|
|
A005230
|
|
Stern's sequence: a(1) = 1, a(n+1) is the sum of the m preceding terms, where m*(m-1)/2 < n <= m*(m+1)/2 or equivalently m = ceiling((sqrt(8*n+1)-1)/2) = A002024(n).
(Formerly M0785)
|
|
8
|
|
|
1, 1, 2, 3, 6, 11, 20, 40, 77, 148, 285, 570, 1120, 2200, 4323, 8498, 16996, 33707, 66844, 132568, 262936, 521549, 1043098, 2077698, 4138400, 8243093, 16419342, 32706116, 65149296, 130298592, 260075635, 519108172, 1036138646, 2068138892, 4128034691
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Limit_{n->oo} a(n)/2^n = 0.11756264240558743281779408719593950494049225979176... - Jon E. Schoenfield, Dec 17 2016
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
M. A. Stern, Aufgaben, J. Reine Angew. Math., 18 (1838), 100.
|
|
FORMULA
|
2*a(n*(n+1)/2 + 1) = a(n*(n+1)/2 + 2) for n>=1; lim_{n->oo} a(n+1)/a(n) = 2. - Paul D. Hanna, Aug 28 2006
|
|
MAPLE
|
A005230[1] := 1: n := 50: for k from 1 to n-1 do: A005230[k+1] := sum('A005230[j]', 'j'=k+1-(ceil((sqrt(8*k+1)-1)/2))..k): od: [seq(A005230[k], k=1..n)]; # UlrSchimke(AT)aol.com, Mar 16 2002
|
|
MATHEMATICA
|
Module[{lst={1, 1}, n=2}, While[n<40, AppendTo[lst, Total[ Take[lst, -Ceiling[ (Sqrt[8n+1]-1)/2]]]]; n++]; lst] (* Harvey P. Dale, Apr 02 2012 *)
|
|
PROG
|
(PARI) a(n)=if(n==1, 1, sum(k=1, ceil((sqrt(8*n-7)-1)/2), a(n-k))) \\ Paul D. Hanna, Aug 28 2006
(PARI) v=vector(10^3); v[1]=v[2]=1; v[3]=2; v[4]=3; u=vector(#v, i, if(i>4, 0, sum(j=1, i, v[j]))); for(i=5, #v, m=ceil((sqrt(8*i-7)-1)/2); v[i]=u[i-1]-u[i-m-1]; u[i]=u[i-1]+v[i]); u=0; v \\ Charles R Greathouse IV, Sep 19 2011
(Python)
from itertools import count, islice
from math import isqrt
def A005230_gen(): # generator of terms
blist = [1]
for n in count(1):
yield blist[-1]
blist.append(sum(blist[-i] for i in range(1, (isqrt(8*n)+3)//2)))
|
|
CROSSREFS
|
|
|
KEYWORD
|
core,easy,nonn,nice
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Name corrected by Mario Szegedy, Sep 15 1996
Name revised by Ulrich Schimke (ulrschimke(AT)aol.com), Mar 16 2002
|
|
STATUS
|
approved
|
|
|
|