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%I #93 Feb 24 2023 08:58:07
%S 1,0,1,1,2,4,4,9,11,26,23,55,64,154,134,323,373,900,781,1885,2174,
%T 5248,4552,10989,12671,30590,26531,64051,73852,178294,154634,373319,
%U 430441,1039176,901273,2175865,2508794,6056764,5253004,12681873,14622323,35301410
%N Sequences A006452 and A216134 interlaced.
%H Colin Barker, <a href="/A216162/b216162.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,6,0,-6,0,-1,0,1)
%F (a(2n) + a(2n - 1)) - (a(2n - 2) + a(2n - 3)) = A000129(n); n>1.
%F It follows that sqrt(2) = lim n --> infinity ((a(2n + 2) + a(2n + 1)) - (a(2n - 2) + a(2n - 3)))/((a(2n + 2) + a(2n + 1)) - (a(2n) + a(2n - 1))).
%F G.f. ( -1-x^3+5*x^4-3*x^5-2*x^6+x^7-2*x^8+x^9 ) / ( (x-1)*(1+x)*(x^4-2*x^2-1)*(x^4+2*x^2-1) ). - _R. J. Mathar_, Sep 08 2012
%o (PARI) Vec((-1-x^3+5*x^4-3*x^5-2*x^6+x^7-2*x^8+x^9)/((x-1)*(1+x)*(x^4-2*x^2-1)*(x^4+2*x^2-1))+O(x^99)) \\ _Charles R Greathouse IV_, Jun 12 2015
%Y Cf. A000129.
%Y For some k in n:
%Y a(2n) = A006452 (k^2 - 1 is triangular).
%Y a(2n + 1) = A216134 (T_k and 2T_k + 1 are triangular).
%Y a(2n + 1) - a(2n) = A006451 (T_k + 1 is square).
%Y a(2n + 1) + a(2n) = A124124 (T_k and (T_k - 1)/2 are triangular).
%Y a(4n + 1) + a(4n + 2) = A001108 (T_k is square).
%Y a(4n + 3) + a(4n + 4) = A001652 (T_k and 2T_k are triangular).
%Y Sum(a(n)) - 1 = A048776 for even n (the second partial summation of the Pell numbers).
%K nonn,easy
%O 0,5
%A _Raphie Frank_, Sep 07 2012
%E Edited by _N. J. A. Sloane_, May 24 2021