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A317323 Multiples of 23 and odd numbers interleaved.

%I A317323 #31 Dec 11 2024 06:27:00
%S A317323 0,1,23,3,46,5,69,7,92,9,115,11,138,13,161,15,184,17,207,19,230,21,
%T A317323 253,23,276,25,299,27,322,29,345,31,368,33,391,35,414,37,437,39,460,
%U A317323 41,483,43,506,45,529,47,552,49,575,51,598,53,621,55,644,57,667,59,690,61,713,63,736,65,759,67,782,69
%N A317323 Multiples of 23 and odd numbers interleaved.
%C A317323 Partial sums give the generalized 27-gonal numbers (A316725).
%C A317323 a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 27-gonal numbers.
%H A317323 Colin Barker, <a href="/A317323/b317323.txt">Table of n, a(n) for n = 0..1000</a>
%H A317323 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A317323 a(2n) = 23*n, a(2n+1) = 2*n + 1.
%F A317323 From _Colin Barker_, Jul 29 2018: (Start)
%F A317323 G.f.: x*(1 + 23*x + x^2) / ((1 - x)^2*(1 + x)^2).
%F A317323 a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
%F A317323 Multiplicative with a(2^e) = 23*2^(e-1), and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023
%F A317323 Dirichlet g.f.: zeta(s-1) * (1 + 21/2^s). - _Amiram Eldar_, Oct 26 2023
%t A317323 With[{nn=40},Riffle[23*Range[0,nn],Range[1,2*nn,2]]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,23,3},80] (* _Harvey P. Dale_, May 05 2019 *)
%o A317323 (PARI) concat(0, Vec(x*(1 + 23*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Jul 29 2018
%Y A317323 Cf. A008605 and A005408 interleaved.
%Y A317323 Column 23 of A195151.
%Y A317323 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 A317323 Cf. A316725.
%K A317323 nonn,easy,mult
%O A317323 0,3
%A A317323 _Omar E. Pol_, Jul 25 2018