A073941 -id:A073941 - cp's OEIS frontend

A120140 a(n) = 13 + floor(Sum_{j=1..n-1} a(j)/2).

13, 19, 29, 43, 65, 97, 146, 219, 328, 492, 738, 1107, 1661, 2491, 3737, 5605, 8408, 12612, 18918, 28377, 42565, 63848, 95772, 143658, 215487, 323230, 484845, 727268, 1090902, 1636353, 2454529, 3681794, 5522691, 8284036, 12426054, 18639081

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 - A120139, A120141 - A120209.

a[n_]:= a[n]= 13 +Quotient[Sum[a[k], {k,n-1}], 2];
Table[a[n], {n,60}] (* G. C. Greubel, May 11 2023 *)

@CachedFunction
def A120140(n): return 13 + sum(A120140(k) for k in range(1,n))//2
[A120140(n) for n in range(1,60)] # G. C. Greubel, May 11 2023

A120141 a(n) = 14 + floor( (1 + Sum_{j=0..n-1} a(j)) / 2).

Original entry on oeis.org

14, 21, 32, 48, 72, 108, 162, 243, 364, 546, 819, 1229, 1843, 2765, 4147, 6221, 9331, 13997, 20995, 31493, 47239, 70859, 106288, 159432, 239148, 358722, 538083, 807125, 1210687, 1816031, 2724046, 4086069, 6129104, 9193656, 13790484

Offset: 1

Graeme McRae, Jun 10 2006

nonn

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

Cf. A073941, A072493, A112088, A120134 - A120140, A120142 - A120209.

nxt[{t_,a_}]:=Module[{a2=14+Floor[(1+t)/2]},{t+a2,a2}]; NestList[nxt,{0,14},60][[All,2]]//Rest (* Harvey P. Dale, Nov 28 2018 *)

@CachedFunction
def A120141(n): return 14 +(1 +sum(A120141(k) for k in range(1,n)))//2
[A120141(n) for n in range(1,60)] # G. C. Greubel, May 11 2023

A120142 a(n) = 16 + floor(Sum_{j=1..n-1} a(j)/2).

Original entry on oeis.org

16, 24, 36, 54, 81, 121, 182, 273, 409, 614, 921, 1381, 2072, 3108, 4662, 6993, 10489, 15734, 23601, 35401, 53102, 79653, 119479, 179219, 268828, 403242, 604863, 907295, 1360942, 2041413, 3062120, 4593180, 6889770, 10334655, 15501982

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 - A120141, A120143 - A120209.

Nest[Append[#,16+Floor[Total[#]/2]]&,{16},40]  (* Harvey P. Dale, Apr 20 2011 *)

@CachedFunction
def A120142(n): return 16 + (sum(A120142(k) for k in range(1,n)))//2
[A120142(n) for n in range(1,60)] # G. C. Greubel, May 11 2023

A120143 a(n) = 17 + floor( (1 + Sum_{j=0..n-1} a(j))/2 ).

Original entry on oeis.org

17, 26, 39, 58, 87, 131, 196, 294, 441, 662, 993, 1489, 2234, 3351, 5026, 7539, 11309, 16963, 25445, 38167, 57251, 85876, 128814, 193221, 289832, 434748, 652122, 978183, 1467274, 2200911, 3301367, 4952050, 7428075, 11142113, 16713169

Offset: 1

Graeme McRae, Jun 10 2006

nonn

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

Cf. A073941, A072493, A112088, A120134 - A120142, A120144 - A120209.

nxt[{t_,a_}]:=Module[{c=17+Floor[(1+t)/2]},{t+c,c}]; NestList[nxt,{17,17},60][[All,2]] (* Harvey P. Dale, Dec 25 2020 *)

@CachedFunction
def A120143(n): return 17 + (1 +sum(A120143(k) for k in range(1,n)))//2
[A120143(n) for n in range(1,60)] # G. C. Greubel, May 11 2023

A120144 a(n) = 19 + floor( Sum_{j=1..n-1} a(j) / 2 ).

Original entry on oeis.org

19, 28, 42, 63, 95, 142, 213, 320, 480, 720, 1080, 1620, 2430, 3645, 5467, 8201, 12301, 18452, 27678, 41517, 62275, 93413, 140119, 210179, 315268, 472902, 709353, 1064030, 1596045, 2394067, 3591101, 5386651, 8079977, 12119965, 18179948

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 - A120143, A120145 - A120209.

a[n_]:= a[n]= 19 +Quotient[Sum[a[k], {k,n-1}], 2];
Table[a[n], {n,60}] (* G. C. Greubel, May 14 2023 *)

@CachedFunction
def A120144(n): return 19 + sum(A120144(k) for k in range(1,n))//2
[A120144(n) for n in range(1,61)] # G. C. Greubel, May 14 2023

A120166 a(n) = 8 + floor((2 + Sum_{j=1..n-1} a(j))/4).

Original entry on oeis.org

8, 10, 13, 16, 20, 25, 31, 39, 49, 61, 76, 95, 119, 149, 186, 232, 290, 363, 454, 567, 709, 886, 1108, 1385, 1731, 2164, 2705, 3381, 4226, 5283, 6603, 8254, 10318, 12897, 16121, 20152, 25190, 31487, 39359, 49199

Offset: 1

Graeme McRae, Jun 10 2006

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

Cf. A072493, A073941, A112088.

Cf. A120160 - A120165, A120167 - A120169.

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) >;
A120166:= func< n | g(n, 8, 2) >;
[A120166(n): n in [1..60]]; // G. C. Greubel, Sep 09 2023

f[n_, p_, q_]:= f[n,p,q]= p +Quotient[q +Sum[f[k,p,q], {k,n-1}], 4];
A120166[n_]:= f[n,8,2];
Table[A120166[n], {n, 60}] (* G. C. Greubel, Sep 09 2023 *)

@CachedFunction
def f(n, p, q): return p + (q +sum(f(k, p, q) for k in range(1, n)))//4
def A120166(n): return f(n, 8, 2)
[A120166(n) for n in range(1, 61)] # G. C. Greubel, Sep 09 2023

A120167 a(n) = 9 + floor((3 + Sum_{j=1..n-1} a(j))/4).

Original entry on oeis.org

9, 12, 15, 18, 23, 29, 36, 45, 56, 70, 88, 110, 137, 171, 214, 268, 335, 418, 523, 654, 817, 1021, 1277, 1596, 1995, 2494, 3117, 3896, 4870, 6088, 7610, 9512, 11890, 14863, 18579, 23223, 29029, 36286, 45358, 56697

Offset: 1

Graeme McRae, Jun 10 2006

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

Cf. A072493, A073941, A112088.

Cf. A120160 - A120166, A120168, A120169.

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) >;
A120167:= func< n | g(n, 9, 3) >;
[A120167(n): n in [1..60]]; // G. C. Greubel, Sep 09 2023

nxt[{t_,a_}]:=Module[{c=Floor[(39+t)/4]},{t+c,c}]; NestList[nxt,{9,9},40][[All,2]] (* Harvey P. Dale, Apr 24 2019 *)

@CachedFunction
def f(n, p, q): return p + (q +sum(f(k, p, q) for k in range(1, n)))//4
def A120167(n): return f(n, 9, 3)
[A120167(n) for n in range(1, 61)] # G. C. Greubel, Sep 09 2023

A120168 a(n) = 11 + floor(Sum_{j-1..n-1} a(j)/4).

Original entry on oeis.org

11, 13, 17, 21, 26, 33, 41, 51, 64, 80, 100, 125, 156, 195, 244, 305, 381, 476, 595, 744, 930, 1163, 1453, 1817, 2271, 2839, 3548, 4435, 5544, 6930, 8663, 10828, 13535, 16919, 21149, 26436, 33045, 41306, 51633, 64541

Offset: 1

Graeme McRae, Jun 10 2006

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

Cf. A072493, A073941, A112088.

Cf. A120160 - A120167, A120169.

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) >;
A120168:= func< n | g(n, 11, 0) >;
[A120168(n): n in [1..60]]; // G. C. Greubel, Sep 09 2023

f[n_, p_, q_]:= f[n,p,q]= p +Quotient[q +Sum[f[k,p,q], {k,n-1}], 4];
A120168[n_]:= f[n, 11, 0];
Table[A120168[n], {n, 60}] (* G. C. Greubel, Sep 09 2023 *)

@CachedFunction
def f(n, p, q): return p + (q +sum(f(k, p, q) for k in range(1, n)))//4
def A120168(n): return f(n, 11, 0)
[A120168(n) for n in range(1, 61)] # G. C. Greubel, Sep 09 2023

A304023 a(n) is the smallest integer with n digits in base 3/2 expressed in base 3/2.

Original entry on oeis.org

0, 20, 210, 2100, 21010, 210110, 2101100, 21011000, 210110000, 2101100010, 21011000110, 210110001100, 2101100011010, 21011000110100, 210110001101000, 2101100011010010, 21011000110100110, 210110001101001100, 2101100011010011010, 21011000110100110100, 210110001101001101010

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 04 2018

Excluding 0, every term starts with 2 and has exactly one 2.
The last digit is always zero.
Removing the last digit produces the sequence A303500 of the smallest even integers in base 3/2.
The value of this sequence in base 10 is A070885.
When subtracting 1 from the value of this sequence we get A304025.
The largest integer with a given number of digits in base 3/2 can be produced directly from this sequence by replacing 21 at the beginning and 0 at the end with 2, and by shifting the rest up by 1, see sequence A304024.

			The number 5 in base 3/2 is 22, and the number 6 is 210. Therefore, 210 is the smallest three-digit integer.

B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.

Cf. A304024, A304025, A070885, A303500, A304272, A024629, A081848, A246435, A005428, A073941.

b:= proc(n) b(n):= `if`(n=1, 1, 3*ceil(b(n-1)/2)) end:
g:= proc(n) g(n):= `if`(n<2, 0, irem(n, 3, 'q')+g(2*q)*10) end:
a:= n-> g(b(n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 13 2021

f(n) = if( n<1, 0, f(n\3 * 2) * 10 + n%3);
a(n) = {my(k=0); while(#Str(f(k)) != n, k++); f(k);} \\ Michel Marcus, Jun 19 2018

def f(n): return 0 if n < 1 else f(n//3*2)*10 + n%3
def a(n):
  k = 0
  while len(str(f(k))) != n: k += 1
  return f(k)
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Feb 12 2021 after Michel Marcus

a(n) = A024629(A070885(n)). - Michel Marcus, Jun 19 2018

More terms from Michel Marcus, Jun 19 2018

A304274 The concatenation of the first n elements is the largest positive even number with n digits when written in base 3/2.

Original entry on oeis.org

2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 09 2018

This sequence is possible due to the fact that the largest even integers are prefixes of each other.

A304272(n) is the largest even integer with n digits.

			Number 8 in base 3/2 is 212, and it is the largest even integer with 3 digits in base 3/2. Its prefix 21 is 4: the largest even integer with 2 digits in base 3/2.

B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.

Cf. A005428, A070885, A073941, A081848, A024629, A246435, A304024, A304025, A303500, A304272, A304273, A305497, A305498.

b:= proc(n) option remember; `if`(n=1, 2,
      (t-> t+irem(t, 2))(b(n-1)*3/2))
    end:
a:= n-> b(n+1)-3/2*b(n)+1:
seq(a(n), n=1..120);  # Alois P. Heinz, Jun 21 2018

b[n_] := b[n] = If[n == 1, 2, Function[t, t + Mod[t, 2]][3/2 b[n-1]]];
a[n_] := b[n+1] - 3/2 b[n] + 1;
Array[a, 120] (* Jean-François Alcover, Dec 13 2018, after Alois P. Heinz *)

a(n) = A304273(n+1) + 1.
From Alois P. Heinz, Jun 21 2018: (Start)
a(n) = A305498(n+1) -3/2*A305498(n) + 1.
Sum_{i=0..n-1} (3/2)^i*a(n-i) = A305497(n). (End)

More terms from Alois P. Heinz, Jun 21 2018

cp's OEIS Frontend

A120140 a(n) = 13 + floor(Sum_{j=1..n-1} a(j)/2).

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A120141 a(n) = 14 + floor( (1 + Sum_{j=0..n-1} a(j)) / 2).

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A120142 a(n) = 16 + floor(Sum_{j=1..n-1} a(j)/2).

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A120143 a(n) = 17 + floor( (1 + Sum_{j=0..n-1} a(j))/2 ).

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A120144 a(n) = 19 + floor( Sum_{j=1..n-1} a(j) / 2 ).

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A120166 a(n) = 8 + floor((2 + Sum_{j=1..n-1} a(j))/4).

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A120167 a(n) = 9 + floor((3 + Sum_{j=1..n-1} a(j))/4).

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A120168 a(n) = 11 + floor(Sum_{j-1..n-1} a(j)/4).

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A304023 a(n) is the smallest integer with n digits in base 3/2 expressed in base 3/2.

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