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%I #48 Sep 08 2022 08:45:59
%S 3,4,6,8,9,12,12,16,15,20,18,24,21,28,24,32,27,36,30,40,33,44,36,48,
%T 39,52,42,56,45,60,48,64,51,68,54,72,57,76,60,80,63,84,66,88,69,92,72,
%U 96,75,100,78,104,81,108,84,112,87,116,90,120,93,124,96,128
%N Multiples of 3 and of 4 interleaved: a(2*n-1) = 3*n, a(2*n) = 4*n.
%C First differences of A195020.
%C 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 [3, 4, 5]. The spiral contains infinitely many Pythagorean triples in which the hypotenuses are the positives A008587. Zero together with partial sums give A195020; the vertices of the spiral.
%H Vincenzo Librandi, <a href="/A195019/b195019.txt">Table of n, a(n) for n = 1..10000</a>
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html#uadgen">Pythagorean triangles and triples</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F pair(3*n, 4*n).
%F a(2*n-1) = 3*n, a(2*n) = 4*n. - _M. F. Hasler_, Sep 08 2011
%F G.f.: x*(3+4*x) / ( (x-1)^2*(1+x)^2 ). - _R. J. Mathar_, Sep 09 2011
%F From _Bruno Berselli_, Sep 12 2011: (Start)
%F a(n) = ((n-3)*(-1)^n + 7*n + 3)/4.
%F a(n) + a(n+1) = A047355(n+2). (End)
%F E.g.f.: (1/4)*((3 + 7*x)*exp(x) - (3 + x)*exp(-x)). - _G. C. Greubel_, Aug 19 2017
%t Table[((n-3)*(-1)^n + 7*n + 3)/4, {n,1,50}] (* _G. C. Greubel_, Aug 19 2017 *)
%o (PARI) a(n)=(n+1)\2*(4-n%2) \\ _M. F. Hasler_, Sep 08 2011
%o (Magma) [((n-3)*(-1)^n+7*n+3)/4: n in [1..60]]; // _Vincenzo Librandi_, Sep 12 2011
%Y Cf. A008585, A008586, A026741, A195020-A195029, A195031, A195033, A195035.
%K nonn,easy
%O 1,1
%A _Omar E. Pol_, Sep 07 2011, Sep 12 2011
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