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A317315 Multiples of 15 and odd numbers interleaved.

%I A317315 #36 Dec 11 2024 05:35:18
%S A317315 0,1,15,3,30,5,45,7,60,9,75,11,90,13,105,15,120,17,135,19,150,21,165,
%T A317315 23,180,25,195,27,210,29,225,31,240,33,255,35,270,37,285,39,300,41,
%U A317315 315,43,330,45,345,47,360,49,375,51,390,53,405,55,420,57,435,59,450,61,465,63,480,65,495,67,510,69
%N A317315 Multiples of 15 and odd numbers interleaved.
%C A317315 Partial sums give the generalized 19-gonal numbers (A303813).
%C A317315 a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 19-gonal numbers.
%H A317315 Colin Barker, <a href="/A317315/b317315.txt">Table of n, a(n) for n = 0..1000</a>
%H A317315 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A317315 a(2n) = 15*n, a(2n+1) = 2*n + 1.
%F A317315 From _Colin Barker_, Jul 29 2018: (Start)
%F A317315 G.f.: x*(1 + 15*x + x^2) / ((1 - x)^2*(1 + x)^2).
%F A317315 a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
%F A317315 Multiplicative with a(2^e) = 15*2^(e-1), and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023
%F A317315 Dirichlet g.f.: zeta(s-1) * (1 + 13/2^s). - _Amiram Eldar_, Oct 25 2023
%t A317315 a[n_] := If[OddQ[n], n, 15*n/2]; Array[a, 70, 0] (* _Amiram Eldar_, Oct 14 2023 *)
%o A317315 (PARI) concat(0, Vec(x*(1 + 15*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Jul 29 2018
%Y A317315 Cf. A008597 and A005408 interleaved.
%Y A317315 Column 15 of A195151.
%Y A317315 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 A317315 Cf. A303813.
%K A317315 nonn,easy,mult
%O A317315 0,3
%A A317315 _Omar E. Pol_, Jul 25 2018