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A114696 Expansion of (1+4*x+x^2)/((x-1)*(x+1)*(x^2+2*x-1)); a Pellian-related sequence. 4

%I

%S 1,6,15,40,97,238,575,1392,3361,8118,19599,47320,114241,275806,665855,

%T 1607520,3880897,9369318,22619535,54608392,131836321,318281038,

%U 768398399,1855077840,4478554081,10812186006,26102926095,63018038200,152139002497,367296043198

%N Expansion of (1+4*x+x^2)/((x-1)*(x+1)*(x^2+2*x-1)); a Pellian-related sequence.

%C Elements of odd index give match to A065113: Sum of the squares of the n-th and the (n+1)st triangular numbers (A000217) is a perfect square.

%C Generating floretion: - 1.5'i + 'j + 'k - .5i' + j' + k' + .5'ii' - .5'jj' - .5'kk' - 'ij' + 'ik' - 'ji' + .5'jk' + 2'ki' - .5'kj' + .5e

%H Colin Barker, <a href="/A114696/b114696.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).

%F a(0)=1, a(1)=6, a(2)=15, a(3)=40, a(n)=2*a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4). - _Harvey P. Dale_, Jan 23 2014

%F a(n) = (-3-(-1)^n+(3-2*sqrt(2))*(1-sqrt(2))^n+(1+sqrt(2))^n*(3+2*sqrt(2)))/2. - _Colin Barker_, May 26 2016

%t CoefficientList[Series[(1+4x+x^2)/((x-1)(x+1)(x^2+2x-1)),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,-2,-1},{1,6,15,40},30] (* _Harvey P. Dale_, Jan 23 2014 *)

%o (PARI) Vec((1+4*x+x^2)/((x-1)*(x+1)*(x^2+2*x-1)) + O(x^30)) \\ _Colin Barker_, May 26 2016

%Y Cf. A100828, A111954, A113224, A114647, A114688, A114689, A114695, A114697, A000129, A005409.

%K easy,nonn

%O 0,2

%A _Creighton Dement_, Feb 18 2006

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Last modified June 24 05:21 EDT 2019. Contains 324318 sequences. (Running on oeis4.)