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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A298357 a(n) = a(n-1) + a(n-2) + a([n/2]) + a([n/3]) + ... + a([n/n]), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 9, 19, 37, 74, 131, 238, 410, 710, 1184, 2014, 3320, 5516, 9044, 14888, 24262, 39698, 64510, 105089, 170545, 277057, 449027, 728502, 1179967, 1912216, 3096110, 5014519, 8116824, 13141430, 21268343, 34425826, 55710704, 90162442, 145899135, 236104060
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.

Programs

  • Mathematica
    a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[a[Floor[n/k]], {k, 2, n}];
Table[a[n], {n, 0, 30}]  (* A298357 *)
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A298357(n):
    if n <= 2:
        return n+1
    c, j = A298357(n-1)+A298357(n-2), 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A298357(k1)
        j, k1 = j2, n//j2
    return c+2*(n-j+1) # Chai Wah Wu, Mar 31 2021

A298369 a(n) = a(n-1) + a(n-2) + 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, 7, 17, 38, 87, 164, 318, 576, 1040, 1773, 3134, 5241, 8877, 14728, 24579, 40298, 66585, 108610, 178004, 289717, 472312, 766643, 1247081, 2021980, 3281557, 5316888, 8619474, 13957420, 22611507, 36603571, 59270152, 95931095, 155290091, 251310597
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.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[k*a[Floor[n/k]], {k, 2, n}];
Table[a[n], {n, 0, 30}]  (* A298369 *)
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A298369(n):
    if n <= 2:
        return 1
    c, j = A298369(n-1)+A298369(n-2), 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2*(j2-1)-j*(j-1))*A298369(k1)//2
        j, k1 = j2, n//j2
    return c+(n*(n+1)-j*(j-1))//2 # Chai Wah Wu, Mar 31 2021

A298370 a(n) = a(n-1) + a(n-2) + 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, 38, 83, 190, 356, 695, 1254, 2267, 3861, 6829, 11417, 19340, 32076, 53545, 87784, 145048, 236589, 387765, 631106, 1028866, 1670013, 2716595, 4404599, 7148426, 11582096, 18776334, 30404300, 49256015, 79735758, 129111774, 208972513, 338277831
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.

Programs

  • Mathematica
    a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[k*a[Floor[n/k]], {k, 2, n}];
Table[a[n], {n, 0, 30}]  (* A298370 *)
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A298370(n):
    if n <= 2:
        return n+1
    c, j = A298370(n-1)+A298370(n-2), 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2*(j2-1)-j*(j-1))*A298370(k1)//2
        j, k1 = j2, n//j2
    return c+2*(n*(n+1)-j*(j-1))//2 # Chai Wah Wu, Mar 31 2021

A298414 a(n) = 2*a(n-1) - a(n-2) + a([n/2]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 12, 19, 30, 45, 67, 96, 137, 190, 262, 353, 474, 625, 821, 1062, 1370, 1745, 2216, 2783, 3487, 4328, 5359, 6580, 8063, 9808, 11906, 14357, 17282, 20681, 24705, 29354, 34824, 41115, 48468, 56883, 66668, 77823, 90723, 105368, 122229, 141306
Offset: 0

Views

Author

Clark Kimberling, Feb 11 2018

Keywords

Links

Crossrefs

Cf. A298338.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + a[Floor[n/2]];
Table[a[n], {n, 0, 30}]  (* A298414 *)

A298416 a(n) = 2*a(n-1) + 2*a(n-2) - a([n/2]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 3, 7, 19, 49, 133, 357, 973, 2641, 7209, 19651, 53671, 146511, 400231, 1093127, 2986359, 8157999, 22287743, 60888843, 166350531, 454471539, 1241636931, 3392197289, 9267648789, 25319638485, 69174520877, 188988172213, 516325239669, 1410626423533
Offset: 0

Views

Author

Clark Kimberling, Feb 11 2018

Keywords

Comments

a(n)/a(n-1) -> 1+sqrt(3).

Links

Crossrefs

Cf. A298338, A298417.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] =+ 2 a[n - 1] + 2 a[n - 2] - a[Floor[n/2]];
Table[a[n], {n, 0, 30}]  (* A298416 *)

A298417 a(n) = 2*a(n-1) + 2*a(n-2) + a([n/2]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 5, 13, 37, 105, 289, 801, 2193, 6025, 16473, 45101, 123253, 336997, 920789, 2516373, 6875125, 18785189, 51322821, 140222045, 383095757, 1046652077, 2859512141, 7812373537, 21343816457, 58312503241, 159312762649, 435250868777, 1189127599849
Offset: 0

Views

Author

Clark Kimberling, Feb 11 2018

Keywords

Comments

a(n)/a(n-1) -> 1+sqrt(3).

Links

Crossrefs

Cf. A298338, A298416.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = 2 a[n - 1] + 2 a[n - 2] + a[Floor[n/2]];
Table[a[n], {n, 0, 30}]  (* A298417 *)

A298412 a(n) = 2*a(n-1) + a(n-2) + a([n/2]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 4, 10, 25, 64, 157, 388, 943, 2299, 5566, 13495, 32620, 78892, 190561, 460402, 1111753, 2684851, 6482398, 15651946, 37788589, 91234690, 220263535, 531775255, 1283827540, 3099462955, 7482786070, 18065113987, 43613092936, 105291490420, 254196264337
Offset: 0

Views

Author

Clark Kimberling, Feb 11 2018

Keywords

Comments

a(n)/a(n-1) -> 1 + sqrt(2).

Links

Crossrefs

Cf. A298338.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = 2 a[n - 1] + a[n - 2] + a[Floor[n/2]];
Table[a[n], {n, 0, 30}]  (* A298412 *)

A298413 a(n) = a(floor(n/2))*a(ceiling(n/2)), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 36, 54, 81, 162, 324, 648, 1296, 1944, 2916, 4374, 6561, 13122, 26244, 52488, 104976, 209952, 419904, 839808, 1679616, 2519424, 3779136, 5668704, 8503056, 12754584, 19131876, 28697814, 43046721, 86093442, 172186884, 344373768, 688747536
Offset: 0

Views

Author

Clark Kimberling, Feb 11 2018

Keywords

Links

Crossrefs

Cf. A298338.

Programs

  • Mathematica
    a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n] = a[Floor[n/2]]*a[Ceiling[n/2]];
t = Table[a[n], {n, 0, 30}]  (* A298413 *)

A298415 a(n) = a(n-1) + 2*a(n-2) + a([n/2]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 4, 7, 16, 34, 70, 145, 292, 598, 1198, 2428, 4858, 9784, 19570, 39283, 78568, 157426, 314854, 630304, 1260610, 2522416, 5044834, 10092094, 20184190, 40373236, 80746474, 161502730, 323005462, 646030492, 1292060986, 2584161253, 5168322508, 10336723582
Offset: 0

Views

Author

Clark Kimberling, Feb 11 2018

Keywords

Links

Crossrefs

Cf. A298338.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = a[n - 1] + 2 a[n - 2] + a[Floor[n/2]];
Table[a[n], {n, 0, 30}]  (* A298415 *)

A298419 a(n) = n*a(n-1) - a(n-2) + a([n/2]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 3, 12, 58, 339, 2318, 18217, 161647, 1598311, 17419832, 207440012, 2679300663, 37302771588, 556862275475, 8872493654229, 150275529864635, 2696087044070848, 51075378307643124, 1018811479110389943, 21343965683012143990, 468548433547174197669
Offset: 0

Views

Author

Clark Kimberling, Feb 11 2018

Keywords

Links

Crossrefs

Cf. A298338.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = n a[n - 1] - a[n - 2] + a[Floor[n/2]];
Table[a[n], {n, 0, 30}]  (* A298419 *)
Previous Showing 31-40 of 42 results. Next

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