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%I #70 Jan 31 2023 16:43:00
%S 1,2,6,14,38,94,246,622,1606,4094,10518,26894,68966,176542,452406,
%T 1158574,2968198,7602494,19475286,49885262,127786406,327327454,
%U 838473078,2147782894,5501675206,14092806782,36099507606,92470734734
%N Expansion of (1+x)/(1-x-4*x^2).
%C This sequence can generated by the following formula: a(n) = a(n-1) + 4*a(n-2) when n > 2; a[1] = 1, a[2] = 2. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 21 2004
%C An elephant sequence, see A175654 and A175655. For the corner squares just one A[5] vector, with decimal value 325, leads to the sequence given above. For the central square this vector leads to a companion sequence that is 4 times this very same sequence with n >= -1. - _Johannes W. Meijer_, Aug 15 2010
%C Equals INVERTi transform of A180168. - _Gary W. Adamson_, Aug 14 2010
%C Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 2 X 2 cells and remove the cells that have one '1' in their modulo 3 coordinates. a(n) is the number of cells after n iterations. Cell configuration converges to a fractal with approximate dimension 1.357. - _Peter Karpov_, Apr 20 2017
%H Nathaniel Johnston, <a href="/A026597/b026597.txt">Table of n, a(n) for n = 0..500</a>
%H Peter Karpov, <a href="http://inversed.ru/InvMem.htm#InvMem_26">InvMem, Item 26</a>
%H Peter Karpov, <a href="/A026597/a026597_1.png">Illustration of initial terms (n = 1..8)</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,4).
%F G.f.: (1+x)/(1-x-4*x^2).
%F a(n) = T(n,0) + T(n,1) + ... + T(n,2*n), T given by A026584.
%F a(n) = Sum_{k=0..n} binomial(floor((2*n-k-1)/2), n-k)*2^k. - _Paul Barry_, Feb 11 2005
%F a(n) = A006131(n) + A006131(n-1), n >= 1. - _R. J. Mathar_, Oct 20 2006
%F a(n) = Sum_{k=0..n} binomial(floor((2*n-k)/2),n-k)*4^floor(k/2). - _Paul Barry_, Feb 02 2007
%F Inverse binomial transform of A007482: (1, 3, 11, 39, 139, 495, ...). - _Gary W. Adamson_, Dec 04 2007
%F a(n) = Sum_{k=0..n+1} A122950(n+1,k)*3^(n+1-k). - _Philippe Deléham_, Jan 04 2008
%F a(n) = (1/2 + 3*sqrt(17)/34)*(1/2 + sqrt(17)/2)^n + (1/2 - 3*sqrt(17)/34)*(1/2 - sqrt(17)/2)^n. - _Antonio Alberto Olivares_, Jun 07 2011
%F a(n) = (2*i)^n*( chebyshevU(n, -i/4) - (i/2)*chebyshevU(n-1, -i/4) ). - _G. C. Greubel_, Dec 08 2021
%F E.g.f.: exp(x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2))/17. - _Stefano Spezia_, Jan 31 2023
%t LinearRecurrence[{1,4},{1,2},40] (* _Harvey P. Dale_, Nov 28 2011 *)
%o (Sage) [(2*i)^n*( chebyshev_U(n, -i/4) - (i/2)*chebyshev_U(n-1, -i/4) ) for n in (0..40)] # _G. C. Greubel_, Dec 08 2021
%o (Magma) [n le 2 select n else Self(n-1) + 4*Self(n-2): n in [1..41]]; // _G. C. Greubel_, Dec 08 2021
%Y Cf. A006131, A006138, A007482, A026581, A026584, A122950, A175654, A175655, A180168.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_
%E Better name from _Ralf Stephan_, Jul 14 2013
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