The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A091820 Number of ways to reach a sum of n by a down-only cascade through Pascal's Triangle, starting at C(0,0)=1 at the apex and shifting left or right by exactly one position at each step. 1
 1, 2, 2, 4, 2, 4, 6, 4, 2, 6, 6, 6, 8, 4, 4, 10, 2, 6, 6, 6, 8, 14, 12, 4, 6, 6, 4, 10, 6, 6, 12, 4, 6, 8, 6, 12, 14, 14, 4, 12, 8, 12, 20, 4, 4, 16, 6, 4, 6, 6, 10, 14, 4, 8, 8, 14, 12, 14, 12, 6, 10, 4, 10, 12, 6, 12, 12, 8, 4, 16, 16, 10, 22, 6, 10, 22, 14, 26, 12, 8, 6, 12, 8, 6, 14, 12, 14, 12, 6, 10, 16, 12, 10, 8, 4, 12, 10, 6, 10, 12 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This always generates a partition of n in increasing order, but we are computing the number of cascades, not the number of distinct partitions. For n > 1 not of the form 2^n+1, there is always a cascade with 1's and then moving over to the C(n,1) column (and its reversal), so a(n) >= 4 except for these cases. 33 can be generated by 1*4+4+10+15; is there any n > 17 with a(n) = 2? More generally, is lim_{n->infinity) a(n) = infinity? a(2049) = 2. - Sean A. Irvine, Nov 30 2010 Computing a(1+2^k) for 11 < k <= 25 finds no additional terms equal to 2. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..15000 EXAMPLE Pascal's triangle (A007318):             1           1   1         1   2   1       1   3   3   1     1   4   6   4   1   1   5  10  10   5   1 Then we can create 7 by 1+1+1+1+1+1+1 on the left and right and 1+1+2+3 gives 4 more possibilities, giving a(7) = 6. Similarly, 10=10*1 (left & right) = 5*1 + 5 (left & right) = 3*1 + 3 + 4 (left & right), giving a(10)=6. MAPLE b:= proc(m, n, k) local h;       if k>n-k then b(m, n, n-k)     else h:= m -binomial(n, k);          if h<0 then 0        elif h=0 or k=0 and h<=n then 1        else b(m, n, k):= b(h, n+1, k) +b(h, n+1, k+1)       fi fi     end: a:= n-> b(n, 0, 0): seq(a(n), n=1..100);  # Alois P. Heinz, Nov 30 2010 MATHEMATICA b[m_, n_, k_] := b[m, n, k] = With[{h = m - Binomial[n, k]}, Which[k == 0 && m-1 <= n, 1, k > n-k , b[m, n, n-k], h < 0, 0, h == 0, 1, True, b[h, n+1, k] + b[h, n+1, k+1] ]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 09 2011, after Alois P. Heinz *) CROSSREFS Cf. A007318. Sequence in context: A331118 A260723 A059214 * A171922 A306743 A140821 Adjacent sequences:  A091817 A091818 A091819 * A091821 A091822 A091823 KEYWORD nonn,look,nice AUTHOR Jon Perry, Mar 08 2004 EXTENSIONS Extended and edited by John W. Layman, Mar 10 2004 Edited by Franklin T. Adams-Watters, Apr 09 2009 More terms from Sean A. Irvine, Nov 30 2010 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 December 8 17:01 EST 2021. Contains 349596 sequences. (Running on oeis4.)