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A114834
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Each term is previous term plus floor of root mean square of two previous terms.
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1
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1, 2, 3, 5, 9, 16, 28, 50, 90, 162, 293, 529, 956, 1728, 3124, 5648, 10211, 18462, 33380, 60352, 109119, 197293, 356716, 644961, 1166123, 2108412, 3812120, 6892514, 12462029, 22532007, 40739059, 73658371, 133178227, 240793271, 435366958, 787166465
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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Eric Weisstein's World of Mathematics, Mean.
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FORMULA
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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]].
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EXAMPLE
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a(3) = 2 + floor[sqrt[(1^2 + 2^2)/2]] = 2 + floor[Sqrt[5/2]] = 2 + 1 = 3.
a(4) = 3 + floor[sqrt[(2^2 + 3^2)/2]] = 4 + floor[Sqrt[13/2]] = 3 + 2 = 5.
a(5) = 5 + floor[sqrt[(3^2 + 5^2)/2]] = 8 + floor[Sqrt[34/2]] = 5 + 4 = 9.
a(6) = 9 + floor[sqrt[(5^2 + 9^2)/2]] = 15 + floor[Sqrt[106/2]] = 9 + 7 = 16.
a(7) = 16 + floor[sqrt[(9^2 + 16^2)/2]] = 15 + floor[Sqrt[337/2]] = 16 + 12 = 28.
a(8) = 28 + floor[sqrt[(16^2 + 28^2)/2]] = 15 + floor[Sqrt[1040/2]] = 28 + 22 = 50.
a(9) = 50 + floor[sqrt[(28^2 + 50^2)/2]] = 50 + floor[Sqrt[3284/2]] = 50 + 40 = 90.
a(10) = 90 + floor[sqrt[(50^2 + 90^2)/2]] = 50 + floor[Sqrt[10600/2]] = 90 + 72 = 162.
a(11) = 162 + floor[sqrt[(90^2 + 162^2)/2]] = 50 + floor[Sqrt[34344/2]] = 162 + 131 = 293.
a(12) = 293 + floor[sqrt[(162^2 + 293^2)/2]] = 293 + floor[Sqrt[112093/2]] = 293 + 236 = 529.
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MAPLE
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rms := proc(a, b)
sqrt((a^2+b^2)/2) ;
end proc:
option remember;
if n<= 2 then
n;
else
procname(n-1)+floor(rms(procname(n-1), procname(n-2))) ;
end if;
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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