Revision History for A081335
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#29 by Charles R Greathouse IV at Thu Sep 08 08:45:09 EDT 2022
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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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#28 by Susanna Cuyler at Fri Aug 02 08:42:19 EDT 2019
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#27 by Jon E. Schoenfield at Fri Aug 02 04:12:55 EDT 2019
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#26 by Jon E. Schoenfield at Fri Aug 02 04:12:53 EDT 2019
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| COMMENTS
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Binomial transform of A034478. 4th binomial transform of (1, 0, 4, 0, 16, 0, 64, ... )., ...).
Case k=4 of the family of recurrences a(n) = 2*k*a(n-1) - (k^2-4)*a(n-2), a(0)=1,, a(1)=k.
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| FORMULA
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a(n) = 8*a(n-1) -) - 12*a(n-2), a(0)=1, a(1)=4.
a(n) = Sum_{k=0..floor(n/2)} Cbinomial(n,2*k) * 4^(n-k) = Sum_{k=0..n} Cbinomial(n,k) * 4^(n-k/2) * (1+(-1)^k)/2 . - _. - _Paul Barry_, Nov 22 2003
a(n) = Sum_{k=0..n} 4^k*A098158(n,k) . - _). - _Philippe Deléham_, Dec 04 2006
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| STATUS
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proposed
editing
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#25 by G. C. Greubel at Fri Aug 02 04:08:32 EDT 2019
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#24 by G. C. Greubel at Fri Aug 02 04:08:05 EDT 2019
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| NAME
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a(n) = (6^n+ + 2^n)/2.
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| COMMENTS
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Binomial transform of A034478. 4th binomial transform of (1,, 0,, 4,, 0,, 16,, 0,, 64,....)., ... ).
Case k=4 of the family of recurrences a(n)=2k) = 2*k*a(n-1)-() - (k^2-4)*a(n-2), a(0)=1,a(1)=k.
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| FORMULA
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a(n) = sum( Sum_{k=0..floor(n/2), )} C(n,2*k) * 4^(n-k) ) = sum( ) = Sum_{k=0..n, } C(n,k) * 4^(n-k/2) * (1+(-1)^k)/2 ). - _ . - _Paul Barry_, Nov 22 2003
a(n) = sum( Sum_{k=0..n, } 4^k*A098158(n,k) ). - _) . - _Philippe Deléham_, Dec 04 2006
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| MATHEMATICA
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CoefficientList[Series[(1 - 4 x) / ((-4x)/((1 - 2 x) (-2x)(1 - 6 x-6x)), {x, , 0, 20, 30}], x] (* Vincenzo Librandi, Aug 08 2013 *)
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| PROG
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(MAGMA) [(6^n+2^n)/2: n in [0..2530]]; // Vincenzo Librandi, Aug 08 2013
(Sage) [2^(n-1)*(3^n + 1) for n in (0..30)] # G. C. Greubel, Aug 02 2019
(GAP) List([0..30], n-> 2^(n-1)*(3^n + 1)); # G. C. Greubel, Aug 02 2019
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| STATUS
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approved
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#23 by Charles R Greathouse IV at Wed Oct 07 16:03:43 EDT 2015
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#22 by Charles R Greathouse IV at Wed Oct 07 16:03:38 EDT 2015
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#21 by Charles R Greathouse IV at Sat Jun 13 00:50:53 EDT 2015
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| LINKS
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<a href="/index/Rec#order_02">Index to sequencesentries withfor linear recurrences with constant coefficients</a>, signature (8,-12).
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Discussion
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| Sat Jun 13
| 00:50
| OEIS Server: https://oeis.org/edit/global/2439
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#20 by R. J. Mathar at Sun Oct 19 15:09:11 EDT 2014
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