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A144459 a(n) = (3*n+1)*(5*n+1).

%I A144459 #31 Sep 17 2023 10:00:07
%S A144459 1,24,77,160,273,416,589,792,1025,1288,1581,1904,2257,2640,3053,3496,
%T A144459 3969,4472,5005,5568,6161,6784,7437,8120,8833,9576,10349,11152,11985,
%U A144459 12848,13741,14664,15617,16600,17613,18656,19729,20832,21965,23128,24321,25544,26797,28080
%N A144459 a(n) = (3*n+1)*(5*n+1).
%C A144459 This appears in a "diagonal" scan through the numerators of the fractions of the hydrogen spectrum: A005563(4), A061037(9), A061039(13), etc.
%C A144459 a(n) mod 9 is a sequence of period length 9: repeat 1, 6, 5, 7, 3, 2, 4, 0, 8 (a permutation of A142069).
%H A144459 Vincenzo Librandi, <a href="/A144459/b144459.txt">Table of n, a(n) for n = 0..10000</a>
%H A144459 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A144459 a(n) = A016777(n)*A016861(n).
%F A144459 a(n) mod 10 = A131579(n+7).
%F A144459 G.f.: (1+21*x+8*x^2) / (1-x)^3 . - _R. J. Mathar_, Jul 01 2011
%F A144459 a(0)=1, a(1)=24, a(2)=77, a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - _Harvey P. Dale_, May 02 2015
%F A144459 E.g.f.: (1 + 23*x + 15*x^2)*exp(x). - _G. C. Greubel_, Sep 20 2018
%F A144459 Sum_{n>=0} 1/a(n) = sqrt(2 + 3/sqrt(5) - sqrt(3 + 6/sqrt(5)))*Pi/(2*sqrt(6)) + sqrt(5)*log(phi)/4 + 5*log(5)/8 - 3*log(3)/4, where phi is the golden ratio (A001622). - _Amiram Eldar_, Sep 17 2023
%t A144459 Table[(3n+1)(5n+1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,24 ,77}, 50] (* _Harvey P. Dale_, Jul 16 2014 *)
%o A144459 (Magma) [(3*n+1)*(5*n+1): n in [0..40]]; // _Vincenzo Librandi_, Aug 07 2011
%o A144459 (PARI) a(n)=(3*n+1)*(5*n+1) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A144459 Cf. A001622, A005563, A016777, A016861, A061037, A061039, A131579, A142069.
%K A144459 nonn,easy
%O A144459 0,2
%A A144459 _Paul Curtz_, Oct 08 2008