triangle), A005169 (compositions of n with c(1)=1 and c(i+1)<=c(i)+1).

Cf. A186505 (antidiagonal sums of triangle A186084).

Cf. A001522, A001523, A001524.

Sequence in context: A004697 A245271 A206743 * A018151 A227374 A124429

Adjacent sequences:  A186082 A186083 A186084 * A186086 A186087 A186088

KEYWORD

nonn,nice

AUTHOR

Joerg Arndt, Feb 12 2011

STATUS

approved

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Last modified March 22 22:17 EDT 2017. Contains 283901 sequences.