Revision History for A266069
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Showing entries 1-10
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| A266069
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Decimal representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.
(history;
published version)
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#39 by Alois P. Heinz at Wed Aug 25 19:39:25 EDT 2021
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#38 by Alois P. Heinz at Wed Aug 25 19:39:22 EDT 2021
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Empirical gG.f.: (1+4*x-17*x^2+45*x^3+16*x^4-64*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)). - Colin Barker, Dec 21 2015 and Apr 18 2019
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proposed
editing
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#37 by Karl V. Keller, Jr. at Wed Aug 25 19:35:00 EDT 2021
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#36 by Karl V. Keller, Jr. at Wed Aug 25 19:34:23 EDT 2021
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a(n) = 2*4^n - 3*2^((n-1)/2) - 1 for odd n; a(n) = 2^(n/2) for even n. - Karl V. Keller, Jr., Aug 25 2021
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(Python) print([2*4**n - 3*2**((n-1)//2) - 1 if n%2 else 2**(n//2) for n in range(30)]) # Karl V. Keller, Jr., Aug 25 2021
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approved
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#35 by Wesley Ivan Hurt at Thu Apr 18 09:58:58 EDT 2019
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#34 by Joerg Arndt at Thu Apr 18 09:33:49 EDT 2019
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#33 by Colin Barker at Thu Apr 18 09:01:08 EDT 2019
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#32 by Colin Barker at Thu Apr 18 09:00:09 EDT 2019
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| LINKS
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<a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,19,0,-50,0,32).
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GEmpirical g.f.: (1+4*x-17*x^2+45*x^3+16*x^4-64*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)). - Colin Barker, Dec 21 2015 and Apr 18 2019
a(n) = ((sqrt(2)+3)*((-1)^n+1)-6)*sqrt(2)^(n-3) - (2*4^n-1)*((-1)^n-1)/2. Therefore: a(n) = 2^(n/2) for even n; otherwise, a(n) = 2^(2*n+1)-3*2^((n-1)/2)-1. [Bruno Berselli, Dec 21 2015]
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| MATHEMATICA
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Table[((Sqrt[2] + 3) ((-1)^n + 1) - 6) Sqrt[2]^(n - 3) - (2 4^n - 1) ((-1)^n - 1)/2, {n, 0, 30}] (* Bruno Berselli, Dec 22 2015 *)
LinearRecurrence[{0, 19, 0, -50, 0, 32}, {1, 4, 2, 121, 4, 2035}, 40] (* Vincenzo Librandi, Dec 22 2015 *)
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(PARI) Vec((1+4*x-17*x^2+45*x^3+16*x^4-64*x^5)/((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)) + O(x^40)) \\ Colin Barker, Dec 21 2015
(MAGMA) [IsEven(n) select 2^(n div 2) else 2^(2*n+1)-3*2^((n-1) div 2)-1: n in [0..30]]; // Bruno Berselli, Dec 22 2015
(Sage) [((sqrt(2)+3)*((-1)^n+1)-6)*sqrt(2)^(n-3)-(2*4^n-1)*((-1)^n-1)/2 for n in (0..30)] # Bruno Berselli, Dec 22 2015
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approved
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#31 by Joerg Arndt at Fri Feb 26 11:34:13 EST 2016
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#30 by Joerg Arndt at Fri Feb 26 11:34:06 EST 2016
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