This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A195161 #76 Sep 08 2022 08:45:59
%S A195161 0,1,8,3,16,5,24,7,32,9,40,11,48,13,56,15,64,17,72,19,80,21,88,23,96,
%T A195161 25,104,27,112,29,120,31,128,33,136,35,144,37,152,39,160,41,168,43,
%U A195161 176,45,184,47,192,49,200,51,208,53,216,55,224,57,232,59
%N A195161 Multiples of 8 and odd numbers interleaved.
%C A195161 A008590 and A005408 interleaved. This is 8*n if n is even, n if n is odd, if n>=0.
%C A195161 Partial sums give the generalized 12-gonal (or dodecagonal) numbers A195162.
%C A195161 The moment generating function of p(x, m=2, n=1, mu=2) = 4*x*E(x, 2, 1), see A163931 and A274181, is given by M(a) = (- 4*log(1-a) - 4 * polylog(2, a))/a^2. The series expansion of M(a) leads to the sequence given above. - _Johannes W. Meijer_, Jul 03 2016
%C A195161 a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 12-gonal numbers. - _Omar E. Pol_, Jul 27 2018
%H A195161 Vincenzo Librandi, <a href="/A195161/b195161.txt">Table of n, a(n) for n = 0..10000</a>
%H A195161 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A195161 a(2n) = 8n, a(2n+1) = 2n+1. [corrected by _Omar E. Pol_, Jul 26 2018]
%F A195161 a(n) = (6*(-1)^n+10)*n/4. - _Vincenzo Librandi_, Sep 27 2011
%F A195161 a(n) = 2*a(n-2)-a(n-4). G.f.: x*(1+8*x+x^2)/((1-x)^2*(1+x)^2). - _Colin Barker_, Aug 11 2012
%F A195161 From _Ilya Gutkovskiy_, Jul 03 2016: (Start)
%F A195161 a(m*2^k) = m*2^(k+2), k>0.
%F A195161 E.g.f.: x*(4*sinh(x) + cosh(x)).
%F A195161 Dirichlet g.f.: 2^(-s)*(2^s + 6)*zeta(s-1). (End)
%F A195161 Multiplicative with a(2^e) = 4*2^e, a(p^e) = p^e for odd prime p. - _Andrew Howroyd_, Jul 23 2018
%F A195161 a(n) = A144433(n-1) for n > 1. - _Georg Fischer_, Oct 14 2018
%p A195161 a := proc(n): (6*(-1)^n+10)*n/4 end: seq(a(n), n=0..59); # _Johannes W. Meijer_, Jul 03 2016
%t A195161 With[{nn=30},Riffle[8*Range[0,nn],2*Range[0,nn]+1]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,8,3},60] (* _Harvey P. Dale_, Nov 24 2013 *)
%o A195161 (Magma) &cat[[8*n, 2*n+1]: n in [0..30]]; // _Vincenzo Librandi_, Sep 27 2011
%o A195161 (PARI) concat(0, Vec(x*(1+8*x+x^2)/((1-x)^2*(1+x)^2) + O(x^99))) \\ _Altug Alkan_, Jul 04 2016
%Y A195161 Column 8 of A195151.
%Y A195161 Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, this sequence, A195312.
%Y A195161 Cf. A144433.
%K A195161 nonn,easy,mult
%O A195161 0,3
%A A195161 _Omar E. Pol_, Sep 10 2011