This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A186085 Number of 1-dimensional sand piles with n grains. 10
 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 Seiichi Manyama, Table of n, a(n) for n = 0..4502 (terms 0..1000 from Alois P. Heinz) 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-François 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 (sand piles by base length). Cf. 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 15 01:40 EDT 2019. Contains 328025 sequences. (Running on oeis4.)