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A317321 Multiples of 21 and odd numbers interleaved.

%I A317321 #28 Dec 11 2024 06:38:10
%S A317321 0,1,21,3,42,5,63,7,84,9,105,11,126,13,147,15,168,17,189,19,210,21,
%T A317321 231,23,252,25,273,27,294,29,315,31,336,33,357,35,378,37,399,39,420,
%U A317321 41,441,43,462,45,483,47,504,49,525,51,546,53,567,55,588,57,609,59,630,61,651,63,672,65,693,67,714,69
%N A317321 Multiples of 21 and odd numbers interleaved.
%C A317321 Partial sums give the generalized 25-gonal numbers (A303304).
%C A317321 a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 25-gonal numbers.
%H A317321 Colin Barker, <a href="/A317321/b317321.txt">Table of n, a(n) for n = 0..1000</a>
%H A317321 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A317321 a(2n) = 21*n, a(2n+1) = 2*n + 1.
%F A317321 Multiplicative with a(2^e) = 21*2^(e-1), and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023
%F A317321 Dirichlet g.f.: zeta(s-1) * (1 + 19/2^s). - _Amiram Eldar_, Oct 26 2023
%t A317321 a[n_] := If[OddQ[n], n, 21*n/2]; Array[a, 70, 0] (* _Amiram Eldar_, Oct 14 2023 *)
%o A317321 (PARI) concat(0, Vec(x*(1 + 21*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Jul 29 2018
%Y A317321 Cf. A008603 and A005408 interleaved.
%Y A317321 Column 21 of A195151.
%Y A317321 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 A317321 Cf. A303304.
%K A317321 nonn,easy,mult
%O A317321 0,3
%A A317321 _Omar E. Pol_, Jul 25 2018