A195031 - OEIS

%I #32 Sep 08 2022 08:45:59

%S 5,12,10,24,15,36,20,48,25,60,30,72,35,84,40,96,45,108,50,120,55,132,

%T 60,144,65,156,70,168,75,180,80,192,85,204,90,216,95,228,100,240,105,

%U 252,110,264,115,276,120,288,125,300,130,312,135,324,140,336,145,348

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

%C First differences of A195032.

%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 [5, 12, 13]. Zero together with partial sums give A195032, the vertices of the spiral.

%H Vincenzo Librandi, <a href="/A195031/b195031.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 From _Bruno Berselli_, Sep 30 2011:  (Start)

%F G.f.: x*(5+12*x)/((1-x)^2*(1+x)^2).

%F a(n) = ((17+7*(-1)^n)/2)*((2*n-(-1)^n+1)/4) = (17*n+(7*n-5)*(-1)^n+5)/4.

%F a(n)*a(n+1) = a(10*s), where s is A002620(n+1).

%F a(n) = 2*a(n-2) - a(n-4). (End)

%t 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 (Magma) &cat[[5*n, 12*n]: n in [1..27]];  // _Bruno Berselli_, Sep 30 2011

%o (PARI) a(n)=(n+1)\2*if(n%2,5,12) \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A195019, A195032, A195033, A195035.

%K nonn,easy

%O 1,1

%A _Omar E. Pol_, Sep 12 2011

%E More terms from _Bruno Berselli_, Sep 30 2011