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A171372 a(n) = Numerator of 1/(2*n)^2 - 1/(3*n)^2 for n > 0, a(0) = 1.

%I A171372 #22 Dec 12 2023 07:39:09
%S A171372 1,5,5,5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,
%T A171372 5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,
%U A171372 5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,5,1,5,5,5,5
%N A171372 a(n) = Numerator of 1/(2*n)^2 - 1/(3*n)^2 for n > 0, a(0) = 1.
%C A171372 The diagonal of a table of numerators of the Rydberg-Ritz spectrum of hydrogen:
%C A171372   1,  1,  1,   1,  1,   1,  1,   1,  1,   1,   1, ... A000012
%C A171372   0,  5,  3,  21,  2,  45, 15,  77,  6, 117,  35, ... A061037
%C A171372   0,  9,  5,  33,  3,  65, 21, 105,  1, 153,  45, ... A061041
%C A171372   0, 13,  7,   5,  4,  85,  1, 133, 10,   7,  55, ... A061045
%C A171372   0, 17,  9,  57,  5, 105, 33, 161,  3, 225,  65, ... A061049
%C A171372   0, 21, 11,  69,  6,   1, 39, 189, 14, 261,   3, ...
%C A171372   0, 25, 13,   1,  7, 145,  5, 217,  1,  11,  85, ...
%C A171372   0, 29, 15,  93,  8, 165, 51,   5, 18, 333,  95, ...
%C A171372   0, 33, 17, 105,  9, 185, 57, 273,  5, 369, 105, ...
%C A171372   0, 37, 19,  13, 10, 205,  7, 301, 22,   5, 115, ...
%C A171372   0, 41, 21, 129, 11,   9, 69, 329,  3, 441,   1, ...
%C A171372 In that respect, constructed similar to A144437.
%H A171372 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,1).
%F A171372 a(n) = numerator of 5/(6*n)^2 .
%F A171372 Period 5: repeat [1,5,5,5,5].
%F A171372 G.f.: (1 + 5*x + 5*x^2 + 5*x^3 + 5*x^4)/((1-x)*(1 + x + x^2 + x^3 + x^4)).
%F A171372 a(n) = 1 + 4*sign(n mod 5). - _Wesley Ivan Hurt_, Sep 26 2018
%F A171372 a(n) = (21-8*cos(2*n*Pi/5)-8*cos(4*n*Pi/5))/5. - _Wesley Ivan Hurt_, Sep 27 2018
%t A171372 Table[If[n==0,1,Numerator[5/(6*n)^2]], {n,0,100}] (* _G. C. Greubel_, Sep 20 2018 *)
%o A171372 (PARI) concat([1], vector(100, n, numerator(5/(6*n)^2))) \\ _G. C. Greubel_, Sep 20 2018
%o A171372 (Magma) [1] cat [Numerator(5/(6*n)^2): n in [1..100]]; // _G. C. Greubel_, Sep 20 2018
%Y A171372 Cf. A171373 (binomial transform), A171408, A105371.
%K A171372 nonn,easy,frac
%O A171372 0,2
%A A171372 _Paul Curtz_, Dec 07 2009
%E A171372 Edited by _R. J. Mathar_, Dec 15 2009