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.

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A298408 a(n) = 2*a(n-1) - a(n-3) + 2*a(floor(n/2)) + 3*a(floor(n/3)) + ... + n*a(floor(n/n)), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 6, 20, 53, 130, 277, 574, 1115, 2126, 3862, 7021, 12341, 21553, 36957, 63111, 106224, 178407, 296638, 492231, 811731, 1335994, 2188950, 3583027, 5847108, 9532980, 15512342, 25226123, 40967842, 66506422, 107869832, 174908573, 283452771, 459264017
Offset: 0

Views

Author

Clark Kimberling, Feb 10 2018

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622), so that (a(n)) has the growth rate of the Fibonacci numbers (A000045). See A298338 for a guide to related sequences.

Links

Crossrefs

Cf. A001622, A000045, A298338, A298409.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + Sum[k*a[Floor[n/k]], {k, 2, n}];
Table[a[n], {n, 0, 90}]  (* A298408  *)
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A298408(n):
    if n <= 2:
        return 1
    c, j = 2*A298408(n-1)-A298408(n-3), 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2*(j2-1)-j*(j-1))*A298408(k1)//2
        j, k1 = j2, n//j2
    return c+(n*(n+1)-j*(j-1))//2 # Chai Wah Wu, Mar 31 2021

A298409 a(n) = 2*a(n-1) - a(n-3) + 2*a(floor(n/2)) + 3*a(floor(n/3)) + ... + n*a(floor(n/n)), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 15, 48, 123, 300, 635, 1316, 2555, 4873, 8850, 16096, 28296, 49424, 84749, 144733, 243607, 409156, 680308, 1128889, 1861633, 3063978, 5020133, 8217296, 13409702, 21862824, 35575784, 57853195, 93954953, 152524643, 247386674, 401132014, 650065133
Offset: 0

Views

Author

Clark Kimberling, Feb 10 2018

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622), so that (a(n)) has the growth rate of the Fibonacci numbers (A000045). See A298338 for a guide to related sequences.

Links

Crossrefs

Cf. A001622, A000045, A298338, A298408.

Programs

  • Mathematica
    a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + Sum[k*a[Floor[n/k]], {k, 2, n}];
Table[a[n], {n, 0, 90}]  (* A298409  *)
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A298409(n):
    if n <= 2:
        return n+1
    c, j = 2*A298409(n-1)-A298409(n-3), 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2*(j2-1)-j*(j-1))*A298409(k1)//2
        j, k1 = j2, n//j2
    return c+2*(n*(n+1)-j*(j-1))//2 # Chai Wah Wu, Mar 31 2021

A298469 a(n) = a(0)*b(n) + a(1)*b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2; b(1) = 4 ; b(2) = 5.

Original entry on oeis.org

1, 3, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 66, 73, 77, 82, 89, 93, 98, 105, 109, 114, 121, 125, 130, 137, 141, 146, 153, 157, 162, 169, 173, 178, 185, 189, 194, 201, 205, 210, 217, 221, 226, 233, 237, 242, 249, 253, 257
Offset: 0

Views

Author

Clark Kimberling, Feb 11 2018

Keywords

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A297830 for a guide to related sequences.

Examples

			a(2) = 1*5 + 3*4 = 17.

Links

Crossrefs

Cf. A298338, A298295.

Programs

  • Mathematica
    mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
aCoeffs = {1, 3}; bCoeffs = {2, 4, 5};
Table[a[n - 1] = #[[n]], {n, Length[#]}] &[aCoeffs];
Table[b[n - 1] = #[[n]], {n, Length[#]}] &[bCoeffs];
a[n_] := Hold[Sum[a[z] b[n - z], {z, 0, Length[aCoeffs] - 1}]]
Table[{a[z] = ReleaseHold[a[z]], b[z + 1] =
    mex[Join[Table[a[n], {n, 0, z}], Table[b[n], {n, 0, z}]], 1]}, {z,
    Length[aCoeffs], 1000}];
Table[a[n], {n, 0, 50}]  (* A298469 *)
Table[b[n], {n, 0, 50}]  (* complement *)
(* Peter J. C. Moses, Jan 19 2018 *)

A298340 a(n) = a(n-1) + a(n-2) + a([n/3]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 15, 25, 41, 69, 113, 185, 303, 493, 801, 1303, 2113, 3425, 5553, 8993, 14561, 23579, 38165, 61769, 99975, 161785, 261801, 423655, 685525, 1109249, 1794887, 2904249, 4699249, 7603683, 12303117, 19906985, 32210405, 52117693, 84328401
Offset: 0

Views

Author

Clark Kimberling, Feb 09 2018

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622), so that (a(n)) has the growth rate of the Fibonacci numbers (A000045). See A298338 for a guide to related sequences.

Links

Crossrefs

Cf. A001622, A000045, A298338.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = a[n - 1] + a[n - 2] + a[Floor[n/3]];
Table[a[n], {n, 0, 30}]  (* A298340 *)

A298341 a(n) = a(n-1) + a(n-2) + a([n/3]), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 7, 12, 21, 36, 60, 99, 166, 272, 445, 729, 1186, 1927, 3134, 5082, 8237, 13355, 21628, 35019, 56707, 91786, 148553, 240438, 389090, 629627, 1018883, 1648676, 2667725, 4316673, 6984670, 11301615, 18286730, 29588790, 47875965, 77465484, 125342178
Offset: 0

Views

Author

Clark Kimberling, Feb 09 2018

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622), so that (a(n)) has the growth rate of the Fibonacci numbers (A000045). See A298338 for a guide to related sequences.

Links

Crossrefs

Cf. A001622, A000045, A298338.

Programs

  • Mathematica
    a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + a[Floor[n/3]];
Table[a[n], {n, 0, 30}]  (* A298341 *)

A298342 a(n) = a(n-1) + a(n-2) + a([2n/3]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 21, 37, 69, 127, 217, 381, 667, 1117, 1911, 3245, 5373, 8999, 15039, 24705, 40861, 67477, 110249, 180971, 296593, 482937, 788529, 1286505, 2090073, 3401283, 5532217, 8974361, 14574055, 23658665, 38342969, 62182605, 100822167, 163301365
Offset: 0

Views

Author

Clark Kimberling, Feb 09 2018

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622), so that (a(n)) has the growth rate of the Fibonacci numbers (A000045). See A298338 for a guide to related sequences.

Links

Crossrefs

Cf. A001622, A000045, A298338.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = a[n - 1] + a[n - 2] + a[Floor[2n/3]];
Table[a[n], {n, 0, 30}]  (* A298342 *)

A298343 a(n) = a(n-1) + a(n-2) + a([2n/3]), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 8, 14, 30, 58, 102, 190, 350, 598, 1050, 1838, 3078, 5266, 8942, 14806, 24798, 41442, 68078, 112598, 185942, 303806, 498690, 817302, 1330798, 2172898, 3545138, 5759478, 9372694, 15244770, 24730062, 40160774, 65194642, 105659222, 171352554, 277829078
Offset: 0

Views

Author

Clark Kimberling, Feb 09 2018

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622), so that (a(n)) has the growth rate of the Fibonacci numbers (A000045). See A298338 for a guide to related sequences.

Links

Crossrefs

Cf. A001622, A000045, A298338.

Programs

  • Maple
    A298343 := proc(n)
    option remember ;
    if n <=2 then
        n+1 ;
    else
        procname(n-1)+procname(n-2)+procname(floor(2*n/3)) ;
    end if;
end proc:
seq(A298343(n),n=0..80) ; # R. J. Mathar, Apr 26 2022
  • Mathematica
    a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + a[Floor[2n/3]];
Table[a[n], {n, 0, 30}]  (* A298343 *)

A298344 a(n) = a(n-1) + a(n-2) + a([n/3]) + a([2n/3]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 4, 7, 16, 31, 55, 103, 193, 331, 583, 1024, 1717, 2941, 5005, 8293, 13897, 23245, 38197, 63190, 104383, 170569, 280012, 458977, 747385, 1220362, 1991185, 3234985, 5264560, 8563066, 13891147, 22558927, 36621226, 59351305, 96253126, 156064432
Offset: 0

Views

Author

Clark Kimberling, Feb 09 2018

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622), so that (a(n)) has the growth rate of the Fibonacci numbers (A000045). See A298338 for a guide to related sequences.

Links

Crossrefs

Cf. A001622, A000045, A298338.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = a[n - 1] + a[n - 2] + a[Floor[n/3]] + a[Floor[2n/3]];
Table[a[n], {n, 0, 30}]  (* A298344 *)

A298345 a(n) = a(n-1) + a(n-2) + a([n/3]) + a([2n/3]), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 10, 18, 40, 79, 140, 262, 491, 842, 1483, 2605, 4368, 7482, 12732, 21096, 35351, 59131, 97166, 160744, 265532, 433898, 712302, 1167558, 1901218, 3104389, 5065229, 8229240, 13392126, 21782952, 35336664, 57385990, 93158035, 150979406, 244851226
Offset: 0

Views

Author

Clark Kimberling, Feb 09 2018

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622), so that (a(n)) has the growth rate of the Fibonacci numbers (A000045). See A298338 for a guide to related sequences.

Links

Crossrefs

Cf. A001622, A000045, A298338.

Programs

  • Mathematica
    a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + a[Floor[n/3]] + a[Floor[2n/3]];
Table[a[n], {n, 0, 30}]  (* A298345 *)

A298346 a(n) = a(n-1) + a(n-2) + 2 a([n/2]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 4, 7, 13, 28, 49, 91, 154, 271, 451, 778, 1285, 2161, 3544, 5887, 9613, 15808, 25729, 42079, 68350, 111331, 180583, 293470, 475609, 771649, 1249828, 2025799, 3279949, 5312836, 8599873, 13924483, 22536130, 36479839, 59035195, 95546650, 154613461
Offset: 0

Views

Author

Clark Kimberling, Feb 09 2018

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622), so that (a(n)) has the growth rate of the Fibonacci numbers (A000045). See A298338 for a guide to related sequences.

Links

Crossrefs

Cf. A001622, A000045, A298338.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = a[n - 1] + a[n - 2] + 2 a[Floor[n/2]];
Table[a[n], {n, 0, 30}]  (* A298346 *)
Previous Showing 11-20 of 42 results. Next

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