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A005230
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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 = ceil((sqrt(8*n+1)-1)/2) = A002024(n).
(Formerly M0785)
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7
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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; internal format)
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OFFSET
| 1,3
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COMMENTS
| A002487 is THE Stern's sequence!
The subsequence of primes in this partial sum begins: 2, 3, 11, a(41) = 262364233421, and no more through a(200). [From Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 18 2010]
The next few primes are a(568), a(657), a(17765), ... with no further primes below a(50000). [Charles R Greathouse IV, Sep 19 2011]
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REFERENCES
| Kreweras, G.; Sur quelques problemes relatifs au vote pondere [Some problems of weighted voting], Math. Sci. Humaines No. 84 (1983), 45-63.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. A. Stern, Aufgaben, J. Reine Angew. Math., 18 (1838), 100.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..200
Index entries for sequences related to Stern's sequences
Index entries for "core" sequences
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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; limit_{n->infty} a(n+1)/a(n) = 2. - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 28 2006
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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)];
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PROG
| (PARI) a(n)=if(n==1, 1, sum(k=1, ceil((sqrt(8*n-7)-1)/2), a(n-k))) - Paul D. Hanna (pauldhanna(AT)juno.com), 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
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CROSSREFS
| Cf. A002487.
Sequence in context: A141435 A096080 A143658 * A030037 A077078 A077079
Adjacent sequences: A005227 A005228 A005229 * A005231 A005232 A005233
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KEYWORD
| core,easy,nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| Description corrected by Mario Szegedy Sep 15 1996; and revised with Maple code by UlrSchimke(AT)aol.com, Mar 16, 2002
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