cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A049933 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = a(3) = 1.

Original entry on oeis.org

1, 1, 1, 4, 8, 19, 35, 70, 140, 349, 663, 1310, 2609, 5214, 10425, 20850, 41700, 104249, 198073, 390935, 779265, 1557231, 3113815, 6227316, 12454423, 24908776, 49817517, 99635018, 199270025, 398540046, 797080089, 1594160178, 3188320356, 7970800889, 15144521689
Offset: 1

Views

Author

Clark Kimberling

Keywords

Examples

			From _Petros Hadjicostas_, Nov 06 2019: (Start)
a(4) = a(2^ceiling(log_2(4-1)) + 2 - 4) + a(1) + a(2) + a(3) = a(2) + a(1) + a(2) + a(3) = 4.
a(5) = a(2^ceiling(log_2(5-1)) + 2 - 5) + a(1) + a(2) + a(3) + a(4) = a(1) + a(1) + a(2) + a(3) + a(4) = 8.
a(6) = a(2^ceiling(log_2(6-1)) + 2 - 6) + a(1) + a(2) + a(3) + a(4) + a(5) = a(4) + a(1) + a(2) + a(3) + a(4) + a(5) = 19.
a(7) =  a(7 - 1 - A006257(7-2)) + Sum_{i = 1..6} a(i) = a(3) +  Sum_{i = 1..6} a(i) = 35.
a(8) =  a(8 - 1 - A006257(8-2)) + Sum_{i = 1..7} a(i) = a(2) +  Sum_{i = 1..7} a(i) = 70. (End)

Crossrefs

Cf. A006257, A049885 (similar, but with minus a(m)), A049937, A049945.

Programs

  • Maple
    s := proc(n) option remember; `if`(n < 1, 0, a(n) + s(n - 1)); end proc;
a := proc(n) option remember;
  `if`(n < 4, 1, s(n - 1) + a(Bits:-Iff(n - 2, n - 2) + 3 - n));
end proc;
seq(a(n), n = 1 .. 30);
  • Mathematica
    b[n_] := Module[{p}, For[p = 0, True, p++, If[2^p < n - 1 <= 2^(p + 1), Return[p]]]];
a[n_] := a[n] = If[n < 4, 1, With[{m = 2^(b[n] + 1) + 2 - n}, Total[ Array[a, n - 1]] + a[m]]];
Array[a, 35] (* Jean-François Alcover, Apr 24 2020 *)

Formula

From Petros Hadjicostas, Nov 06 2019: (Start)
a(n) = a(2^ceiling(log_2(n-1)) + 2 - n) + Sum_{i = 1..n-1} a(i) for n >= 4.
a(n) = a(n - 1 - A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 4. (End)

Extensions

Name edited by and more terms from Petros Hadjicostas, Nov 06 2019

A049945 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n-1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 4.

Original entry on oeis.org

1, 1, 4, 7, 14, 34, 65, 127, 254, 634, 1206, 2381, 4742, 9477, 18951, 37899, 75798, 189494, 360040, 710606, 1416477, 2830593, 5660011, 11319450, 22638520, 45276913, 90553764, 181107497, 362214974, 724429941, 1448859879, 2897719755, 5795439510, 14488598774, 27528337672, 54332245406
Offset: 1

Views

Author

Clark Kimberling

Keywords

Examples

			From _Petros Hadjicostas_, Nov 06 2019: (Start)
a(4) = a(2^ceiling(log_2(4-1)) + 2 - 4) + a(1) + a(2) + a(3) = a(2) + a(1) + a(2) + a(3) = 7.
a(5) = a(2^ceiling(log_2(5-1)) + 2 - 5) + a(1) + a(2) + a(3) + a(4) = a(1) + a(1) + a(2) + a(3) + a(4) = 14.
a(6) = a(2^ceiling(log_2(6-1)) + 2 - 6) + a(1) + a(2) + a(3) + a(4) + a(5) = a(4) + a(1) + a(2) + a(3) + a(4) + a(5) = 34.
a(7) =  a(7 - 1 - A006257(7-2)) + Sum_{i = 1..6} a(i) = a(3) +  Sum_{i = 1..6} a(i) = 65.
a(8) =  a(8 - 1 - A006257(8-2)) + Sum_{i = 1..7} a(i) = a(2) +  Sum_{i = 1..7} a(i) = 127. (End)

Crossrefs

Cf. A006257, A049933, A049937.

Programs

  • Maple
    s:= proc(n) option remember; `if`(n<1, 0, a(n)+s(n-1)) end:
a:= proc(n) option remember; `if`(n<4, [1, 1, 4][n],
      s(n-1)+a(Bits:-Iff(n-2$2)+3-n))
    end:
seq(a(n), n=1..36); # Petros Hadjicostas, Nov 06 2019

Formula

From Petros Hadjicostas, Nov 06 2019: (Start)
a(n) = a(2^ceiling(log_2(n-1)) + 2 - n) + Sum_{i = 1..n-1} a(i) for n >= 4.
a(n) = a(n - 1 - A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 4. (End)

Extensions

Name edited by and more terms from Petros Hadjicostas, Nov 06 2019

A049929 a(n) = a(1) + a(2) + ... + a(n-1) - a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n-1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 4.

Original entry on oeis.org

1, 3, 4, 5, 12, 20, 41, 83, 168, 254, 550, 1121, 2250, 4507, 9015, 18031, 36064, 54098, 117212, 238932, 480121, 961371, 1923313, 3846922, 7693930, 15387945, 30775932, 61551885, 123103778, 246207563, 492415127, 984830255, 1969660512
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Empirical: Lim_{n->infinity} a(n+1)/a(n) = 2. - Iain Fox, Dec 05 2017

Examples

			For n = 4, 2^p < 3 <= 2^(p+1), so p = 1, m = 2^2 + 2 - 4 = 2, and a(n) = a(1) + a(2) + a(3) - a(2) = 1 + 3 + 4 - 3 = 5.
For n = 6, 2^p < 5 <= 2^(p+1), so p = 2, m = 2^3 + 2 - 6 = 4, and a(n) = a(1) + a(2) + a(3) + a(4) + a(5) - a(4) = 1 + 3 + 4 + 5 + 12 - 5 = 20.

Links

Crossrefs

Cf. A080079, A049933, A049937, A049945.

Programs

  • Mathematica
    Fold[Append[#1, Total@ #1 - #1[[2^Ceiling@ Log2@ #2 + 1 - #2]] ] &, {1, 3, 4}, Range[3, 32]] (* Michael De Vlieger, Dec 06 2017 *)
  • PARI
    first(n)= my(res = vector(n), s = 8); res[1]=1; res[2]=3; res[3]=4; for(x=4, n, res[x] = s - res[2*2^logint(x-2, 2)+2-x]; s += res[x]); res; \\ Iain Fox, Dec 05 2017

Formula

a(n) = (Sum_{i=1..n-1} a(i)) - a(2^ceiling(log_2(n-1)) + 2 - n) for n > 3. - Iain Fox, Dec 06 2017
For n > 3, a(n) is the sum of all previous terms except a(A080079(n-2)). - Iain Fox, Dec 13 2017

Extensions

Name edited by Petros Hadjicostas, Nov 06 2019

A049977 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n -1 <= 2^(p+1), with a(1) = 1, a(2) = 3, and a(3) = 4.

Original entry on oeis.org

1, 3, 4, 11, 20, 50, 93, 185, 368, 920, 1748, 3453, 6876, 13743, 27479, 54957, 109912, 274780, 522082, 1030428, 2053989, 4104555, 8207405, 16413982, 32827412, 65654641, 131309190, 262618337, 525236644, 1050473279, 2100946551, 4201893101, 8403786200, 21009465500, 39917984450
Offset: 1

Views

Author

Clark Kimberling

Keywords

Examples

			From _Petros Hadjicostas_, Nov 07 2019: (Start)
a(4) = a(2^ceiling(log_2(4-1)) + 2 - 4) + a(1) + a(2) + a(3) = a(2) + a(1) + a(2) + a(3) = 11.
a(5) = a(2^ceiling(log_2(5-1)) + 2 - 5) + a(1) + a(2) + a(3) + a(4) = a(1) + a(1) + a(2) + a(3) + a(4) = 20.
a(6) = a(2^ceiling(log_2(6-1)) + 2 - 6) + a(1) + a(2) + a(3) + a(4) + a(5) = a(4) + a(1) + a(2) + a(3) + a(4) + a(5) = 50.
a(7) =  a(7 - 1 - A006257(7-2)) + Sum_{i = 1..6} a(i) = a(3) +  Sum_{i = 1..6} a(i) = 93.
a(8) =  a(8 - 1 - A006257(8-2)) + Sum_{i = 1..7} a(i) = a(2) +  Sum_{i = 1..7} a(i) = 185. (End)

Crossrefs

Cf. A006257, A049933, A049937, A049945.

Programs

  • Maple
    s := proc(n) option remember; `if`(n < 1, 0, a(n) + s(n - 1)) end proc:
a := proc(n) option remember; `if`(n < 2, 1, `if`(n < 3, 3,
       `if`(n < 4, 4, s(n - 1) + a(Bits:-Iff(n - 2, n - 2) + 3 - n))))
     end proc:
seq(a(n), n = 1 .. 40); # Petros Hadjicostas, Nov 07 2019

Formula

From Petros Hadjicostas, Nov 07 2019: (Start)
a(n) = a(2^ceiling(log_2(n-1)) + 2 - n) + Sum_{i = 1..n-1} a(i) for n >= 4.
a(n) = a(n - 1 - A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 4. (End)

Extensions

Name edited by and more terms from Petros Hadjicostas, Nov 07 2019

A049909 a(n) = a(1) + a(2) + ... + a(n-1) - a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the smallest integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 3.

Original entry on oeis.org

1, 2, 3, 4, 9, 15, 31, 63, 127, 192, 416, 848, 1702, 3409, 6819, 13639, 27279, 40920, 88660, 180730, 363167, 727188, 1454808, 2909840, 5819745, 11639554, 23279140, 46558296, 93116598, 186233201, 372466403, 744932807, 1489865615
Offset: 1

Views

Author

Clark Kimberling

Keywords

Examples

			From _Petros Hadjicostas_, Nov 07 2019: (Start)
a(4) = -a(2^ceiling(log_2(4-1)) + 2 - 4) + a(1) + a(2) + a(3) = -a(2) + a(1) + a(2) + a(3) = 4.
a(5) = -a(2^ceiling(log_2(5-1)) + 2 - 5) + a(1) + a(2) + a(3) + a(4) = -a(1) + a(1) + a(2) + a(3) + a(4) = 9.
a(6) = -a(2^ceiling(log_2(6-1)) + 2 - 6) + a(1) + a(2) + a(3) + a(4) + a(5) = -a(4) + a(1) + a(2) + a(3) + a(4) + a(5) = 15.
a(7) =  -a(7 - 1 - A006257(7-2)) + Sum_{i = 1..6} a(i) = -a(3) +  Sum_{i = 1..6} a(i) = 31.
a(8) =  -a(8 - 1 - A006257(8-2)) + Sum_{i = 1..7} a(i) = -a(2) +  Sum_{i = 1..7} a(i) = 63. (End)

Crossrefs

Cf. A006257, A049933, A049937, A049945, A049977.

Programs

  • Maple
    s:= proc(n) option remember; `if`(n<1, 0, a(n)+s(n-1)) end:
a:= proc(n) option remember; `if`(n<4, [1, 2, 3][n],
      s(n-1) - a(Bits:-Iff(n-2$2) + 3 - n))
    end:
seq(a(n), n=1..40); # Petros Hadjicostas, Nov 07 2019

Formula

From Petros Hadjicostas, Nov 07 2019: (Start)
a(n) = -a(2^ceiling(log_2(n-1)) + 2 - n) + Sum_{i = 1..n-1} a(i) for n >= 4.
a(n) = -a(n - 1 - A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 4. (End)

Extensions

Name edited by Petros Hadjicostas, Nov 07 2019

A049949 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 1.

Original entry on oeis.org

1, 2, 1, 6, 11, 27, 49, 99, 197, 492, 934, 1846, 3676, 7347, 14689, 29379, 58757, 146892, 279094, 550846, 1098021, 2194212, 4387512, 8774582, 17548869, 35097640, 70195230, 140390438, 280780860, 561561715, 1123123425, 2246246851
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A006257, A049933, A049937, A049945.

Programs

  • Maple
    s := proc(n) option remember; `if`(n < 1, 0, a(n) + s(n - 1)) end proc:
a := proc(n) option remember;
`if`(n < 4, [1, 2, 1][n], s(n - 1) + a(2^ceil(log[2](n - 1)) + 2 - n)):
end proc:
seq(a(n), n = 1..34); # Petros Hadjicostas, Nov 12 2019

Extensions

Name edited by Petros Hadjicostas, Nov 12 2019

A049957 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 3.

Original entry on oeis.org

1, 2, 3, 8, 15, 37, 69, 137, 273, 682, 1296, 2560, 5098, 10189, 20373, 40745, 81489, 203722, 387072, 763960, 1522829, 3043120, 6084976, 12169338, 24338267, 48676398, 97352728, 194705424, 389410826, 778821645, 1557643285, 3115286569, 6230573137, 15576432842, 29595222400
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A006257, A049933, A049937, A049945.

Programs

  • Maple
    s := proc(n) option remember; `if`(n < 1, 0, a(n) + s(n - 1)) end proc:
a := proc(n) option remember;
`if`(n < 4, [1, 2, 3][n], s(n - 1) + a(2^ceil(log[2](n - 1)) + 2 - n)):
end proc:
seq(a(n), n = 1..34); # Petros Hadjicostas, Nov 11 2019

Extensions

Name edited by and more terms from Petros Hadjicostas, Nov 11 2019

A049973 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = a(3) = 3.

Original entry on oeis.org

1, 3, 3, 10, 18, 45, 83, 166, 330, 825, 1567, 3096, 6165, 12322, 24637, 49274, 98546, 246365, 468093, 923871, 1841585, 3680101, 7358673, 14716604, 29432713, 58865262, 117730441, 235460844, 470921661, 941843314, 1883686621
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A049933, A049937, A049945.

Programs

  • Maple
    s := proc(n) option remember; `if`(n < 1, 0, a(n) + s(n - 1)) end proc:
a := proc(n) option remember;
`if`(n < 4, [1, 3, 3][n], s(n - 1) + a(2^ceil(log[2](n - 1)) + 2 - n)):
end proc:
seq(a(n), n = 1..40); # Petros Hadjicostas, Nov 14 2019

Extensions

Name edited by Petros Hadjicostas, Nov 14 2019
Showing 1-8 of 8 results.

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