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%I #27 Oct 26 2023 08:30:44
%S 0,1,19,3,38,5,57,7,76,9,95,11,114,13,133,15,152,17,171,19,190,21,209,
%T 23,228,25,247,27,266,29,285,31,304,33,323,35,342,37,361,39,380,41,
%U 399,43,418,45,437,47,456,49,475,51,494,53,513,55,532,57,551,59,570,61,589,63,608,65,627,67,646,69
%N Multiples of 19 and odd numbers interleaved.
%C Partial sums give the generalized 23-gonal numbers (A303303).
%C a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 23-gonal numbers.
%H Colin Barker, <a href="/A317319/b317319.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F a(2n) = 19*n, a(2n+1) = 2*n + 1.
%F From _Colin Barker_, Jul 29 2018: (Start)
%F G.f.: x*(1 + 19*x + x^2) / ((1 - x)^2*(1 + x)^2).
%F a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
%F Multiplicative with a(2^e) = 19*2^(e-1), and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023
%F Dirichlet g.f.: zeta(s-1) * (1 + 17/2^s). - _Amiram Eldar_, Oct 26 2023
%t a[n_] := If[OddQ[n], n, 19*n/2]; Array[a, 70, 0] (* _Amiram Eldar_, Oct 14 2023 *)
%o (PARI) concat(0, Vec(x*(1 + 19*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Jul 29 2018
%Y Cf. A008601 and A005408 interleaved.
%Y Column 19 of A195151.
%Y 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 Cf. A303303.
%K nonn,easy,mult
%O 0,3
%A _Omar E. Pol_, Jul 25 2018
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