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%I #37 Oct 25 2023 09:29:51
%S 0,1,9,3,18,5,27,7,36,9,45,11,54,13,63,15,72,17,81,19,90,21,99,23,108,
%T 25,117,27,126,29,135,31,144,33,153,35,162,37,171,39,180,41,189,43,
%U 198,45,207,47,216,49,225,51,234,53,243,55,252,57,261,59,270,61
%N Multiples of 9 and odd numbers interleaved.
%C Partial sums give the generalized 13-gonal (or tridecagonal) numbers A195313.
%C a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized 13-gonal numbers. - _Omar E. Pol_, Jul 27 2018
%H Vincenzo Librandi, <a href="/A195312/b195312.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F From _Bruno Berselli_, Sep 15 2011: (Start)
%F G.f.: x*(1+9*x+x^2)/((1-x)^2*(1+x)^2).
%F a(n) = (7*(-1)^n+11)*n/4.
%F a(n) + a(n-1) = A175885(n).
%F Sum_{i=0..n} a(i) = A195313(n). (End)
%F Multiplicative with a(2^e) = 9*2^(e-1), a(p^e) = p^e for odd prime p. - _Andrew Howroyd_, Jul 23 2018
%F Dirichlet g.f.: zeta(s-1) * (1 + 7/2^s). - _Amiram Eldar_, Oct 25 2023
%t With[{nn=30},Riffle[9Range[0,nn],Range[1,2nn+1,2]]] (* _Harvey P. Dale_, Sep 24 2011 *)
%o (Magma) /* By definition */ &cat[[9*n,2*n+1]: n in [0..33]]; // _Bruno Berselli_, Sep 16 2011
%o (PARI) a(n)=(7*(-1)^n+11)*n/4 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Column 9 of A195151.
%Y Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, A195161, this sequence.
%Y Cf. A195313, A175885.
%K nonn,easy,mult
%O 0,3
%A _Omar E. Pol_, Sep 14 2011
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