A005230 - OEIS

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)

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

A002487 is THE Stern's sequence!

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000 [extending prior submission by T. D. Noe]

Jaegug Bae and Sungjin Choi, A generalization of a subset-sum-distinct sequence, J. Korean Math. Soc. 40 (2003), no. 5, 757-768. MR1996839 (2004d:05198). See d_1(n).

G. Kreweras, Sur quelques problèmes relatifs au vote pondéré, [Some problems of weighted voting], Math. Sci. Humaines No. 84 (1983), 45-63.

M. A. Stern, Aufgaben, J. Reine Angew. Math., 18 (1838), 100.

Index entries for sequences related to Stern's sequences

Index entries for "core" sequences

FORMULA

Partial sums give Conway-Guy sequence A005318. Cf. A066777.

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)))

A005230_list = list(islice(A005230_gen(), 30)) # Chai Wah Wu, Feb 02 2022

CROSSREFS

Cf. A002487.

Sequence in context: A096080 A329665 A143658 * A030037 A077078 A077079