A096436 -id:A096436 - cp's OEIS frontend

A005831 a(n+1) = a(n) * (a(n-1) + 1).

0, 1, 1, 2, 4, 12, 60, 780, 47580, 37159980, 1768109008380, 65702897157329640780, 116169884340604934905464739377180, 7632697963609645128663145969343357330533515068777580, 886689639639303288926299195509965193299034793881606681727875910370940270908216401980

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

A discrete analog of the derivative of t(x) = tetration base e, since t'(x) = t(x) * t(x-1) * t(x-2) * ... y = y * exp(y) * exp(exp(y)) * ... * t(x) This sequence satisfies almost the same equation but the derivative is replaced by a difference, comparable to the relations between differential equations and their associated difference equations. - Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008

			a(5) = 12 since 12 = 1*2*4 + 4.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..19
M. E. Mays, Iterating the division algorithm, Fib. Quart., 25 (1987), 204-213.

Cf. A007807.

Cf. A000058, A001622, A001113, A102575, A096436, A111129.

a005831 n = a005831_list !! n
a005831_list = 0:1:zipWith (*) (tail a005831_list) (map succ a005831_list)
-- Reinhard Zumkeller, Mar 19 2011

a=0;b=1;lst={a,b};Do[c=a*b+b;AppendTo[lst,c];a=b;b=c,{n,18}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 13 2009 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-1](a[n-2]+1)},a,{n,15}] (* Harvey P. Dale, Aug 17 2013 *)

a(0) = a(1) = 1, a(2) = 2; a(n) = a(n-1)*a(n-2)*a(n-3)*... + a(n-1). - Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008

The sequence grows like a doubly exponential function, similar to Sylvester's sequence. In fact we have the asymptotic form : a(n) ~ e ^ (Phi ^ n) where e and Phi are the best possible constants. - Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A141435 a(1) = 1, a(2) = 2; a(n) = a(n-a(1)) + a(n-a(2)) + a(n-a(3)) + a(n-a(4)) + ...

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 38, 71, 132, 247, 461, 861, 1609, 3005, 5613, 10485, 19584, 36581, 68330, 127632, 238404, 445314, 831798, 1553712, 2902170, 5420945, 10125754, 18913838, 35329048, 65990929, 123264078, 230244265, 430071949, 803328933

Offset: 1

Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008

Thus we get a self-reference sequence that grows exponentially. a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-6) + a(n-11) + a(n-20) + ...
A Fibonacci-like sequence, even closer to the tribonacci numbers.
Lim n-> oo log (a(n))/n converges.

			a(6) = 20 because 20 = a(5) + a(4) + a(3) = 11 + 6 + 3
a(8) = 71 because 71 = a(7) + a(6) + a(5) + a(2) = 38 + 20 + 11 + 2

Cf. A058265, A008937, A001590, A000213, A000073, A096436, A102575, A111129.

A141435 := proc(n) option remember; local a,i; if n <= 3 then RETURN(n); else a :=0 ; for i from 1 to n-1 do if n-procname(i) < 1 then RETURN(a); else a := a+procname(n-procname(i)) ; fi; od; RETURN(a); fi; end: for n from 1 to 80 do printf("%d,",A141435(n)) ; od: # R. J. Mathar, Nov 03 2008

def A141435(terms):
    seq = [1, 2]
    for n in range(3, terms):
        s = 0
        for m in seq:
            if (n - m) > 0:
                s += seq[n - m - 1] #fix for python indexing
        seq.append(s)
    return seq
print(A141435(40)) # Andres Cruz y Corro A, Jun 19 2019

More terms from R. J. Mathar, Nov 03 2008

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A098066 Duplicate of A096436.

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A005831 a(n+1) = a(n) * (a(n-1) + 1).

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A141435 a(1) = 1, a(2) = 2; a(n) = a(n-a(1)) + a(n-a(2)) + a(n-a(3)) + a(n-a(4)) + ...

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