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A195033 Multiples of 21 and of 20 interleaved: a(2n-1) = 21n, a(2n) = 20n.

%I A195033 #37 Feb 16 2025 08:33:15
%S A195033 21,20,42,40,63,60,84,80,105,100,126,120,147,140,168,160,189,180,210,
%T A195033 200,231,220,252,240,273,260,294,280,315,300,336,320,357,340,378,360,
%U A195033 399,380,420,400,441,420,462,440,483,460,504,480,525,500,546,520,567,540
%N A195033 Multiples of 21 and of 20 interleaved: a(2n-1) = 21n, a(2n) = 20n.
%C A195033 First differences of A195034.
%C A195033 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 [21, 20, 29]. Zero together with partial sums give A195034, the vertices of the spiral.
%H A195033 Vincenzo Librandi, <a href="/A195033/b195033.txt">Table of n, a(n) for n = 1..10000</a>
%H A195033 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html#uadgen">Pythagorean triangles and Triples</a>
%H A195033 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%H A195033 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A195033 From _Bruno Berselli_, Sep 29 2011:  (Start)
%F A195033 G.f.: x*(21+20*x)/((1-x)^2*(1+x)^2).
%F A195033 a(n) = A010693(n)*A010718(n)*A029578(n+1) = (41*n-(n+21)*(-1)^n+21)/4.
%F A195033 a(n) = 2*a(n-2) - a(n-4). (End)
%t A195033 Riffle[21*#, 20*#] & [Range[50]] (* _Paolo Xausa_, Mar 21 2024 *)
%o A195033 (Magma) &cat[[21*n,20*n]: n in [1..27]];  // _Bruno Berselli_, Sep 29 2011
%o A195033 (PARI) a(n)=(n+1)\2*if(n%2,21,20) \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A195033 Cf. A010693, A010718, A029578.
%Y A195033 Cf. A195019, A195031, A195034, A195035.
%K A195033 nonn,easy
%O A195033 1,1
%A A195033 _Omar E. Pol_, Sep 12 2011
%E A195033 More terms from _Bruno Berselli_, Sep 29 2011