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#25 by Ray Chandler at Sun Oct 01 13:43:58 EDT 2023
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#24 by Ray Chandler at Sun Oct 01 13:43:52 EDT 2023
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| LINKS
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<a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1, 3, -3, -2, 2, -2, 2, 3, -3, -1, 1).
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approved
editing
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#23 by Charles R Greathouse IV at Thu Sep 08 08:45:10 EDT 2022
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(MAGMAMagma) [(6*n^4 +108*n^3 +666*n^2 +1620*n +1251 +(4*n^3 +54*n^2 +236*n +333)*(-1)^n -48*(-1)^Floor((6*n -1 +(-1)^n)/4))/1536: n in [0..50]]; // Vincenzo Librandi, Oct 23 2014
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Discussion
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| Thu Sep 08
| 08:45
| OEIS Server: https://oeis.org/edit/global/2944
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#22 by Michael Somos at Tue May 05 16:13:04 EDT 2015
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#21 by Michael Somos at Tue May 05 16:12:38 EDT 2015
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| COMMENTS
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Euler transform of length 4 sequence [2,3,-1,1]. - Michael Somos, Feb 15 2006
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| FORMULA
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Euler transform of length 4 sequence [ 2, 3, -1, 1]. - Michael Somos, Feb 15 2006
G.f.: ( .: (1+ + x+ + x^2 ) / ( () / ((1+ + x^2)*() * (1+ + x)^4*( * (1- - x)^5 ).).
a(n) = a(-9-n) = a() for all n). in Z.
a(2*n) = A070893(n+1). a(2*n + 1) = A082289(n+4).
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| EXAMPLE
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G.f. = 1 + 2*x + 6*x^2 + 9*x^3 + 19*x^4 + 26*x^5 + 46*x^6 + 59*x^7 + ...
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| PROG
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(PARI) ) {a(n)=) = if(( n<-8, , a(-9-n), ), polcoeff((( (1+ + x+ + x^2)/(() / ((1+ + x^2)*() *(1+ + x)^4*( * (1- - x)^5)+) + x* * O(x^n), ), n))))};
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| CROSSREFS
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Cf. A070893(n)=a(2n-2); A082289(n)=a(2n-7).
Cf. A070893, A082289.
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| STATUS
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approved
editing
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Discussion
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| Tue May 05
| 16:13
| Michael Somos: Added more info. Light and space edits. Moved comment to formula.
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#20 by Bruno Berselli at Sun Oct 26 17:25:23 EDT 2014
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#19 by Bruno Berselli at Sun Oct 26 17:25:16 EDT 2014
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| FORMULA
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G.f.: (.: ( 1+x+x^2)/(( ) / ( (1+x^2)()*(1+x)^4(*(1-x)^5). ).
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#18 by Bruno Berselli at Sun Oct 26 17:23:57 EDT 2014
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| COMMENTS
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Euler transform of length 4 sequence [2,3,-1,1]. - Michael Somos, Feb 15 2006
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| FORMULA
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G.f.: (1+x+x^2)/((1+x^2)(1+x)^4(1-x)^5). a(n) = 3*a(n-2) - 2*a(n-4) - 2*a(n-6) + 3*a(n-8) - a(n-10) + 3. a(-9-n) = a(n).
Euler transform of length 4 sequence [2,3,-1,1]. - Michael Somos, Feb 15 2006
a(n) = 3*a(n-2) - 2*a(n-4) - 2*a(n-6) + 3*a(n-8) - a(n-10) + 3.
a(-9-n) = a(n).
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| MATHEMATICA
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Table[(6 n^4 + 108 n^3 + 666 n^2 + 1620 n + 1251 + (4 n^3 + 54 n^2 + 236 n + 333) (-1)^n - 48 (-1)^((6 n - 1 + (-1)^n)/4))/1536, {n, 0, 50}] (* after Luce ETIENNE; or, by definition: *) CoefficientList[Series[(1 + x + x^2)/((1 + x^2)*(1 + x)^4*(1 - x)^5), {x, 0, 50}], x] (* Bruno Berselli, Oct 26 2014 *)
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(MAGMA) [Floor(() [(6*n^4 +108*n^3 +666*n^2 +1620*n +1251 +(4*n^3 +54*n^2 +236*n +333)*(-1)^n -48*(-1)^(()^Floor((6*n -1 +(-1)^n)/4))/1536): : n in [0..50]]; // Vincenzo Librandi, Oct 23 2014
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| CROSSREFS
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Cf. A070893(n)=a(2n-2). ); A082289(n)=a(2n-7).
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| STATUS
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proposed
editing
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#17 by Vincenzo Librandi at Thu Oct 23 12:52:05 EDT 2014
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Discussion
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| Thu Oct 23
| 12:56
| Michel Marcus: in pari, I did not have to use floor
without floor what do you get ? reals ?
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| Sun Oct 26
| 17:15
| Bruno Berselli: No, Vincenzo, Magma approximates due to a complicated exponent, the Luce's formula is correct ---
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#16 by Vincenzo Librandi at Thu Oct 23 12:50:37 EDT 2014
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(MAGMA) [Floor((6*n^4 +108*n^3 +666*n^2 +1620*n +1251 +(4*n^3 +54*n^2 +236*n +333)*(-1)^n -48*(-1)^((6*n -1 +(-1)^n)/4))/1536): n in [0..50]]; // Vincenzo Librandi, Oct 23 2014
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| STATUS
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reviewed
editing
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Discussion
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| Thu Oct 23
| 12:51
| Vincenzo Librandi: Michel: Better Floor( ---) see Magma.
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