A065094 -id:A065094 - cp's OEIS frontend

A241772 First differences of A065094 and also arithmetic means of initial terms of A065094.

1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 15, 19, 24, 30, 37, 45, 55, 68, 82, 99, 119, 143, 171, 203, 241, 286, 338, 398, 468, 548, 642, 749, 873, 1015, 1177, 1364, 1577, 1821, 2099, 2415, 2775, 3184, 3647, 4173, 4768, 5441, 6201, 7058, 8025, 9113, 10337, 11713, 13258

Offset: 1

Reinhard Zumkeller, Apr 28 2014

nonn

a(n) = A065094(n+1) - A065094(n) = (Sum_{k=1..n} A065094(k)) / n.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A065094.

a241772 n = a241772_list !! (n-1)
a241772_list = zipWith (-) (tail a065094_list) a065094_list

A065095 a(1) = 1, a(n+1) is the sum of a(n) and ceiling( arithmetic mean of a(1) ... a(n) ).

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 23, 33, 46, 62, 83, 110, 144, 186, 238, 303, 383, 481, 600, 744, 918, 1128, 1380, 1681, 2039, 2464, 2968, 3563, 4264, 5088, 6054, 7184, 8503, 10040, 11827, 13901, 16304, 19082, 22289, 25986, 30240, 35128, 40736, 47161, 54512

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

It seems that a(n) is asymptotic to C*BesselI(0,2*sqrt(n)) where C is a constant C = 0.78... and BesselI(b,x) is the modified Bessel function of the first kind. Can someone prove this?

Numerically, a(n) ~ c * exp(2*sqrt(n)) / n^(1/4), where c = 0.2214496835182522607818590241239262909281832289078... It follows that the constant above is equal to C = 0.78501868866746800511978860290796656518270697588... - Vaclav Kotesovec, Oct 12 2024

			a(5) = a(4) + ceiling((a(1)+a(2)+a(3)+a(4))/4) = 7 + ceiling((1+2+4+7)/4) = 7 + floor(14/4) = 7 + 4 = 11.

Cf. A065094, A376995.

a[1] := 1: summe := 0: flip := 1: for j from 1 to 100 do: print (j, a[flip]); summe := summe + a[flip]: a[1-flip] := a[flip] + ceil(summe/j): flip := 1-flip: od:

a[1] = 1; a[n_] := a[n] = a[n - 1] + Ceiling[ Sum[ a[i], {i, 1, n - 1} ]/(n - 1) ]; Table[ a[ n], {n, 1, 45} ]
Nest[Append[#,Last[#]+Ceiling[Mean[#]]]&,{1},44] (* James C. McMahon, Oct 10 2024 *)

{ for (n=1, 1000, if (n==1, s=0; a=1, s+=a; a+=ceil(s/(n - 1))); write("b065095.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 06 2009

a(1) = 1, a(n+1) = a(n) + ceiling((a(1) + a(2) + ... + a(n))/n).

A114829 Each term is previous term plus floor of geometric mean of all previous terms.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 29, 36, 44, 53, 63, 74, 87, 101, 117, 135, 155, 177, 201, 227, 256, 287, 321, 358, 398, 442, 489, 540, 595, 654, 717, 785, 858, 936, 1019, 1107, 1201, 1301, 1408, 1521, 1641, 1768, 1903, 2046, 2197, 2356, 2524, 2701, 2888, 3085, 3292, 3510, 3739, 3979, 4231

Offset: 1

Jonathan Vos Post, Feb 19 2006

What is this sequence, asymptotically?

			a(2) = 1 + floor(1^(1/1)) = 1 + 1 = 2.
a(3) = 2 + floor[(1*2)^(1/2)] = 2 + floor[sqrt(2)] = 2 + 1 = 3.
a(4) = 3 + floor[(1*2*3)^(1/3)] = 3 + floor[CubeRoot(6)] = 3 + 1 = 4.
a(5) = 4 + floor[(1*2*3*4)^(1/4)] = 4 + floor[4thRoot(24)] = 4 + 2 = 6.
a(6) = 6 + floor[(1*2*3*4*6)^(1/5)] = 6 + floor[5thRoot(144)] = 6 + 2 = 8.
a(7) = 8 + floor[(1*2*3*4*6*8)^(1/6)] = 6 + floor[6thRoot(1152)] = 8 + 3 = 11.

Eric Weisstein's World of Mathematics, Geometric Mean.

Cf. A065094, A065095.

A114829 := proc(n)
    option remember;
    if n= 1 then
        1;
    else
        mul(procname(i),i=1..n-1) ;
        procname(n-1)+floor(root[n-1](%)) ;
    end if;
end proc:
seq(A114829(n),n=1..60) ; # R. J. Mathar, Jun 23 2014

s={1};Do[AppendTo[s,Last[s]+Floor[GeometricMean[s]]],{n,58}];s (* James C. McMahon, Aug 19 2024 *)

a(1) = 1, a(n+1) = a(n) + floor(GeometricMean[a(1),a(2),...,a(n)]).

a(n+1) = a(n) + floor((Product_{k=1..n} a(k))^(1/n)).

A376995 a(n) = floor(b(n)), where b(1) = 1 and b(n) = b(n-1) + Sum_{k=1..n-1} b(k)/(n-1).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 18, 25, 35, 48, 64, 85, 111, 143, 184, 234, 296, 371, 463, 575, 709, 872, 1066, 1298, 1575, 1903, 2293, 2752, 3294, 3931, 4677, 5550, 6570, 7757, 9138, 10741, 12597, 14744, 17222, 20078, 23365, 27141, 31474, 36438, 42118, 48607, 56012, 64451, 74057

Offset: 1

Vaclav Kotesovec, Oct 12 2024

nonn

Cf. A065094, A160617, A160618.

b[1]=1; b[n_]:=b[n] = b[n-1] + Sum[b[i], {i, 1, n-1}]/(n-1); Table[b[n], {n, 1, 50}] // Floor
Clear[b]; RecurrenceTable[{b[n] == 2*b[n-1] - (1-1/(n-1))*b[n-2], b[1]==1, b[2]==2}, b, {n, 1, 50}] // Floor
Table[HypergeometricPFQ[{n}, {1}, 1]/E, {n, 1, 50}] // Floor
Rest[CoefficientList[Series[x*E^(x/(1-x))/(1-x), {x, 0, 50}], x]] // Floor
nmax = 50; Rest[CoefficientList[Series[Integrate[Exp[t]*BesselI[0, 2*Sqrt[t]], {t, 0, x}], {x, 0, nmax}], x] * Range[0, nmax]!] // Floor
Table[LaguerreL[n-1, -1], {n, 1, 50}] // Floor

a(n) = floor(b(n)), where b(n) = 2*b(n-1) - (1 - 1/(n-1))*b(n-2), b(1)=1, b(2)=2.
a(n) ~ exp(2*sqrt(n) - 1/2) / (2*sqrt(Pi)*n^(1/4)).
a(n) ~ exp(-1/2) * BesselI(0, 2*sqrt(n)).
a(n) = floor(A160617(n-1)/A160618(n-1)).

A114830 Each term is previous term plus ceiling of geometric mean of all previous terms.

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 18, 24, 31, 39, 48, 59, 71, 85, 101, 119, 139, 162, 187, 215, 246, 280, 318, 359, 404, 453, 507, 565, 628, 697, 771, 851, 937, 1029, 1128, 1234, 1348, 1470, 1600, 1738, 1885, 2042, 2209, 2386, 2574, 2773, 2984, 3207, 3443, 3692, 3955, 4232, 4524, 4831, 5154, 5494, 5851, 6226, 6620

Offset: 1

Jonathan Vos Post, Feb 19 2006

What is this sequence, asymptotically?

			a(2) = 1 + ceiling(1^(1/1)) = 1 + 1 = 2.
a(3) = 2 + ceiling[(1*2)^(1/2)] = 2 + ceiling[sqrt(2)] = 2 + 2 = 4.
a(4) = 4 + ceiling[(1*2*4)^(1/3)] = 4 + ceiling[CubeRoot(8)] = 4 + 2 = 6.
a(5) = 6 + ceiling[(1*2*4*6)^(1/4)] = 6 + floor[4thRoot(48)] = 6 + 3 = 9.
a(6) = 9 + ceiling[(1*2*4*6*9)^(1/5)] = 9 + ceiling[5thRoot(432)] = 9 + 4 = 13.
a(7) = 13 + ceiling[(1*2*4*6*9*13)^(1/6)] = 6 + floor[6thRoot(5616)] = 13 + 5 = 18.
a(25) = 359 + ceiling[(1 * 2 * 4 * 6 * 9 * 13 * 18 * 24 * 31 * 39 * 48 * 59 * 71 * 85 * 101 * 119 * 139 * 162 * 187 * 215 * 246 * 280 * 318 * 359)^(1/24)] = 359 + ceiling[44.8074289] = 359 + 45 = 404.

Eric Weisstein's World of Mathematics, Geometric Mean.

Cf. A065094, A065095.

A114830 := proc(n)
    option remember;
    if n= 1 then
        1;
    else
        mul(procname(i),i=1..n-1) ;
        procname(n-1)+ceil(root[n-1](%)) ;
    end if;
end proc:
seq(A114830(n),n=1..60) ; # R. J. Mathar, Jun 23 2014

Nest[Append[#,Last[#]+Ceiling@GeometricMean[#]]&,{1},58] (* James C. McMahon, Aug 20 2024 *)

a(1) = 1, a(n+1) = a(n) + ceiling(GeometricMean[a(1),a(2),...,a(n)]).

a(n+1) = a(n) + ceiling((Product_{k=1..n} a(k))^(1/n)).

A114831 Each term is previous term plus floor of harmonic mean of two previous terms.

Original entry on oeis.org

1, 2, 3, 5, 8, 14, 24, 41, 71, 122, 211, 365, 632, 1094, 1895, 3282, 5684, 9845, 17052, 29534, 51154, 88601, 153461, 265802, 460382, 797405, 1381145, 2392213, 4143434, 7176638, 12430301, 21529913, 37290903, 64589738, 111872708, 193769214, 335618123, 581307641, 1006854369, 1743922922

Offset: 1

Jonathan Vos Post, Feb 19 2006

For two numbers x and y, HarmonicMean[x,y] = [(GeometricMean[x,y])^2] / Arithmetic Mean[x,y].

What is this sequence, asymptotically?

			a(3) = 2 + floor(2*1*2/(1+2)) = 2 + floor(4/3) = 2 + 1 = 3.
a(4) = 3 + floor(2*2*3/(2+3)) = 3 + floor(12/5) = 3 + 2 = 5.
a(5) = 5 + floor(2*3*5/(3+5)) = 5 + floor(30/8) = 5 + 3 = 8.
a(6) = 8 + floor(2*5*8/(5+8)) = 8 + floor(80/13) = 8 + 6 = 14.
a(7) = 14 + floor(2*8*14/(8+14)) = 14 + floor(112/11) = 14 + 10 = 24.

Eric Weisstein's World of Mathematics, Harmonic Mean.
Eric Weisstein's World of Mathematics, Geometric Mean.

Cf. A065094, A065095.

hMean := proc(a,b)
    2*a*b/(a+b) ;
end proc:
A114831 := proc(n)
    option remember;
    if n<= 2 then
        n;
    else
        procname(n-1)+floor(hMean(procname(n-1),procname(n-2))) ;
    end if;
end proc:
seq(A114831(n),n=1..60) ; # R. J. Mathar, Jun 23 2014

a(1) = 1, a(2) = 2, for n>2: a(n+1) = a(n) + floor(HarmonicMean[a(n),a(n-1)]). a(n+1) = a(n) + floor[(2*a(n)*a(n-1))/(a(n)+a(n-1))].

Corrected by R. J. Mathar, Jun 23 2014

Typo in a(40) corrected by Seth A. Troisi, May 13 2022

A114833 Each term is previous term plus ceiling of root mean square of two previous terms.

Original entry on oeis.org

0, 1, 2, 4, 8, 15, 28, 51, 93, 168, 304, 550, 995, 1799, 3253, 5882, 10635, 19229, 34767, 62861, 113656, 205497, 371550, 671782, 1214618, 2196094, 3970654, 7179153, 12980288, 23469047, 42433278, 76721609, 138716724, 250807167, 453472612, 819902445, 1482426947, 2680306255, 4846135343

Offset: 0

Jonathan Vos Post, Feb 19 2006

			a(3) = 2 + ceiling[sqrt[(1^2 + 2^2)/2]] = 2 + ceiling[Sqrt[5/2]] = 2 + 2 = 4.
a(4) = 4 + ceiling[sqrt[(2^2 + 4^2)/2]] = 4 + ceiling[Sqrt[20/2]] = 4 + 4 = 8.
a(5) = 8 + ceiling[sqrt[(4^2 + 8^2)/2]] = 8 + ceiling[Sqrt[80/2]] = 8 + 7 = 15.
a(6) = 15 + ceiling[sqrt[(8^2 + 15^2)/2]] = 15 + ceiling[Sqrt[289/2]] = 15 + 13 = 28.
a(7) = 28 + ceiling[sqrt[(15^2 + 28^2)/2]] = 28 + ceiling[Sqrt[1009/2]] = 28 + 23 = 51.
a(8) = 51 + ceiling[sqrt[(28^2 + 51^2)/2]] = 51 + ceiling[Sqrt[3385/2]] = 51 + 42 = 93.
a(9) = 93 + ceiling[sqrt[(51^2 + 93^2)/2]] = 93 + ceiling[Sqrt[11250/2]] = 93 + 75 = 168 [the 75 is an exact value].
a(10) = 168 + ceiling[sqrt[(93^2 + 168^2)/2]] = 168 + ceiling[Sqrt[36873/2]] = 168 + 136 = 304.
a(11) = 304 + ceiling[sqrt[(168^2 + 304^2)/2]] = 304 + ceiling[Sqrt[120640/2]] = 304 + 246 = 550.
a(12) = 550 + ceiling[sqrt[(304^2 + 550^2)/2]] = 550 + ceiling[Sqrt[394916/2]] = 550 + 445 = 995.

Robert G. Wilson v, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Root-Mean-Square.
Eric Weisstein's World of Mathematics, Mean.

Cf. A065094, A065095.

a[n_] := a[n] = a[n - 1] + Ceiling[ Sqrt[(a[n - 1]^2 + a[n - 2]^2)/2]]; a[0] = 0; a[1] = 1; Array[a, 39, 0] (* Robert G. Wilson v, Jun 22 2014 *)
nxt[{a_,b_}]:={b,b+Ceiling[Sqrt[(a^2+b^2)/2]]}; Transpose[NestList[nxt,{0,1},40]][[1]] (* Harvey P. Dale, May 12 2015 *)

a(1) = 1, a(2) = 2, for n>2: a(n+1) = a(n) + ceiling(RMS[a(n),a(n-1)]). a(n+1) = a(n) + ceiling[Sqrt[[a(n)^2]+[a(n-1)^2]/2]].

It can easily be proved via induction that a(n)<=2^n. On the other hand we can derive a lower bound: We derive another sequence of the form b(n) = a*c^n, where "a" and "c" are real numbers. If b(1)<=a(1) and b(2)<=a(2) and a(n+1) = a(n)+Ceiling(Sqrt((a(n)^2+a(n-1)^2)/2)) >= b(n)+Sqrt((b(n)^2+b(n-1)^2)/2) >= b(n+1) then, via induction we can safely conclude that a(n)>=b(n). With this method we can derive that a(n) >= 1.80805^(n-1) (where 1.80... is the positive solution of x^2 = x+Sqrt((x^2+1)/2)). Hence we have 1.80805 < a(n)^(1/n) < 2. Can these bounds be improved? - Stefan Steinerberger, Feb 21 2006

a(0) and a(13) onward from Robert G. Wilson v, Jun 22 2014

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

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Feb 19 2006

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.

			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.

Eric Weisstein's World of Mathematics, Root-Mean-Square.
Eric Weisstein's World of Mathematics, Mean.

Cf. A065094, A065095.

rms := proc(a,b)
    sqrt((a^2+b^2)/2) ;
end proc:
A114834 := proc(n)
    option remember;
    if n<= 2 then
        n;
    else
        procname(n-1)+floor(rms(procname(n-1),procname(n-2))) ;
    end if;
end proc: # R. J. Mathar, Jun 23 2014

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]].

A375606 a(1) = 1; a(n+1) = a(n) + floor(harmonic mean of previous terms).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 14, 17, 20, 23, 27, 31, 35, 40, 45, 50, 55, 61, 67, 73, 80, 87, 94, 102, 110, 118, 126, 135, 144, 153, 163, 173, 183, 193, 204, 215, 226, 238, 250, 262, 275, 288, 301, 314, 328, 342, 356, 371, 386, 401, 417, 433, 449, 465, 482, 499, 516

Offset: 1

James C. McMahon, Aug 20 2024

nonn

			a(6) = 5 + floor(harmonic mean(1,2,3,4,5)) = 5 + floor(300/137) = 7.

James C. McMahon, Table of n, a(n) for n = 1..10000

Cf. A065094, A114829.

Nest[Append[#,Last[#]+Floor@HarmonicMean[#]]&,{1},58]

A114832 Each term is previous term plus ceiling of harmonic mean of two previous terms.

Original entry on oeis.org

1, 2, 4, 7, 13, 23, 40, 70, 121, 210, 364, 631, 1093, 1894, 3281, 5683, 9844, 17050, 29532, 51151, 88597, 153455, 265792, 460366, 797377, 1381098, 2392132, 4143295, 7176398, 12429886, 21529195, 37289660, 64587586, 111868981, 193762759

Offset: 1

Jonathan Vos Post, Feb 19 2006

For two numbers x and y, HarmonicMean[x,y] = [(GeometricMean[x,y])^2] / Arithmetic Mean[x,y]. What is this sequence, asymptotically? a(n) is prime for n = 2, 4, 5, 6, 12, ... are there an infinite number of prime values?

			a(3) = 2 + ceiling(2*1*2/(1+2)) = 2 + ceiling(4/3) = 2 + 2 = 4.
a(4) = 4 + ceiling(2*2*4/(2+4)) = 4 + ceiling(16/6) = 4 + 3 = 7.
a(5) = 7 + ceiling(2*4*7/(4+7)) = 7 + ceiling(56/8) = 7 + 6 = 13.
a(6) = 13 + ceiling(2*7*13/(7+13)) = 13 + ceiling(182/13) = 13 + 10 = 23.
a(7) = 23 + ceiling(2*13*23/(13+23)) = 23 + ceiling(598/36) = 23 + 17 = 40.
a(8) = 40 + ceiling(2*23*40/(23+40)) = 40 + ceiling(1840/63) = 40 + 30 = 70.
a(9) = 70 + ceiling(2*40*70/(40+70)) = 70 + ceiling(5600/110) = 70 + 51 = 121.
a(10) = 121 + ceiling(2*70*121/(70+121)) = 121 + ceiling(16940/191) = 121 + 89 = 210.
a(11) = 210 + ceiling(2*121*210/(121+210)) = 121 + ceiling(50820/331) = 210 + 154 = 364.
a(12) = 364 + ceiling(2*210*364/(210+364)) = 364 + ceiling(152880/574) = 364 + 267 = 631.

Eric Weisstein's World of Mathematics, Harmonic Mean.
Eric Weisstein's World of Mathematics, Geometric Mean.

Cf. A065094, A065095.

a[1]:=1: a[2]:=2: for n from 2 to 40 do a[n+1]:=a[n]+ceil((2*a[n]*a[n-1])/(a[n]+a[n-1])) od: seq(a[n],n=1..40); # Emeric Deutsch, Mar 03 2006

a(1) = 1, a(2) = 2, for n > 2: a(n+1) = a(n) + ceiling(HarmonicMean(a(n), a(n-1))). a(n+1) = a(n) + ceiling((2*a(n)*a(n-1))/(a(n) + a(n-1))).

More terms from Emeric Deutsch, Mar 03 2006

cp's OEIS Frontend

A241772 First differences of A065094 and also arithmetic means of initial terms of A065094.

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A065095 a(1) = 1, a(n+1) is the sum of a(n) and ceiling( arithmetic mean of a(1) ... a(n) ).

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A114829 Each term is previous term plus floor of geometric mean of all previous terms.

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A376995 a(n) = floor(b(n)), where b(1) = 1 and b(n) = b(n-1) + Sum_{k=1..n-1} b(k)/(n-1).

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A114830 Each term is previous term plus ceiling of geometric mean of all previous terms.

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Maple

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A114831 Each term is previous term plus floor of harmonic mean of two previous terms.

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Maple

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A114833 Each term is previous term plus ceiling of root mean square of two previous terms.

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

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