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%I #35 Oct 25 2023 09:29:34
%S 0,1,14,3,28,5,42,7,56,9,70,11,84,13,98,15,112,17,126,19,140,21,154,
%T 23,168,25,182,27,196,29,210,31,224,33,238,35,252,37,266,39,280,41,
%U 294,43,308,45,322,47,336,49,350,51,364,53,378,55,392,57,406,59,420,61,434,63,448,65,462,67,476,69
%N Multiples of 14 and odd numbers interleaved.
%C Partial sums give the generalized 18-gonal numbers (A274979).
%C a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 18-gonal numbers.
%H Colin Barker, <a href="/A317314/b317314.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) = 14*n, a(2n+1) = 2*n + 1.
%F From _Colin Barker_, Jul 29 2018: (Start)
%F G.f.: x*(1 + 14*x + x^2) / ((1 - x)^2*(1 + x)^2).
%F a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
%F a(n) = 4*n + 3*n*(-1)^n. - _Wesley Ivan Hurt_, Nov 25 2021
%F Multiplicative with a(2^e) = 7*2^e, and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023
%F Dirichlet g.f.: zeta(s-1) * (1 + 3*2^(2-s)). - _Amiram Eldar_, Oct 25 2023
%t Table[4 n + 3 n (-1)^n, {n, 0, 80}] (* _Wesley Ivan Hurt_, Nov 25 2021 *)
%o (PARI) a(n) = if(n%2==0, return(14*n/2), return(n)) \\ _Felix Fröhlich_, Jul 26 2018
%o (PARI) concat(0, Vec(x*(1 + 14*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Jul 29 2018
%Y Cf. A008596 and A005408 interleaved.
%Y Column 14 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), A317311 (k=15), A317312 (k=16), A317313 (k=17).
%Y Cf. A274979.
%K nonn,easy,mult
%O 0,3
%A _Omar E. Pol_, Jul 25 2018
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