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A236203 Interleave A005563(n), A028347(n).

%I A236203 #46 Aug 21 2022 04:18:57
%S A236203 0,0,3,5,8,12,15,21,24,32,35,45,48,60,63,77,80,96,99,117,120,140,143,
%T A236203 165,168,192,195,221,224,252,255,285,288,320,323,357,360,396,399,437,
%U A236203 440,480,483,525,528,572,575,621,624,672,675,725,728,780,783,837,840,896
%N A236203 Interleave A005563(n), A028347(n).
%C A236203 A175628 gives the numerators of interleaved Lyman and Balmer series, i.e., A005563(n)/A000290(n+1) and A061037(n+2)/A061038(n+2).
%C A236203 Difference table of a(n):
%C A236203    -1,  -3,  0,  0,  3,   5,   8,  12,  15,  21,  24, ...
%C A236203    -2,   3,  0,  3,  2,   3,   4,   3,   6,   3,   8, ...
%C A236203     5,  -3,  3, -1,  1,   1,  -1,   3,  -3,   5,  -5, ...
%C A236203    -8,   6, -4,  2,  0,  -2,   4,  -6,   8, -10,  12, ...
%C A236203    14, -10,  6, -2, -2,   6, -10,  14, -18,  22, -26, ...
%C A236203   -24,  16, -8,  0,  8, -16,  24, -32,  40, -48,  56, ... .
%C A236203 a(n+2) gives the numerators of 0/1, 0/16, 3/4, 5/36, 8/9, 12/64, 15/16, 21/100, 24/25, 32/144, ... . The denominators are A097362(n+1)^2. (Compare A097362 to A029578.)
%C A236203 Note the particular distribution of a(-n). Example:
%C A236203 a(n-9) = 12,15, 5,8, 0,3, -3,0, -4,-1, -3,0, 0,3, 5,8, 12,15, ... .
%C A236203 a(2n) + a(2n+1) = a(-2n-1) + a(-2n-2) = -4,0,8,20,36,56,80,... = 4*A000096(n-1).
%C A236203 a(2n) + a(2n-1) = a(-2n) + a(-2n-1) = -5,-3,3,13,... = A001105(n) - A010716(n).
%H A236203 Vincenzo Librandi, <a href="/A236203/b236203.txt">Table of n, a(n) for n = 2..1000</a>
%H A236203 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A236203 a(n+2) = (period 8: repeat 1, 16, 1, 1, 1, 4, 1, 1)*A175628(n+1).
%F A236203 a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
%F A236203 a(n+4) - a(n-4) = 0, 8, 8, ... = A168397.
%F A236203 From _Colin Barker_, Jan 26 2014: (Start)
%F A236203 a(n) = (n^2 -4)/4 for n even, a(n) = (n^2 +2*n -15)/4 for n odd.
%F A236203 G.f.: x^4*(3 + 2*x - 3*x^2)/ ((1-x)^3*(1+x)^2). (End)
%F A236203 a(n) = (2*n^2 + 2*n - 19 - (2*n - 11)*(-1)^n)/8. - _Luce ETIENNE_, Jul 26 2014
%F A236203 Sum_{n>=4} (-1)^n/a(n) = 11/48. - _Amiram Eldar_, Aug 21 2022
%p A236203 seq( (2*n^2 +2*n -19 -(2*n-11)*(-1)^n)/8, n=2..60); # _G. C. Greubel_, Dec 04 2019
%t A236203 CoefficientList[Series[x^2(3x^2-2x-3)/((x-1)^3(x+1)^2), {x, 0, 60}], x] (* _Vincenzo Librandi_, Jul 27 2014 *)
%t A236203 LinearRecurrence[{1,2,-2,-1,1},{0,0,3,5,8},60] (* _Harvey P. Dale_, Aug 30 2018 *)
%o A236203 (PARI) concat([0,0], Vec(x^4*(3*x^2-2*x-3)/((x-1)^3*(x+1)^2) + O(x^60))) \\ _Colin Barker_, Jan 26 2014
%o A236203 (Magma) [(2*n^2 + 2*n - 19 - (2*n - 11)*(-1)^n)/8: n in [2..60]]; // _Vincenzo Librandi_, Jul 27 2014
%o A236203 (Sage) [(2*n^2 +2*n -19 -(2*n-11)*(-1)^n)/8 for n in (2..60)] # _G. C. Greubel_, Dec 04 2019
%o A236203 (GAP) List([2..60], n-> (2*n^2 +2*n -19 -(2*n-11)*(-1)^n)/8 ); # _G. C. Greubel_, Dec 04 2019
%Y A236203 Cf. A005843, A008590, A016825, A109613, A147658.
%K A236203 nonn,easy
%O A236203 2,3
%A A236203 _Paul Curtz_, Jan 20 2014
%E A236203 More terms from _Colin Barker_, Jan 26 2014