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%I #30 Apr 25 2023 15:29:03
%S 1,3,8,24,64,192,512,1536,4096,12288,32768,98304,262144,786432,
%T 2097152,6291456,16777216,50331648,134217728,402653184,1073741824,
%U 3221225472,8589934592,25769803776,68719476736,206158430208,549755813888
%N Expansion of (1+3*x)/(1-8*x^2).
%C From _R. J. Mathar_, Oct 12 2010: (Start)
%C Apparently the row n=3 of an array T(n,k) counting walks with k steps on an n X n board starting at an edge position next to a corner, each step to one of the <= 4 adjacent squares:
%C 1, 3, 8, 24, 64, 192, 512, 1536, 4096, 12288, 32768, 98304, 262144, ...
%C 1, 3, 9, 29, 93, 301, 973, 3149, 10189, 32973, 106701, 345293, 1117389, ...
%C 1, 3, 9, 30, 99, 342, 1161, 4050, 13851, 48438, 165969, 580770, 1990899, ...
%C 1, 3, 9, 30, 100, 349, 1216, 4329, 15381, 55187, 197714, 711458, 2557699, ...
%C 1, 3, 9, 30, 100, 350, 1224, 4400, 15776, 57552, 209088, 768768, 2812160, ...
%C 1, 3, 9, 30, 100, 350, 1225, 4409, 15865, 58091, 212586, 786708, 2909166, ...
%C 1, 3, 9, 30, 100, 350, 1225, 4410, 15875, 58200, 213300, 791700, 2936375, ...
%C 1, 3, 9, 30, 100, 350, 1225, 4410, 15876, 58211, 213431, 792623, 2943291, ...
%C 1, 3, 9, 30, 100, 350, 1225, 4410, 15876, 58212, 213443, 792778, 2944460, ...
%C 1, 3, 9, 30, 100, 350, 1225, 4410, 15876, 58212, 213444, 792791, 2944641, ...
%C ...
%C (End)
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,8).
%F G.f.: (1+3*x)/(1-8*x^2).
%F a(n) = (1 + (-1)^n)*8^floor((n+1)/2)/2 + 3*(1-(-1)^n)*8^floor(n/2)/2.
%F a(n) = 2^(3*n/2)*(3*sqrt(2)/8 + 1/2 - (3*sqrt(2)/8 - 1/2)*(-1)^n).
%F a(2n+1) = 2*a(2n) + 2*a(2n-1) + 2*a(2n-2).
%F a(2n) = 2*a(2n-1) + 2*a(2n-2).
%F a(n+3) = a(n+2)*a(n+1)/a(n). - _Reinhard Zumkeller_, Mar 04 2011
%t CoefficientList[Series[(1+3x)/(1-8x^2),{x,0,30}],x] (* or *) LinearRecurrence[{0,8},{1,3},30] (* _Harvey P. Dale_, Apr 25 2023 *)
%Y Cf. A038754.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jul 14 2004
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