A039941
Alternately add and multiply.
Original entry on oeis.org
0, 1, 1, 1, 2, 2, 4, 8, 12, 96, 108, 10368, 10476, 108615168, 108625644, 11798392572168192, 11798392680793836, 139202068568601556987554268864512, 139202068568601568785946949658348, 19377215893777651167043206536157390321290709180447278572301746176
Offset: 0
- Reinhard Zumkeller, Table of n, a(n) for n = 0..27
- A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
- A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
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a039941 n = a039941_list !! (n-1)
a039941_list = 0 : 1 : zipWith3 ($)
(cycle [(+),(*)]) a039941_list (tail a039941_list)
-- Reinhard Zumkeller, May 07 2012
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nxt[{n_,a_,b_}]:={n+1,b,If[EvenQ[n],a+b,a*b]}; Join[{0},Transpose[ NestList[ nxt,{0,0,1},20]][[3]]] (* Harvey P. Dale, Aug 23 2013 *)
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a(n)=if(n<2,n>0, if(n%2,a(n-1)*a(n-2),a(n-1)+a(n-2)))
A001697
a(n+1) = a(n)(a(0) + ... + a(n)).
Original entry on oeis.org
1, 1, 2, 8, 96, 10368, 108615168, 11798392572168192, 139202068568601556987554268864512, 19377215893777651167043206536157390321290709180447278572301746176
Offset: 0
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- John Cerkan, Table of n, a(n) for n = 0..12
- A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
- A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
- Daniel Duverney, Takeshi Kurosawa, Iekata Shiokawa, Transformation formulas of finite sums into continued fractions, arXiv:1912.12565 [math.NT], 2019.
- Index entries for sequences of form a(n+1)=a(n)^2 + ...
- Index to divisibility sequences
a(n) =
A039941(2*n+1); first differences of
A001696 give this sequence.
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a001697 n = a001697_list !! n
a001697_list = 1 : 1 : f [1,1] where
f xs@(x:_) = y : f (y : xs) where y = x * sum xs
-- Reinhard Zumkeller, Apr 29 2013
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[n le 2 select 1 else Self(n-1)^2*(1+1/Self(n-2)): n in [1..12]]; // Vincenzo Librandi, Nov 25 2015
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a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 1]^2*(1 + 1/a[n - 2]); Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Jul 02 2013 *)
nxt[{t_,a_}]:={t+t*a,t*a}; Transpose[NestList[nxt,{1,1},10]][[2]] (* Harvey P. Dale, Jan 24 2016 *)
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a(n)=if(n<2,n >= 0,a(n-1)^2*(1+1/a(n-2)))
A045761
Define polynomials Pn by P0 = 0, P1 = x, P2 = P1 + P0, P3 = P2 * P1, P4 = P3 + P2, etc. alternately adding or multiplying. For even n > 2k, then first k coefficients of Pn remain unchanged and their values are the first k terms of the sequence.
Original entry on oeis.org
0, 1, 1, 1, 2, 3, 6, 12, 24, 50, 107, 232, 508, 1124, 2513, 5665, 12858, 29356, 67371, 155345, 359733, 836261, 1950829, 4565305, 10714501, 25212843, 59474318, 140609809, 333126672, 790764280, 1880489541, 4479494059, 10687448937, 25536624382, 61102431113
Offset: 0
James Boudinot (jboudinot(AT)yahoo.com)
The sequence of polynomials is 0, x, x, x^2, x^2 + x, x^4 + x^3, x^4 + x^3 + x^2 + x, ..., and after this all the even polynomials end with x^3 + x^2 + x (+ 0), so the first 4 terms of the sequence are these coefficients (in ascending order): 0, 1, 1, 1. - _Michael B. Porter_, Aug 09 2016
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k = 32; P[0] = 0; P[1] = x;
P[n_] := P[n] = If[EvenQ[n], P[n-1] + P[n-2], P[n-1]*P[n-2]] + O[x]^(2k+1) // Normal;
CoefficientList[P[2k], x][[1 ;; k+1]] (* Jean-François Alcover, Aug 07 2016 *)
A057268
a(n-1)+k=a(n) => a(n)*k=a(n+1).
Original entry on oeis.org
1, 3, 6, 18, 216, 42768, 1819863936, 3311826913612597248, 10968197499701667312529793029329125376, 120301356392461906290006219878693096559247086148184247242087281541542576128
Offset: 0
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nxt[{a_,b_}]:={b,b(b-a)}; NestList[nxt,{1,3},9][[All,1]] (* Harvey P. Dale, Sep 05 2017 *)
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