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 A195161 Multiples of 8 and odd numbers interleaved. 26

%I

%S 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 25,104,27,112,29,120,31,128,33,136,35,144,37,152,39,160,41,168,43,

%U 176,45,184,47,192,49,200,51,208,53,216,55,224,57,232,59

%N Multiples of 8 and odd numbers interleaved.

%C A008590 and A005408 interleaved. This is 8*n if n is even, n if n is odd, if n>=0.

%C Partial sums give the generalized 12-gonal (or dodecagonal) numbers A195162.

%C 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 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 Vincenzo Librandi, <a href="/A195161/b195161.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).

%F a(2n) = 8n, a(2n+1) = 2n+1. [corrected by _Omar E. Pol_, Jul 26 2018]

%F a(n) = (6*(-1)^n+10)*n/4. - _Vincenzo Librandi_, Sep 27 2011

%F 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 From _Ilya Gutkovskiy_, Jul 03 2016: (Start)

%F a(m*2^k) = m*2^(k+2), k>0.

%F E.g.f.: x*(4*sinh(x) + cosh(x)).

%F Dirichlet g.f.: 2^(-s)*(2^s + 6)*zeta(s-1). (End)

%F Multiplicative with a(2^e) = 4*2^e, a(p^e) = p^e for odd prime p. - _Andrew Howroyd_, Jul 23 2018

%F a(n) = A144433(n-1) for n > 1. - _Georg Fischer_, Oct 14 2018

%p a := proc(n): (6*(-1)^n+10)*n/4 end: seq(a(n), n=0..59); # _Johannes W. Meijer_, Jul 03 2016

%t 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 (MAGMA) &cat[[8*n, 2*n+1]: n in [0..30]]; // _Vincenzo Librandi_, Sep 27 2011

%o (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 Column 8 of A195151.

%Y 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 Cf. A144433.

%K nonn,easy,mult

%O 0,3

%A _Omar E. Pol_, Sep 10 2011

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Last modified August 24 03:20 EDT 2019. Contains 326260 sequences. (Running on oeis4.)