A112088 -id:A112088 - cp's OEIS frontend

A120160 a(n) = ceiling(Sum_{i=1..n-1} a(i)/4) for n >= 2 starting with a(1) = 1.

1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 10, 12, 15, 19, 24, 30, 37, 47, 58, 73, 91, 114, 142, 178, 222, 278, 347, 434, 543, 678, 848, 1060, 1325, 1656, 2070, 2588, 3235, 4043, 5054, 6318, 7897, 9871, 12339, 15424, 19280, 24100, 30125, 37656, 47070, 58838, 73547

Offset: 1

Graeme McRae, Jun 10 2006

nonn

From Petros Hadjicostas, Jul 21 2020: (Start)
Conjecture 1: a(n) equals the number of multiples of 4 whose representation in base 5/4 (see A024634) has n-1 digits. For example, a(8) = 3 because there are three multiples of 4 with n-1 = 7 digits in their representation in base 5/4: 36 = 4321031, 40 = 4321420, and 44 = 4321424.
Conjecture 2: a(n) equals 1/5 times the number of nonnegative integers with the property that their 5/4-expansion has n digits (assuming that the 5/4-expansion of 0 has 1 digit). For example, a(7)*5 = 10 because the following 10 numbers have 5/4 expansions with n = 7 digits: 35 = 4321030, 36 = 4321031, 37 = 4321032, 38 = 4321033, 39 = 4321034, 40 = 4321420, 41 = 4321421, 42 = 4321422, 43 = 4321423, and 44 = 4321424. (End)
From Petros Hadjicostas, Jul 23 2020: (Start)
Starting at a(11) = 5, we conjecture that this sequence gives all those numbers m for which when we place m persons on a circle, label them 1 through m, start counting at person number 1, and remove every 5th person, the last survivors have numbers in {1, 2, 3, 4}.
When m = 6, 12, 15, 37, 58, 142, 222, 347, ... the last survivor is person 1.
When m = 5, 19, 91, 434, 1656, 2070, 5054, ... the last survivor is person 2.
When m = 8, 10, 24, 30, 73, 114, 178, 278, ... the last survivor is person 3.
When m = 47, 543, 2588, 3235, 6318, 58838, ... the last survivor is person 4. (End)

			From _Petros Hadjicostas_, Jul 23 2020: (Start)
We explain why 6 and 8 belong to the sequence related to the Josephus problem (for the case k = 5) while 7 does not.
When we place m = 6 people on a circle, label them 1 through 6, start counting at person 1, and remove every 5th person, the list of people we remove is 5 -> 4 -> 6 -> 2 -> 3. Thus the last survivor is person 1, so 6 belongs to this sequence.
When we place m = 7 people on a circle, label them 1 through 7, start counting at person 1, and remove every 5th person, the list of people we remove is 5 -> 3 -> 2 -> 4 -> 7 -> 1. Thus the last survivor is person 6 (not in {1, 2, 3, 4}), so 7 does not belong to this sequence.
When we place m = 8 people on a circle, label them 1 through 8, start the counting at person 1, and remove every 5th person, the list of people we remove is 5 -> 2 -> 8 -> 7 -> 1 -> 4 -> 6. Thus the last survivor is 3, so 8 belongs to this sequence. (End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
K. Burde, Das Problem der Abzählreime und Zahlentwicklungen mit gebrochenen Basen [The problem of counting rhymes and number expansions with fractional bases], J. Number Theory 26(2) (1987), 192-209. [The author deals with the representation of n in fractional bases k/(k-1) and its relation to counting-off games (variations of Josephus problem). Here k = 5. See the review in MathSciNet (MR0889384) by R. G. Stoneham.]
Nicolas Thériault, Generalizations of the Josephus problem, Utilitas Mathematica, 58 (2000), 161-173 (MR1801309).
Nicolas Thériault, Generalizations of the Josephus problem, Utilitas Mathematica, 58 (2000), 161-173 (MR1801309).
Index entries for sequences related to the Josephus Problem

Cf. A011782, A024634, A072493, A073941, A112088, A120170, A120178, A120186, A120194, A120202.

function f(n, a, b)
  t:=0;
    for k in [1..n-1] do
      t+:= a+Ceiling((b+t)/4);
    end for;
  return t;
end function;
g:= func< n, a, b | f(n+1, a, b)-f(n, a, b) >;
A120160:= func< n | n le 4 select 1 else g(n-4, 1, 0) >;
[A120160(n): n in [1..60]]; // G. C. Greubel, Sep 01 2023

f[s_]:= Append[s, Ceiling[Plus @@ s/4]]; Nest[f,1,53] (* Robert G. Wilson v *)
(* Second program *)
f[n_]:= f[n]= 1 + Quotient[Sum[f[k], {k,n-1}], 4];
A120160[n_]:= If[n==1, 1, f[n-1]];
Table[A120160[n], {n, 60}] (* G. C. Greubel, Sep 01 2023 *)

/* PARI program for the general case of Josephus problem. We use the Burde-Thériault algorithm. Here k = 5. To get the corresponding last survivors, modify the program to get the vector j. */
lista(nn, k) = {my(j=vector(nn)); my(f=vector(nn)); my(N=vector(nn));
j[1]=1; f[1]=0; N[1] = 1;
for(n=1, nn-1, f[n+1] = ((j[n]-N[n]-1) % (k-1)) + 1 - j[n];
j[n+1] = j[n] + f[n+1]; N[n+1] = (k*N[n] + f[n+1])/(k-1); );
for(n=1, nn, if(N[n] > k-1, print1(N[n], ", "))); } \\ Petros Hadjicostas, Jul 23 2020

@CachedFunction
def A120160(n): return 1 + ceil( sum(A120160(k) for k in range(1,n))//4 )
[A120160(n) for n in range(1,61)] # G. C. Greubel, Sep 01 2023

Edited and extended by Robert G. Wilson v, Jul 07 2006

Appended the first missing term. - Tom Edgar, Jul 18 2014

A120134 a(n) = 4 + floor(Sum_{k=1..n-1} a(k) / 2).

Original entry on oeis.org

4, 6, 9, 13, 20, 30, 45, 67, 101, 151, 227, 340, 510, 765, 1148, 1722, 2583, 3874, 5811, 8717, 13075, 19613, 29419, 44129, 66193, 99290, 148935, 223402, 335103, 502655, 753982, 1130973, 1696460, 2544690, 3817035, 5725552, 8588328, 12882492

Offset: 1

Graeme McRae, Jun 10 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A073941, A072493, A112088, A120135 - A120209, A172161.

a := proc(n) option remember; 4 + iquo(add(a(k), k = 1..n-1), 2) end:
seq(a(n), n = 1..37); # Peter Luschny, May 07 2023

f[s_]:= Append[s, 4+Floor[Plus @@ s/2]]; Nest[f, {4}, 37] (* Robert G. Wilson v, Jun 10 2006 *)

@CachedFunction
def A120134(n): return 4 + sum(A120134(k) for k in range(1,n))//2
print([A120134(n) for n in range(1,41)])  # G. C. Greubel, May 07 2023

a(n) ~ c * (3/2)^n, where c = 3.9320886310283094862025842648411406926537121258867950257873967568454133192... - Vaclav Kotesovec, May 07 2023

Name edited by Peter Luschny, May 07 2023

A120209 a(1)=8; a(n)=floor((79+sum(a(1) to a(n-1)))/9).

Original entry on oeis.org

8, 9, 10, 11, 13, 14, 16, 17, 19, 21, 24, 26, 29, 32, 36, 40, 44, 49, 55, 61, 68, 75, 84, 93, 103, 115, 127, 142, 157, 175, 194, 216, 240, 266, 296, 329, 365, 406, 451, 501, 557, 619, 688, 764, 849, 943, 1048, 1164, 1294, 1438, 1597, 1775, 1972, 2191, 2435, 2705, 3006, 3340, 3711

Offset: 1

Graeme McRae, Jun 10 2006

nonn

Cf. A073941, A112088, A072493.

nxt[{t_,a_}]:=Module[{c=Floor[(79+t)/9]},{t+c,c}]; NestList[nxt,{8,8},60][[;;,2]] (* Harvey P. Dale, May 03 2024 *)

More terms from Alonso del Arte, Dec 25 2013

A120170 a(n) = ceiling( Sum_{i=1..n-1} a(i)/5 ), a(1)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 12, 14, 17, 21, 25, 30, 36, 43, 52, 62, 74, 89, 107, 128, 154, 185, 222, 266, 319, 383, 460, 552, 662, 795, 954, 1144, 1373, 1648, 1977, 2373, 2847, 3417, 4100, 4920, 5904, 7085, 8502, 10202, 12243, 14691, 17630

Offset: 1

Graeme McRae, Jun 10 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A011782, A073941, A072493, A112088, A120160, A120178, A120186, A120194, A120202.

function f(n, a, b)
  t:=0;
    for k in [1..n-1] do
      t+:= a+Ceiling ((b+t)/5);
    end for;
  return t;
end function;
g:= func< n, a, b | f(n+1, a, b)-f(n, a, b) >;
A120170:= func< n | n eq 1 select 1 else g(n-1, 1, -4) >;
[A120170(n): n in [1..60]]; // G. C. Greubel, Dec 25 2023

f[s_] := Append[s, Ceiling[Plus @@ s/5]]; Nest[f, {1}, 57] (* Robert G. Wilson v, Jul 07 2006 *)

@CachedFunction
def a(n):
    if (n==1): return 1
    else: return ceil(sum(a(k)/5 for k in (1..n-1)))
[a(n) for n in (1..60)] # G. C. Greubel, Aug 19 2019

Edited and extended by Robert G. Wilson v, Jul 07 2006

A120186 a(n) = ceiling( Sum_{i=1..n-1} a(i)/7 ), a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23, 27, 30, 35, 40, 45, 52, 59, 68, 77, 88, 101, 115, 132, 151, 172, 197, 225, 257, 294, 336, 384, 439, 501, 573, 655, 748, 855, 977, 1117, 1277, 1459, 1667, 1906, 2178, 2489, 2845

Offset: 1

Graeme McRae, Jun 10 2006

nonn

Cf. A112088, A072493, A011782, A073941, A072493, A120160, A120170, A120178, A120186, A120194, A120202.

f[s_] := Append[s, Ceiling[Plus @@ s/7]]; Nest[f, {1}, 64] (* Robert G. Wilson v, Jul 07 2006 *)

Edited and extended by Robert G. Wilson v, Jul 07 2006

A120194 a(n) = ceiling( Sum_{i=1..n-1} a(i)/8 ), a(1)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 25, 28, 32, 36, 40, 45, 51, 57, 64, 72, 81, 91, 103, 116, 130, 146, 165, 185, 208, 234, 264, 297, 334, 376, 423, 475, 535, 602, 677, 762, 857, 964, 1084, 1220, 1372, 1544

Offset: 1

Graeme McRae, Jun 10 2006

nonn

Cf. A112088, A011782, A073941, A072493, A120160, A120170, A120178, A120186, A120194, A120202.

f[s_] := Append[s, Ceiling[Plus @@ s/8]]; Nest[f, {1}, 67] (* Robert G. Wilson v *)

Edited and extended by Robert G. Wilson v, Jul 07 2006

A120202 a(n) = ceiling( Sum_{i=1..n-1} a(i)/9), a(1)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, 20, 22, 25, 28, 31, 34, 38, 42, 47, 52, 58, 64, 71, 79, 88, 98, 109, 121, 134, 149, 166, 184, 205, 227, 253, 281, 312, 347, 385, 428, 476, 528, 587, 652, 725, 805, 895

Offset: 1

Graeme McRae, Jun 10 2006

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A112088, A011782, A073941, A072493, A120160, A120170, A120178, A120186, A120194.

N:= 100: # to get a(1) to a(N)
S:= 0: A[1]:= 1:
for n from 2 to N do
  S:= S + A[n-1];
  A[n]:= ceil(S/9);
od:
seq(A[n],n=1..N); # Robert Israel, Jul 14 2014

a[s_] := Append[s, Ceiling[Plus @@ s/9]]; Nest[a, {1}, 70] (* Robert G. Wilson v, Jul 07 2006 *)

Edited and extended by Robert G. Wilson v, Jul 07 2006

Typo in name corrected by Tom Edgar, Jul 14 2014

A120178 a(n)=ceiling( sum_{i=1..n-1} a(i)/6), a(1)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 20, 23, 27, 32, 37, 43, 50, 59, 69, 80, 93, 109, 127, 148, 173, 202, 235, 275, 320, 374, 436, 509, 594, 693, 808, 943, 1100, 1283, 1497, 1747, 2038, 2377, 2774, 3236, 3775, 4404, 5138, 5995, 6994

Offset: 1

Graeme McRae, Jun 10 2006

nonn

Cf. A112088, A072493, A011782, A073941, A072493, A120160, A120170, A120186, A120194, A120202.

f[s_] := Append[s, Ceiling[Plus @@ s/5]]; Nest[f, {1}, 61] (* Robert G. Wilson v *)

Edited and extended by Robert G. Wilson v, Jul 07 2006

A120169 a(n) = 12 + floor((1 + Sum_{j=1..n-1} a(j))/4).

Original entry on oeis.org

12, 15, 19, 23, 29, 36, 45, 57, 71, 89, 111, 139, 173, 217, 271, 339, 423, 529, 661, 827, 1033, 1292, 1615, 2018, 2523, 3154, 3942, 4928, 6160, 7700, 9625, 12031, 15039, 18798, 23498, 29372, 36715, 45894, 57368, 71710

Offset: 1

Graeme McRae, Jun 10 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A072493, A073941, A112088.

Cf. A120149, A120160 - A120168, A120171.

function f(n, a, b)
  t:=0;
    for k in [1..n-1] do
      t+:= a+Floor((b+t)/4);
    end for;
  return t;
end function;
g:= func< n, a, b | f(n+1, a, b)-f(n, a, b) >;
A120169:= func< n | g(n, 12, 1) >;
[A120169(n): n in [1..60]]; // G. C. Greubel, Sep 09 2023

nxt[{t_,a_}]:=Module[{c=Floor[(t+49)/4]},{t+c,c}]; NestList[nxt,{12,12},40][[All,2]] (* Harvey P. Dale, Jun 21 2017 *)

@CachedFunction
def f(n, p, q): return p + (q +sum(f(k, p, q) for k in range(1, n)))//4
def A120169(n): return f(n, 12, 1)
[A120169(n) for n in range(1, 61)] # G. C. Greubel, Sep 09 2023

A120135 a(n) = 5 + floor((1 + Sum_{j=1..n-1} a(j)) / 2).

Original entry on oeis.org

5, 8, 12, 18, 27, 40, 60, 90, 135, 203, 304, 456, 684, 1026, 1539, 2309, 3463, 5195, 7792, 11688, 17532, 26298, 39447, 59171, 88756, 133134, 199701, 299552, 449328, 673992, 1010988, 1516482, 2274723, 3412084, 5118126, 7677189, 11515784

Offset: 1

Graeme McRae, Jun 10 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A073941, A072493, A112088, A120134, A120136 - A120209.

a[n_]:= a[n]= 5 +Floor[(1+Sum[a[k], {k,n-1}])/2];
Table[a[n], {n,60}] (* G. C. Greubel, May 07 2023 *)

@CachedFunction
def A120135(n): return 5 + (1 + sum(A120135(k) for k in range(1,n)))//2
[A120135(n) for n in range(1,61)] # G. C. Greubel, May 07 2023

a(n) ~ c * (3/2)^n, where c = 3.514931952760438754899508881646642282344325354834703833076259269449577... - Vaclav Kotesovec, May 07 2023

cp's OEIS Frontend

A120160 a(n) = ceiling(Sum_{i=1..n-1} a(i)/4) for n >= 2 starting with a(1) = 1.

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A120134 a(n) = 4 + floor(Sum_{k=1..n-1} a(k) / 2).

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A120209 a(1)=8; a(n)=floor((79+sum(a(1) to a(n-1)))/9).

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A120170 a(n) = ceiling( Sum_{i=1..n-1} a(i)/5 ), a(1)=1.

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Magma

Mathematica

SageMath

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A120186 a(n) = ceiling( Sum_{i=1..n-1} a(i)/7 ), a(1) = 1.

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Mathematica

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A120194 a(n) = ceiling( Sum_{i=1..n-1} a(i)/8 ), a(1)=1.

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A120202 a(n) = ceiling( Sum_{i=1..n-1} a(i)/9), a(1)=1.

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A120178 a(n)=ceiling( sum_{i=1..n-1} a(i)/6), a(1)=1.

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