A112088 -id:A112088 - cp's OEIS frontend

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

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

A306470 a(0) = 0; for n > 0, if the number of blanks already in the sequence is greater than or equal to n, n is filled in at the n-th blank counting from the end of the sequence; otherwise n is added to the end of the sequence followed by n blanks.

Original entry on oeis.org

0, 1, 3, 2, 9, 5, 4, 120, 8, 14, 7, 6, 22, 13, 34, 12, 21, 11, 10, 52, 20, 33, 19, 79, 18, 32, 17, 51, 16, 15, 31, 181, 30, 50, 29, 78, 28, 49, 27, 119, 26, 48, 25, 77, 24, 23, 47, 615, 46, 76, 45, 118, 44, 75, 43, 180, 42, 74, 41, 117, 40, 73

Offset: 0

Jan Koornstra, Feb 17 2019

The consecutive number of blanks added to the sequence (1, 2, 4, 6, 10, 15, ...) appear to form the sequence A112088 - 1. The unique values for the length of the sequence (1, 3, 6, 11, 18, 29, 45, 69, 105, ...) then becomes 1 + the partial sums of A112088.

			For n < 3, n is added to the sequence along with n blanks (denoted by -1): [0, 1, -1, 2, -1, -1]. There are now three blanks in the sequence, hence n = 3 is filled in at the third blank counting from the end of the sequence: [0, 1, 3, 2, -1, -1].

Jan Koornstra, Table of n, a(n) for n = 0..10000

Cf. A112088.

TakeWhile[Nest[Function[{a, n, b}, If[b >= n, ReplacePart[a, Position[a, -1][[-n]] -> n ], Join[a, Prepend[ConstantArray[-1, n], n]]]] @@ {#, Max@ # + 1, Count[#, -1]} &, {0}, 10^3], # > -1 &] (* Michael De Vlieger, Mar 24 2019 *)

seq = [0]
for n in range(1, 1387):
  num_blanks = seq.count(-1)
  if num_blanks >= n: seq[[index for index, value in enumerate(seq) if value == -1][num_blanks - n]] = n
  else: seq += [n] + [-1] * n
print(seq[:100])

A120145 a(n) = 20 + floor( (1 + Sum_{j=1..n-1} a(j)) / 2 ).

Original entry on oeis.org

20, 30, 45, 68, 102, 153, 229, 344, 516, 774, 1161, 1741, 2612, 3918, 5877, 8815, 13223, 19834, 29751, 44627, 66940, 100410, 150615, 225923, 338884, 508326, 762489, 1143734, 1715601, 2573401, 3860102, 5790153, 8685229, 13027844

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 - A120144, A120146 - A120209.

a[n_]:= a[n]= 20 +Quotient[1 +Sum[a[k], {k,n-1}], 2];
Table[a[n], {n,60}] (* G. C. Greubel, May 14 2023 *)

@CachedFunction
def A120145(n): return 20 + (1+sum(A120145(k) for k in range(1,n)))//2
[A120145(n) for n in range(1,61)] # G. C. Greubel, May 14 2023

A120146 a(n) = 22 + floor( Sum_{j=1..n-1} a(j)/2 ).

Original entry on oeis.org

22, 33, 49, 74, 111, 166, 249, 374, 561, 841, 1262, 1893, 2839, 4259, 6388, 9582, 14373, 21560, 32340, 48510, 72765, 109147, 163721, 245581, 368372, 552558, 828837, 1243255, 1864883, 2797324, 4195986, 6293979, 9440969, 14161453

Offset: 1

Graeme McRae, Jun 10 2006

nonn

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

Cf. A073941, A112088, A072493.

A120146[n_]:= 22 + Quotient[Sum[a[k], {k,n-1}], 2];
Table[A120146[n], {n,60}] (* G. C. Greubel, May 30 2023 *)

@CachedFunction
def A120146(n): return 22 + sum(A120146(k) for k in range(1,n))//2
[A120146(n) for n in range(1, 61)] # G. C. Greubel, May 30 2023

A120149 a(n) = 2 + floor((1 + Sum_{j=1..n-1} a(j))/3).

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 38, 50, 67, 89, 119, 159, 212, 282, 376, 502, 669, 892, 1189, 1586, 2114, 2819, 3759, 5012, 6682, 8910, 11880, 15840, 21120, 28160, 37546, 50062, 66749, 88999, 118665, 158220, 210960, 281280, 375040, 500053, 666738

Offset: 1

Graeme McRae, Jun 10 2006

nonn

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

Cf. A072493, A073941, A112088.

f[s_] := Append[s, Floor[(7 + Plus @@ s)/3]]; Nest[f, {2}, 44] (*  Robert G. Wilson v, Jul 08 2006 *)

@CachedFunction
def A120149(n): return 2 + (1+sum(A120149(k) for k in range(1,n)))//3
[A120149(n) for n in range(1, 61)] # G. C. Greubel, Jun 04 2023

More terms from Robert G. Wilson v, Jul 08 2006

A120164 a(n) = 6 + floor( Sum_{j=1..n-1} a(j)/4 ).

Original entry on oeis.org

6, 7, 9, 11, 14, 17, 22, 27, 34, 42, 53, 66, 83, 103, 129, 161, 202, 252, 315, 394, 492, 615, 769, 961, 1202, 1502, 1878, 2347, 2934, 3667, 4584, 5730, 7163, 8953, 11192, 13990, 17487, 21859, 27324, 34155

Offset: 1

Graeme McRae, Jun 10 2006

nonn

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

Cf. A072493, A073941, A112088.

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) >;
A120164:= func< n | g(n, 6, 0) >;
[A120164(n): n in [1..60]]; // G. C. Greubel, Sep 05 2023

f[n_, p_, q_]:= f[n, p, q]= p +Quotient[q +Sum[f[k,p,q], {k,n-1}], 4];
A120164[n_]:= f[n,6,0];
Table[A120164[n], {n, 60}] (* G. C. Greubel, Sep 05 2023 *)

@CachedFunction
def f(n,p,q): return p + (q +sum(f(k,p,q) for k in range(1, n)))//4
def A120164(n): return f(n, 6, 0)
[A120164(n) for n in range(1, 61)] # G. C. Greubel, Sep 05 2023

A120165 a(n) = 7 + floor((1 + Sum_{j=1..n-1} a(j))/4).

Original entry on oeis.org

7, 9, 11, 14, 17, 21, 27, 33, 42, 52, 65, 81, 102, 127, 159, 199, 248, 310, 388, 485, 606, 758, 947, 1184, 1480, 1850, 2312, 2890, 3613, 4516, 5645, 7056, 8820, 11025, 13782, 17227, 21534, 26917, 33647, 42058

Offset: 1

Graeme McRae, Jun 10 2006

nonn

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

Cf. A072493, A073941, A112088.

Cf. A120160 - A120164, A120166 - 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) >;
A120165:= func< n | g(n, 7, 1) >;
[A120165(n): n in [1..60]]; // G. C. Greubel, Sep 09 2023

A[1]:= 7: S:= 7:
for n from 2 to 100 do A[n]:= floor((29 + S)/4); S:= S + A[n] od:
seq(A[i],i=1..100); # Robert Israel, Mar 20 2017

a = {7}; Do[AppendTo[a, Floor[(29 + Total@ a)/4]], {i, 2, 40}]; a (* Michael De Vlieger, Mar 20 2017 *)

@CachedFunction
def f(n, p, q): return p + (q +sum(f(k, p, q) for k in range(1, n)))//4
def A120165(n): return f(n, 7, 1)
[A120165(n) for n in range(1, 61)] # G. C. Greubel, Sep 09 2023

a(n) ~ c (5/4)^n with c approximately 5.5905081519. - Robert Israel, Mar 20 2017

A120171 a(n) = 2 + floor((1 + Sum_{j=1..n-1} a(j))/5).

Original entry on oeis.org

2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 12, 14, 17, 20, 24, 29, 35, 42, 50, 60, 72, 87, 104, 125, 150, 180, 216, 259, 311, 373, 448, 537, 645, 774, 929, 1114, 1337, 1605, 1926, 2311, 2773, 3328, 3993, 4792, 5750, 6900, 8280, 9936, 11923, 14308, 17170, 20604, 24724, 29669

Offset: 1

Graeme McRae, Jun 10 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A073941, A072493, A112088.

function f(n, a, b)
  t:=0;
    for k in [1..n-1] do
      t+:= a+Floor((b+t)/5);
    end for;
  return t;
end function;
g:= func< n, a, b | f(n+1, a, b)-f(n, a, b) >;
A120171:= func< n | g(n, 2, 1) >;
[A120171(n): n in [1..60]]; // G. C. Greubel, Dec 25 2023

f[s_] := Append[s, Floor[(11 + Plus @@ s)/5]]; Nest[f, {2}, 53] (* Robert G. Wilson v, Jul 08 2006 *)

@CachedFunction
def f(n, p, q): return p + (q +sum(f(k, p, q) for k in range(1, n)))//5
def A120171(n): return f(n, 2, 1)
[A120171(n) for n in range(1, 61)] # G. C. Greubel, Dec 25 2023

More terms from Robert G. Wilson v, Jul 08 2006

A120184 a(1)=8; a(n)=floor((48+sum(a(1) to a(n-1)))/6).

Original entry on oeis.org

8, 9, 10, 12, 14, 16, 19, 22, 26, 30, 35, 41, 48, 56, 65, 76, 89, 104, 121, 141, 165, 192, 224, 261, 305, 356, 415, 484, 565, 659, 769, 897, 1047, 1221, 1425, 1662, 1939, 2262, 2639, 3079

Offset: 1

Graeme McRae, Jun 10 2006

nonn

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

Cf. A073941, A112088, A072493.

nxt[{t_,a_}]:=With[{c=Floor[(48+t)/6]},{t+c,c}]; NestList[nxt,{8,8},40][[All,2]] (* Harvey P. Dale, Apr 07 2019 *)

cp's OEIS Frontend

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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A306470 a(0) = 0; for n > 0, if the number of blanks already in the sequence is greater than or equal to n, n is filled in at the n-th blank counting from the end of the sequence; otherwise n is added to the end of the sequence followed by n blanks.

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A120145 a(n) = 20 + floor( (1 + Sum_{j=1..n-1} a(j)) / 2 ).

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A120146 a(n) = 22 + floor( Sum_{j=1..n-1} a(j)/2 ).

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A120149 a(n) = 2 + floor((1 + Sum_{j=1..n-1} a(j))/3).

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A120164 a(n) = 6 + floor( Sum_{j=1..n-1} a(j)/4 ).

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Magma

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SageMath

A120165 a(n) = 7 + floor((1 + Sum_{j=1..n-1} a(j))/4).

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