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A317312 Multiples of 12 and odd numbers interleaved.

%I A317312 #39 Dec 11 2024 05:35:30
%S A317312 0,1,12,3,24,5,36,7,48,9,60,11,72,13,84,15,96,17,108,19,120,21,132,23,
%T A317312 144,25,156,27,168,29,180,31,192,33,204,35,216,37,228,39,240,41,252,
%U A317312 43,264,45,276,47,288,49,300,51,312,53,324,55,336,57,348,59,360,61,372,63,384,65,396,67,408,69
%N A317312 Multiples of 12 and odd numbers interleaved.
%C A317312 Partial sums give the generalized 16-gonal numbers (A274978).
%C A317312 a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 16-gonal numbers.
%H A317312 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A317312 a(2n) = 12*n, a(2n+1) = 2*n + 1.
%F A317312 From _Michael De Vlieger_, Jul 26 2018: (Start)
%F A317312 G.f.: x*(1 + 12*x + x^2) / ((1 - x)^2*(1 + x)^2).
%F A317312 a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
%F A317312 Multiplicative with a(2^e) = 3*2^(e+1), and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023
%F A317312 Dirichlet g.f.: zeta(s-1) * (1 + 5*2^(1-s)). - _Amiram Eldar_, Oct 25 2023
%F A317312 a(n) = (7 + 5*(-1)^n)*n/2. - _Aaron J Grech_, Aug 20 2024
%t A317312 {0}~Join~Riffle[2 Range@ # - 1, 12 Range@ #] &@ 35 (* or *)
%t A317312 CoefficientList[Series[x (1 + 12 x + x^2)/((1 - x)^2*(1 + x)^2), {x, 0, 69}], x] (* or *)
%t A317312 LinearRecurrence[{0, 2, 0, -1}, {0, 1, 12, 3}, 70] (* _Michael De Vlieger_, Jul 26 2018 *)
%Y A317312 Cf. A008594 and A005408 interleaved.
%Y A317312 Column 12 of A195151.
%Y A317312 Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14), A317311 (k=15).
%Y A317312 Cf. A274978.
%K A317312 nonn,easy,mult
%O A317312 0,3
%A A317312 _Omar E. Pol_, Jul 25 2018