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A026567 a(n) = Sum_{i=0..2*n} Sum_{j=0..i} T(i, j), where T is given by A026552.

%I A026567 #33 Sep 08 2022 08:44:49
%S A026567 1,4,13,31,85,193,517,1165,3109,6997,18661,41989,111973,251941,671845,
%T A026567 1511653,4031077,9069925,24186469,54419557,145118821,326517349,
%U A026567 870712933,1959104101,5224277605,11754624613,31345665637,70527747685
%N A026567 a(n) = Sum_{i=0..2*n} Sum_{j=0..i} T(i, j), where T is given by A026552.
%H A026567 Vincenzo Librandi, <a href="/A026567/b026567.txt">Table of n, a(n) for n = 0..1000</a>
%H A026567 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-6).
%F A026567 a(n) = Sum_{i=0..2*n} Sum_{j=0..i} A026552(i, j).
%F A026567 G.f.: (1+3*x+3*x^2)/((1-x)*(1-6*x^2)). - _Ralf Stephan_, Feb 03 2004
%F A026567 a(n) = 6*a(n-2) + 7. - _Philippe Deléham_, Feb 24 2014
%F A026567 a(2*k) = A233325(k). - _Philippe Deléham_, Feb 24 2014
%F A026567 From _Colin Barker_, Nov 25 2016: (Start)
%F A026567 a(n) = (2^(n/2+2) * 3^(n/2+1) - 7)/5 for n even.
%F A026567 a(n) = (2^((n-1)/2) * 3^((n+5)/2) - 7)/5 for n odd. (End)
%F A026567 a(n) = (1/10)*(2*(1+(-1)^n)*6^((n+2)/2) + 27*(1-(-1)^n)*6^((n-1)/2) - 14). - _G. C. Greubel_, Dec 19 2021
%t A026567 CoefficientList[Series[(1 +3x +3x^2)/((1-x)(1-6x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 25 2014 *)
%t A026567 LinearRecurrence[{1,6,-6},{1,4,13},30] (* _Harvey P. Dale_, Aug 23 2014 *)
%o A026567 (Magma) [Truncate((2*(1+(-1)^n)*6^((n+2)/2) + 27*(1-(-1)^n)*6^((n-1)/2) -14)/10): n in [0..40]]; // _G. C. Greubel_, Dec 19 2021
%o A026567 (Sage) [(1/10)*(2*(1+(-1)^n)*6^((n+2)/2) +27*(1-(-1)^n)*6^((n-1)/2) -14) for n in (0..40)] # _G. C. Greubel_, Dec 19 2021
%Y A026567 Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A027272, A027273, A027274, A027275, A027276.
%Y A026567 Cf. A026565, A233325.
%K A026567 nonn,easy
%O A026567 0,2
%A A026567 _Clark Kimberling_