A037576 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,3.
1, 7, 29, 119, 477, 1911, 7645, 30583, 122333, 489335, 1957341, 7829367, 31317469, 125269879, 501079517, 2004318071, 8017272285, 32069089143, 128276356573, 513105426295, 2052421705181, 8209686820727, 32838747282909
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Eric Weisstein's World of Mathematics, Rule 190
- Index entries for sequences related to cellular automata
- Index to Elementary Cellular Automata
- Index entries for linear recurrences with constant coefficients, signature (4,1,-4).
Programs
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Magma
I:=[1, 7, 29]; [n le 3 select I[n] else 4*Self(n-1)+Self(n-2)-4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012
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Mathematica
CoefficientList[Series[(1+3*x)/((x-1)*(4*x-1)*(1+x)),{x,0,30}],x] (*or*) LinearRecurrence[{4,1,-4},{1,7,29},40] (* Vincenzo Librandi, Jun 22 2012 *) -
PARI
my(x='x+O('x^99)); Vec(x*(1+3*x)/((1-x)*(1-4*x)*(1+x))) \\ Altug Alkan, Sep 21 2018 -
Python
print([7*4**n//15 for n in range(1,30)]) # Karl V. Keller, Jr., Mar 09 2021
Formula
G.f.: x*(1+3*x)/((1-x)*(1-4*x)*(1+x)). - Vincenzo Librandi, Jun 22 2012
a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3). - Vincenzo Librandi, Jun 22 2012
a(n) = (7*4^n + 3*(-1)^n - 10)/15. - Bruno Berselli, Jun 22 2012, corrected by Klaus Purath, Mar 18 2021.
a(n) = floor(7*4^n/15). - Karl V. Keller, Jr., Mar 09 2021
From Klaus Purath, Mar 18 2021: (Start)
a(n) = 16*a(n-2) - 3*(-1)^n + 10, assuming that a(0) = 0.
a(n) = 4*a(n-1) + 2 + (-1)^n.
a(n) = 5*a(n-1) - 4*a(n-2) + 2*(-1)^n, n > 2. (End)
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