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A317317 Multiples of 17 and odd numbers interleaved.

%I A317317 #33 Dec 11 2024 05:35:08
%S A317317 0,1,17,3,34,5,51,7,68,9,85,11,102,13,119,15,136,17,153,19,170,21,187,
%T A317317 23,204,25,221,27,238,29,255,31,272,33,289,35,306,37,323,39,340,41,
%U A317317 357,43,374,45,391,47,408,49,425,51,442,53,459,55,476,57,493,59,510,61,527,63,544,65,561,67,578,69
%N A317317 Multiples of 17 and odd numbers interleaved.
%C A317317 Partial sums give the generalized 21-gonal numbers (A303298).
%C A317317 a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 21-gonal numbers.
%H A317317 Colin Barker, <a href="/A317317/b317317.txt">Table of n, a(n) for n = 0..1000</a>
%H A317317 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A317317 a(2n) = 17*n, a(2n+1) = 2*n + 1.
%F A317317 From _Colin Barker_, Jul 29 2018: (Start)
%F A317317 G.f.: x*(1 + 17*x + x^2) / ((1 - x)^2*(1 + x)^2).
%F A317317 a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
%F A317317 Multiplicative with a(2^e) = 17*2^(e-1), and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023
%F A317317 Dirichlet g.f.: zeta(s-1) * (1 + 15/2^s). - _Amiram Eldar_, Oct 25 2023
%t A317317 With[{nn=40},Riffle[17*Range[0,nn],2*Range[0,nn]+1]] (* or *) LinearRecurrence[ {0,2,0,-1},{0,1,17,3},80] (* _Harvey P. Dale_, Jun 06 2020 *)
%o A317317 (PARI) concat(0, Vec(x*(1 + 17*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Jul 29 2018
%Y A317317 Cf. A008599 and A005408 interleaved.
%Y A317317 Column 17 of A195151.
%Y A317317 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 A317317 Cf. A303298.
%K A317317 nonn,easy,mult
%O A317317 0,3
%A A317317 _Omar E. Pol_, Jul 25 2018