|
%I #34 Oct 25 2023 09:29:48
%S 0,1,10,3,20,5,30,7,40,9,50,11,60,13,70,15,80,17,90,19,100,21,110,23,
%T 120,25,130,27,140,29,150,31,160,33,170,35,180,37,190,39,200,41,210,
%U 43,220,45,230,47,240,49,250,51,260,53,270,55,280,57,290,59,300
%N Multiples of 10 and odd numbers interleaved.
%C A008592 and A005408 interleaved.
%C Partial sums give the generalized 14-gonal (or tetradecagonal) numbers A195818.
%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 14-gonal numbers. - _Omar E. Pol_, Jul 27 2018
%H Vincenzo Librandi, <a href="/A195817/b195817.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 a(n) = (2*(-1)^n+3)*n. - _Vincenzo Librandi_, Sep 30 2011
%F From _Bruno Berselli_, Sep 30 2011: (Start)
%F G.f.: x*(1+10*x+x^2))/((1-x)^2*(1+x)^2).
%F a(n) = -a(-n) = a(n-2)*n/(n-2) = 2*a(n-2)-a(n-4).
%F a(n) * a(n+1) = a(n(n+1)).
%F a(n) + a(n+1) = A091998(n+1). (End)
%F a(0)=0, a(1)=1, a(2)=10, a(3)=3, a(n)=2*a(n-2)-a(n-4). - _Harvey P. Dale_, Nov 24 2013
%F Multiplicative with a(2^e) = 5*2^e, a(p^e) = p^e for odd prime p. - _Andrew Howroyd_, Jul 23 2018
%F Dirichlet g.f.: zeta(s-1) * (1 + 2^(3-s)). - _Amiram Eldar_, Oct 25 2023
%t With[{nn=30},Riffle[10*Range[0,nn],Range[1,2*nn+1,2]]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,10,3},70] (* _Harvey P. Dale_, Nov 24 2013 *)
%o (Magma) [(2*(-1)^n+3)*n: n in [0..60]]; // _Vincenzo Librandi_, Sep 30 2011
%o (PARI) a(n) = (2*(-1)^n+3)*n; \\ _Andrew Howroyd_, Jul 23 2018
%Y Column 10 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, A195312, this sequence.
%Y Cf. A091998, A195152.
%K nonn,easy,mult
%O 0,3
%A _Omar E. Pol_, Sep 29 2011
|
