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A195031 Multiples of 5 and of 12 interleaved: a(2n-1) = 5n, a(2n) = 12n.

%I A195031 #33 Feb 16 2025 08:33:15
%S A195031 5,12,10,24,15,36,20,48,25,60,30,72,35,84,40,96,45,108,50,120,55,132,
%T A195031 60,144,65,156,70,168,75,180,80,192,85,204,90,216,95,228,100,240,105,
%U A195031 252,110,264,115,276,120,288,125,300,130,312,135,324,140,336,145,348
%N A195031 Multiples of 5 and of 12 interleaved: a(2n-1) = 5n, a(2n) = 12n.
%C A195031 First differences of A195032.
%C A195031 a(n) is also the length of the n-th edge of a square spiral in which the first two edges are the legs of the primitive Pythagorean triple [5, 12, 13]. Zero together with partial sums give A195032, the vertices of the spiral.
%H A195031 Vincenzo Librandi, <a href="/A195031/b195031.txt">Table of n, a(n) for n = 1..10000</a>
%H A195031 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html#uadgen">Pythagorean triangles and Triples</a>
%H A195031 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%H A195031 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A195031 From _Bruno Berselli_, Sep 30 2011:  (Start)
%F A195031 G.f.: x*(5+12*x)/((1-x)^2*(1+x)^2).
%F A195031 a(n) = ((17+7*(-1)^n)/2)*((2*n-(-1)^n+1)/4) = (17*n+(7*n-5)*(-1)^n+5)/4.
%F A195031 a(n)*a(n+1) = a(10*s), where s is A002620(n+1).
%F A195031 a(n) = 2*a(n-2) - a(n-4). (End)
%t A195031 With[{nn=30},Riffle[5Range[nn],12Range[nn]]] (* or *) LinearRecurrence[ {0,2,0,-1},{5,12,10,24},60] (* _Harvey P. Dale_, Aug 18 2012 *)
%o A195031 (Magma) &cat[[5*n, 12*n]: n in [1..27]];  // _Bruno Berselli_, Sep 30 2011
%o A195031 (PARI) a(n)=(n+1)\2*if(n%2,5,12) \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A195031 Cf. A195019, A195032, A195033, A195035.
%K A195031 nonn,easy
%O A195031 1,1
%A A195031 _Omar E. Pol_, Sep 12 2011
%E A195031 More terms from _Bruno Berselli_, Sep 30 2011