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A317318 Multiples of 18 and odd numbers interleaved.

%I A317318 #30 Dec 11 2024 05:35:05
%S A317318 0,1,18,3,36,5,54,7,72,9,90,11,108,13,126,15,144,17,162,19,180,21,198,
%T A317318 23,216,25,234,27,252,29,270,31,288,33,306,35,324,37,342,39,360,41,
%U A317318 378,43,396,45,414,47,432,49,450,51,468,53,486,55,504,57,522,59,540,61,558,63,576,65,594,67,612,69
%N A317318 Multiples of 18 and odd numbers interleaved.
%C A317318 Partial sums give the generalized 22-gonal numbers (A303299).
%C A317318 a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 22-gonal numbers.
%H A317318 Colin Barker, <a href="/A317318/b317318.txt">Table of n, a(n) for n = 0..1000</a>
%H A317318 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A317318 a(2n) = 18*n, a(2n+1) = 2*n + 1.
%F A317318 From _Colin Barker_, Jul 29 2018: (Start)
%F A317318 G.f.: x*(1 + 18*x + x^2) / ((1 - x)^2*(1 + x)^2).
%F A317318 a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
%F A317318 Multiplicative with a(2^e) = 9*2^e, and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023
%F A317318 Dirichlet g.f.: zeta(s-1) * (1 + 2^(4-s)). - _Amiram Eldar_, Oct 25 2023
%t A317318 a[n_] := If[OddQ[n], n, 9*n]; Array[a, 70, 0] (* _Amiram Eldar_, Oct 14 2023 *)
%o A317318 (PARI) concat(0, Vec(x*(1 + 18*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Jul 29 2018
%Y A317318 Cf. A008600 and A005408 interleaved.
%Y A317318 Column 18 of A195151.
%Y A317318 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).
%Y A317318 Cf. A303299.
%K A317318 nonn,easy,mult
%O A317318 0,3
%A A317318 _Omar E. Pol_, Jul 25 2018