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A317320 Multiples of 20 and odd numbers interleaved.

%I A317320 #35 Dec 11 2024 05:34:58
%S A317320 0,1,20,3,40,5,60,7,80,9,100,11,120,13,140,15,160,17,180,19,200,21,
%T A317320 220,23,240,25,260,27,280,29,300,31,320,33,340,35,360,37,380,39,400,
%U A317320 41,420,43,440,45,460,47,480,49,500,51,520,53,540,55,560,57,580,59,600,61,620,63,640,65,660,67,680,69
%N A317320 Multiples of 20 and odd numbers interleaved.
%C A317320 Partial sums give the generalized 24-gonal numbers (A303814).
%C A317320 a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 24-gonal numbers.
%H A317320 Colin Barker, <a href="/A317320/b317320.txt">Table of n, a(n) for n = 0..1000</a>
%H A317320 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A317320 a(n) = n, if n is odd.
%F A317320 a(n) = 10*n, if n is even.
%F A317320 a(2n) = 20*n, a(2n+1) = 2*n + 1.
%F A317320 From _Colin Barker_, Jul 29 2018: (Start)
%F A317320 G.f.: x*(1 + 20*x + x^2) / ((1 - x)^2*(1 + x)^2).
%F A317320 a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
%F A317320 Multiplicative with a(2^e) = 5*2^(e+1), and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023
%F A317320 Dirichlet g.f.: zeta(s-1) * (1 + 9*2^(1-s)). - _Amiram Eldar_, Oct 26 2023
%t A317320 With[{nn=40},Riffle[20*Range[0,nn],Range[1,2*nn+1,2]]] (* _Harvey P. Dale_, Feb 16 2020 *)
%o A317320 (PARI) concat(0, Vec(x*(1 + 20*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Jul 29 2018
%Y A317320 Cf. A008602 and A005408 interleaved.
%Y A317320 Column 20 of A195151.
%Y A317320 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 A317320 Cf. A303814.
%K A317320 nonn,easy,mult
%O A317320 0,3
%A A317320 _Omar E. Pol_, Jul 25 2018