A049960 -id:A049960 - cp's OEIS frontend

A049939 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n-1 <= 2^(p+1), with a(1) = a(2) = 1 and a(3) = 2.

1, 1, 2, 5, 14, 24, 52, 123, 345, 568, 1140, 2299, 4697, 9839, 21409, 50358, 141235, 232113, 464230, 928479, 1857057, 3714559, 7430849, 14869238, 29778995, 59739745, 120175856, 243137792, 497430263, 1039731033, 2262860113

Offset: 1

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Sep 24 2019: (Start)
a(4) = a(1 + A006257(4-2)) + a(1) + a(2) + a(3) = a(2) + a(1) + a(2) + a(3) = 1 + 1 + 1 + 2 = 5.
a(5) = a(1 + A006257(5-2)) + a(1) + a(2) + a(3) + a(4) = a(4) + a(1) + a(2) + a(3) + a(4) = 5 + 1 + 1 + 2 + 5 = 14.
a(6) = a(1 + A006257(6-2)) + a(1) + a(2) + a(3) + a(4) + a(5) = a(2) +  a(1) + a(2) + a(3) + a(4) + a(5) = 1 + 1 + 1 + 2 + 5 + 14 = 24.
(End)

Cf. A006257, A049890 (similar, but with minus a(m/2)), A049891 (similar, but with minus a(m)), A049938 (similar, but with plus a(m/2)), A049940, A049960, A049964.

a := proc(n) local i; option remember; if n < 4 then return [1, 1, 2][n]; end if; add(a(i), i = 1 .. n - 1) + a(2*n - 3 - Bits:-Iff(n - 2, n - 2)); end proc;
seq(a(n), n = 1 .. 37); # Petros Hadjicostas, Sep 24 2019, courtesy of Peter Luschny

a(n) = a(1 + A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 4 with a(1) = a(2) = 1 and a(3) = 2.

Name edited by Petros Hadjicostas, Sep 24 2019

A049964 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) 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) = 1.

Original entry on oeis.org

1, 3, 1, 6, 12, 24, 48, 107, 250, 453, 906, 1823, 3682, 7566, 15788, 34352, 80810, 145833, 291666, 583343, 1166722, 2333646, 4667948, 9338672, 18689450, 37443922, 75098700, 151072456, 305646138, 625313778, 1307036806, 2844621050, 6690632768, 12074228731, 24148457462, 48296914935, 96593829906

Offset: 1

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Sep 24 2019: (Start)
a(4) = a(1) + a(2) + a(3) + a(m=1) = 1 + 3 + 1 + 1 = 6 because m = A006257(4-2) = 2*4 - 3 - 2^ceiling(log_2(4-1)) = 1.
a(5) = a(1) + a(2) + a(3) + a(4) + a(m=3) = 1 + 3 + 1 + 6 + 1 = 12 because m = A006257(5-2) = 2*5 - 3 - 2^ceiling(log_2(5-1)) = 3.
a(6) = a(1) + a(2) + a(3) + a(4) + a(5) + a(m=1) = 1 + 3 + 1 + 6 + 12 + 1 = 24 because m = A006257(6-2) = 2*6 - 3 - 2^ceiling(log_2(6-1)) = 1.
(End)

Cf. A006257, A049939, A049940, A049960.

a := proc(n) option remember; if n<4 then return [1,3,1][n] fi; add(a(i), i=1..n-1) + a(2*(n-2) - Bits:-Iff(n-2, n-2)) end: seq(a(n), n=1..37); # Petros Hadjicostas, Sep 24 2019, courtesy of Peter Luschny

a(n) = a(2*n - 3 - 2^ceiling(log_2(n-1))) + Sum_{i = 1..n-1} a(i) = a(A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 4 with a(1) = 1, a(2) = 3, and a(3) = 1. - Petros Hadjicostas, Sep 24 2019

Name edited by and more terms from Petros Hadjicostas, Sep 23 2019

A049940 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 3, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1) with a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 3, 6, 14, 26, 54, 119, 278, 503, 1008, 2027, 4094, 8412, 17554, 38194, 89848, 162143, 324288, 648587, 1297214, 2594652, 5190034, 10383154, 20779768, 41631830, 83498100, 167969126, 339831072, 695251878, 1453222088, 3162777148, 7438945312, 13424668537, 26849337076, 53698674163

Offset: 1

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Sep 24 2019: (Start)
a(3) = a(1) + a(2) + a(m=1) = 1 + 1 + 1 = 3 because m = A006257(3-2) = 2*3 - 3 - 2^ceiling(log[2](3-1)) = 1.
a(4) = a(1) + a(2) + a(3) + a(m=1) = 1 + 1 + 3 + 1 = 6 because m = A006257(4-2) = 2*4 - 3 - 2^ceiling(log[2](4-1)) = 1.
a(5) = a(1) + a(2) + a(3) + a(4) + a(m=3) = 1 + 1 + 3 + 6 + 3 = 14 because m = A006257(5-2) = 2*5 - 3 - 2^ceiling(log[2](5-1)) = 3.
a(6) = a(1) + a(2) + a(3) + a(4) + a(5) + a(m=1) = 1 + 1 + 3 + 6 + 14 + 1 = 26 because m = A006257(6-2) = 2*6 - 3 - 2^ceiling(log[2](6-1)) = 1.
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..3320

Cf. A006257, A049939, A049960, A049964.

a := proc(n) local vv, i; option remember; if n = 1 then vv := 1; end if; if n = 2 then vv := 1; end if; if 3 <= n then vv := 0; for i to n - 1 do vv := vv + a(i); end do; vv := vv + a(2*n - 3 - 2^ceil(log[2](n - 1))); end if; vv; end proc; # Petros Hadjicostas, Sep 24 2019
# second Maple program:
s:= proc(n) option remember; `if`(n<1, 0, a(n)+s(n-1)) end:
a:= proc(n) option remember; `if`(n<3, 1,
      s(n-1)+a(2*(n-2^ilog2(n-2))-3))
    end:
seq(a(n), n=1..36);  # Alois P. Heinz, Sep 24 2019

s[n_] := s[n] = If[n < 1, 0, a[n] + s[n-1]];
a[n_] := a[n] = If[n < 3, 1, s[n-1] + a[2(n - 2^Floor@Log[2, n-2]) - 3]];
Array[a, 36] (* Jean-François Alcover, Apr 23 2020, after Alois P. Heinz *)

a(n) = a(2*n - 3 - 2^ceiling(log_2(n-1))) + Sum_{i = 1..n-1} a(i) = a(A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 3 with a(1) = a(2) = 1. - Petros Hadjicostas, Sep 24 2019

Name edited by and more terms from Petros Hadjicostas, Sep 24 2019

A049938 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = n - 1 - 2^p and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), with a(1) = a(2) = 1 and a(3) = 2.

Original entry on oeis.org

1, 1, 2, 5, 10, 20, 40, 81, 165, 326, 652, 1305, 2613, 5231, 10472, 20964, 41969, 83858, 167716, 335433, 670869, 1341743, 2683496, 5367012, 10734065, 21468214, 42936589, 85873504, 171747661, 343496630, 686995878, 1373996997, 2748004486, 5495988009, 10991976018, 21983952037, 43967904077

Offset: 1

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Oct 01 2019: (Start)
a(4) = a(4 - 1 - 2^ceiling(-1 + log_2(3))) + a(1) + a(2) + a(3) = a(1) + a(1) + a(2) + a(3) = 5.
a(5) = a(5 - 1 - 2^ceiling(-1 + log_2(4))) + a(1) + a(2) + a(3) + a(4) = a(2) + a(1) + a(2) + a(3) + a(4) = 10.
a(6) = a(6 - 1 - 2^ceiling(-1 + log_2(5))) + a(1) + a(2) + a(3) + a(4) + a(5) = a(1) + a(1) + a(2) + a(3) + a(4) + a(5) = 20.
(End)

Cf. A006257, A049890 (similar, but with minus a(m)), A049891 (similar, but with minus a(2*m)), A049939 (similar, but with plus a(2*m)), A049940, A049960, A049964, A049978.

a := proc(n) local i; option remember; if n < 4 then return [1, 1, 2][n]; end if; add(a(i), i = 1 .. n - 1) + a(n - 3/2 - 1/2*Bits:-Iff(n - 2, n - 2)); end proc; # Petros Hadjicostas, Oct 01 2019

lista(nn) = { nn = max(nn, 3); my(va = vector(nn)); va[1] = 1; va[2] = 1; va[3] = 2; my(sa = vecsum(va)); for (n=4, nn, va[n] = sa + va[n - 1 - 2^ceil(-1 + log(n-1)/log(2))]; sa += va[n]; ); va; } \\ Petros Hadjicostas, Apr 27 2020

a(n) = a(n - 1 - 2^ceiling(-1 + log_2(n-1))) + Sum_{i = 1..n-1} a(i) = a((1 + A006257(n-2))/2) + Sum_{i = 1..n-1} a(i) for n >= 4 with a(1) = a(2) = 1 and a(3) = 2. - Petros Hadjicostas, Oct 01 2019

More terms from Petros Hadjicostas, Oct 01 2019

A049963 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n-1 <= 2^(p+1), with a(1) = 1, a(2) = 2 and a(3) = 4.

Original entry on oeis.org

1, 2, 4, 9, 25, 43, 93, 220, 617, 1016, 2039, 4112, 8401, 17598, 38292, 90070, 252612, 415156, 830319, 1660672, 3321521, 6643838, 13290772, 26595030, 53262532, 106850150, 214945816, 434874798, 889700788, 1859656696

Offset: 1

Clark Kimberling

nonn

The number m in the definition of the sequence equals 2*n - 2 - x, where x is the smallest power of 2 >= n-1. It turns out that m = 1 + A006257(n-2), where the sequence b(n) = A006257(n) satisfies b(2*n) = 2*b(n) - 1 and b(2*n + 1) = 2*b(n) + 1, and it is related to the so-called Josephus problem. - Petros Hadjicostas, Sep 25 2019

			From _Petros Hadjicostas_, Sep 25 2019: (Start)
a(4) = a(1 + A006257(4-2)) + a(1) + a(2) + a(3) = a(2) + a(1) + a(2) + a(3) = 9.
a(7) = a(1 + A006257(7-2)) + a(1) + a(2) + a(3) + a(4) + a(5) + a(6) = a(4) + a(1) + a(2) + a(3) + a(4) + a(5) + a(6) = 93.
(End)

Index entries for sequences related to the Josephus Problem

Cf. A006257, A049920, A049939, A049960, A049964, A049979.

Cf. A049914 (similar, but with minus a(m/2)), A049915 (similar, but with minus a(m)), A049962 (similar, but with plus a(m/2)).

a := proc(n) local i; option remember; if n < 4 then return [1, 2, 4][n]; end if; add(a(i), i = 1 .. n - 1) + a(2*n - 3 - Bits:-Iff(n - 2, n - 2)); end proc;
seq(a(n), n = 1..40); # Petros Hadjicostas, Sep 25 2019, courtesy of Peter Luschny

a(n) = a(1 + A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 4 with a(1) = 1, a(2) = 2 and a(3) = 4. - Petros Hadjicostas, Sep 25 2019

Name edited by Petros Hadjicostas, Sep 25 2019

A049920 a(n) = a(1) + a(2) + ... + a(n-1) - a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) 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) = 2.

Original entry on oeis.org

1, 3, 2, 5, 9, 19, 37, 67, 106, 248, 495, 983, 1938, 3807, 7225, 13007, 20727, 48678, 97355, 194703, 389378, 778687, 1556985, 3112527, 6219767, 12426032, 24775436, 49258849, 97350091, 190037400, 361519131, 650463607, 1036758174

Offset: 1

Clark Kimberling

nonn

The number m in the definition of the sequence equals 2*n - 3 - x, where x is the smallest power of 2 >= n-1. It turns out that m = A006257(n-2), where the sequence b(n) = A006257(n) satisfies b(2*n) = 2*b(n) - 1 and b(2*n + 1) = 2*b(n) + 1, and it is related to the so-called Josephus problem. - Petros Hadjicostas, Sep 25 2019

			From _Petros Hadjicostas_, Sep 25 2019: (Start)
a(4) = -a(A006257(4-2)) + a(1) + a(2) + a(3) = -a(1) + a(1) + a(2) + a(3) = 5.
a(5) = -a(A006257(5-2)) + a(1) + a(2) + a(3) + a(4) = -a(3) + a(1) + a(2) + a(3) + a(4) = 9.
a(6) = -a(A006257(6-2)) + a(1) + a(2) + a(3) + a(4) + a(5) = 19.
a(7) = -a(A006257(7-2)) + a(1) + a(2) + a(3) + a(4) + a(5) + a(6) = 37.
(End)

Robert Israel, Table of n, a(n) for n = 1..3323
Index entries for sequences related to the Josephus Problem

Cf. A006257, A049939, A049960, A049964, A049979.

A[1]:= 1: A[2]:= 3: A[3]:= 2:
for n from 4 to 100 do
  q:= ceil(log[2](n-1));
  m:= 2*n-3-2^q;
  A[n]:= add(A[i],i=1..n-1)-A[m];
od:
seq(A[i],i=1..100); # Robert Israel, Feb 27 2017

a(n) = -a(A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 4 with a(1) = 1, a(2) = 3, and a(3) = 2.

Name edited by Petros Hadjicostas, Sep 25 2019

A049979 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) 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, 30, 52, 112, 265, 743, 1224, 2456, 4953, 10119, 21197, 46123, 108490, 304273, 500059, 1000126, 2000293, 4000799, 8002557, 16008843, 32033930, 64155153, 128701875, 258903984, 523810232, 1071651837, 2239971619

Offset: 1

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Sep 24 2019: (Start)
a(4) = a(1 + A006257(4-2)) + a(1) + a(2) + a(3) = a(2) + a(1) + a(2) + a(3) = 3 + 1 + 3 + 4 = 11.
a(7) = a(1 + A006257(7-2)) + a(1) + ... + a(6) = a(4) + a(1) + ... + a(6) = 11 + 1 + 3 + 4 + 11 + 30 + 52 = 112.
(End)

Cf. A006257, A049939, A049940, A049960, A049964.

a := proc(n) local i; option remember; if n < 4 then return [1, 3, 4][n]; end if; add(a(i), i = 1 .. n - 1) + a(2*n - 3 - Bits:-Iff(n - 2, n - 2)); end proc;
seq(a(n), n = 1 .. 37); # Petros Hadjicostas, Sep 24 2019, courtesy of Peter Luschny

a(n) = a(1 + A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 4 with a(1) = 1, a(2) = 3, and a(3) = 4. - Petros Hadjicostas, Sep 24 2019

Name edited by Petros Hadjicostas, Sep 24 2019

A049978 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = n - 1 - 2^p 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, 9, 20, 38, 78, 157, 319, 630, 1262, 2525, 5055, 10121, 20260, 40560, 81199, 162242, 324486, 648973, 1297951, 2595913, 5191844, 10383728, 20767535, 41535232, 83070775, 166142182, 332285627, 664573784, 1329152634, 2658315407, 5316651114, 10633261669, 21266523340, 42533046681, 85066093367, 170132186745

Offset: 1

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Sep 27 2019: (Start)
a(4) = a(4-1-2^ceiling(-1 + log_2(4-1))) + a(1) + a(2) + a(3) = a(1) + a(1) + a(2) + a(3) = 9.
a(5) = a(5-1-2^ceiling(-1 + log_2(5-1))) + a(1) + a(2) + a(3) + a(4) = a(2) + a(1) + a(2) + a(3) + a(4) = 20.
a(6) = a(6-1-2^ceiling(-1 + log_2(6-1))) + a(1) + a(2) + a(3) + a(4) + a(5) = a(1) + a(1) + a(2) + a(3) + a(4) + a(5) = 38.
(End)

Cf. A006257, A049939, A049940, A049960, A049964.

a := proc(n) local i; option remember; if n < 4 then return [1, 3, 4][n]; end if; add(a(i), i = 1 .. n - 1) + a(n - 3/2 - 1/2*Bits:-Iff(n - 2, n - 2)); end proc;
seq(a(n), n = 1 .. 37); # Petros Hadjicostas, Sep 27 2019 using a modification of a program by Peter Luschny

a(n) = a(n - 1 - 2^ceiling(-1 + log_2(n-1))) + Sum_{i = 1..n-1} a(i) = a((1 + A006257(n-2))/2) + Sum_{i = 1..n-1} a(i) for n >= 4 with a(1) = 1, a(2) = 3, and a(3) = 4. - Petros Hadjicostas, Sep 27 2019

Name edited by and more terms from Petros Hadjicostas, Sep 27 2019

cp's OEIS Frontend

A049939 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n-1 <= 2^(p+1), with a(1) = a(2) = 1 and a(3) = 2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A049964 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) 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) = 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A049940 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 3, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1) with a(1) = a(2) = 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A049938 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = n - 1 - 2^p and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), with a(1) = a(2) = 1 and a(3) = 2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A049963 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n-1 <= 2^(p+1), with a(1) = 1, a(2) = 2 and a(3) = 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A049920 a(n) = a(1) + a(2) + ... + a(n-1) - a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) 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) = 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A049979 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) 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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A049978 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = n - 1 - 2^p 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.

Views