cp's OEIS Frontend

A317316 Multiples of 16 and odd numbers interleaved.

%I A317316 #31 Dec 11 2024 05:35:14
%S A317316 0,1,16,3,32,5,48,7,64,9,80,11,96,13,112,15,128,17,144,19,160,21,176,
%T A317316 23,192,25,208,27,224,29,240,31,256,33,272,35,288,37,304,39,320,41,
%U A317316 336,43,352,45,368,47,384,49,400,51,416,53,432,55,448,57,464,59,480,61,496,63,512,65,528,67,544,69
%N A317316 Multiples of 16 and odd numbers interleaved.
%C A317316 Partial sums give the generalized 20-gonal numbers (A218864).
%C A317316 a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 20-gonal numbers.
%H A317316 Colin Barker, <a href="/A317316/b317316.txt">Table of n, a(n) for n = 0..1000</a>
%H A317316 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A317316 a(2n) = 16*n, a(2n+1) = 2*n + 1.
%F A317316 From _Colin Barker_, Jul 29 2018: (Start)
%F A317316 G.f.: x*(1 + 16*x + x^2) / ((1 - x)^2*(1 + x)^2).
%F A317316 a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
%F A317316 Multiplicative with a(2^e) = 2^(e+3), and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023
%F A317316 Dirichlet g.f.: zeta(s-1) * (1 + 7*2^(1-s)). - _Amiram Eldar_, Oct 25 2023
%t A317316 a[n_] := If[OddQ[n], n, 8*n]; Array[a, 70, 0] (* _Amiram Eldar_, Oct 14 2023 *)
%o A317316 (PARI) concat(0, Vec(x*(1 + 16*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Jul 29 2018
%Y A317316 Cf. A008598 and A005408 interleaved.
%Y A317316 Column 16 of A195151.
%Y A317316 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 A317316 Cf. A218864.
%K A317316 nonn,easy,mult
%O A317316 0,3
%A A317316 _Omar E. Pol_, Jul 25 2018