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A114834 Each term is previous term plus floor of root mean square of two previous terms.

%I A114834 #9 Feb 16 2025 08:33:00
%S A114834 1,2,3,5,9,16,28,50,90,162,293,529,956,1728,3124,5648,10211,18462,
%T A114834 33380,60352,109119,197293,356716,644961,1166123,2108412,3812120,
%U A114834 6892514,12462029,22532007,40739059,73658371,133178227,240793271,435366958,787166465
%N A114834 Each term is previous term plus floor of root mean square of two previous terms.
%C A114834 What is this sequence and the ratio of adjacent terms, asymptotically? Primes in this sequence include 2, 3, 5, 293. Squares in this sequence include 9, 16, 529.
%H A114834 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Root-Mean-Square.html">Root-Mean-Square.</a>
%H A114834 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Mean.html">Mean.</a>
%F A114834 a(1) = 1, a(2) = 2, for n>2: a(n+1) = a(n) + floor(RMS[a(n),a(n-1)]). a(n+1) = a(n) + floor[Sqrt[[a(n)^2]+[a(n-1)^2]/2]].
%e A114834 a(3) = 2 + floor[sqrt[(1^2 + 2^2)/2]] = 2 + floor[Sqrt[5/2]] = 2 + 1 = 3.
%e A114834 a(4) = 3 + floor[sqrt[(2^2 + 3^2)/2]] = 4 + floor[Sqrt[13/2]] = 3 + 2 = 5.
%e A114834 a(5) = 5 + floor[sqrt[(3^2 + 5^2)/2]] = 8 + floor[Sqrt[34/2]] = 5 + 4 = 9.
%e A114834 a(6) = 9 + floor[sqrt[(5^2 + 9^2)/2]] = 15 + floor[Sqrt[106/2]] = 9 + 7 = 16.
%e A114834 a(7) = 16 + floor[sqrt[(9^2 + 16^2)/2]] = 15 + floor[Sqrt[337/2]] = 16 + 12 = 28.
%e A114834 a(8) = 28 + floor[sqrt[(16^2 + 28^2)/2]] = 15 + floor[Sqrt[1040/2]] = 28 + 22 = 50.
%e A114834 a(9) = 50 + floor[sqrt[(28^2 + 50^2)/2]] = 50 + floor[Sqrt[3284/2]] = 50 + 40 = 90.
%e A114834 a(10) = 90 + floor[sqrt[(50^2 + 90^2)/2]] = 50 + floor[Sqrt[10600/2]] = 90 + 72 = 162.
%e A114834 a(11) = 162 + floor[sqrt[(90^2 + 162^2)/2]] = 50 + floor[Sqrt[34344/2]] = 162 + 131 = 293.
%e A114834 a(12) = 293 + floor[sqrt[(162^2 + 293^2)/2]] = 293 + floor[Sqrt[112093/2]] = 293 + 236 = 529.
%p A114834 rms := proc(a,b)
%p A114834     sqrt((a^2+b^2)/2) ;
%p A114834 end proc:
%p A114834 A114834 := proc(n)
%p A114834     option remember;
%p A114834     if n<= 2 then
%p A114834         n;
%p A114834     else
%p A114834         procname(n-1)+floor(rms(procname(n-1),procname(n-2))) ;
%p A114834     end if;
%p A114834 end proc: # _R. J. Mathar_, Jun 23 2014
%Y A114834 Cf. A065094, A065095.
%K A114834 easy,nonn
%O A114834 1,2
%A A114834 _Jonathan Vos Post_, Feb 19 2006