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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A298347 a(n) = a(n-1) + a(n-2) + 2 a([n/2]), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 9, 18, 33, 69, 120, 225, 381, 672, 1119, 1929, 3186, 5355, 8781, 14586, 23817, 39165, 63744, 104253, 169341, 275832, 447411, 727101, 1178370, 1911843, 3096585, 5019138, 8126433, 13163133, 21307128, 34499433, 55835733, 90382800, 146266167, 236727297
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] + 2 a[Floor[n/2]];
Table[a[n], {n, 0, 30}]  (* A298347 *)

A298348 a(n) = a(n-1) + a(n-2) + a([(n+1)/2]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 19, 35, 59, 105, 175, 299, 493, 827, 1355, 2241, 3655, 6001, 9761, 15937, 25873, 42109, 68281, 110883, 179657, 291367, 471851, 764573, 1237779, 2004593, 3244613, 5252861, 8501129, 13759991, 22267121, 36036873, 58313755, 94366565, 152696257
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] + a[Floor[(n+1)/2]];
Table[a[n], {n, 0, 30}]  (* A298348 *)

A298349 a(n) = a(n-1) + a(n-2) + a([(n+1)/2]), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 8, 14, 30, 52, 96, 162, 288, 480, 820, 1352, 2268, 3716, 6146, 10024, 16458, 26770, 43708, 70958, 115486, 187264, 304102, 492718, 799088, 1294074, 2096878, 3394668, 5497692, 8898506, 14406222, 23314752, 37737432, 61068642, 98832844, 159928256
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] + a[Floor[(n+1)/2]];
Table[a[n], {n, 0, 30}]  (* A298349 *)

A298350 a(n) = a(n-1) + a(n-2) + 2 a(ceiling(n/2)), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 4, 7, 19, 34, 67, 115, 220, 373, 661, 1102, 1897, 3133, 5260, 8623, 14323, 23386, 38455, 62587, 102364, 166273, 270841, 439318, 713953, 1157065, 1877284, 3040615, 4928419, 7979554, 12925219, 20922019, 33875884, 54826549, 88749205, 143622526
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] + a[Ceiling[n/2]];
Table[a[n], {n, 0, 30}]  (* A298350 *)

A298351 a(n) = a(n-1) + a(n-2) + 2 a(ceiling(n/2)), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 11, 20, 53, 95, 188, 323, 617, 1046, 1853, 3089, 5318, 8783, 14747, 24176, 40157, 65567, 107816, 175475, 286997, 466178, 759353, 1231709, 2001698, 3244043, 5263307, 8524916, 13817717, 22372127, 36238196, 58658675, 94977185, 153716174, 248824493
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] + a[Ceiling[n/2]];
Table[a[n], {n, 0, 30}]  (* A298351 *)

A298352 a(n) = a(n-1) + a(n-2) + a([(n-1)/2]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 15, 27, 45, 77, 127, 213, 349, 577, 941, 1545, 2513, 4103, 6661, 10841, 17579, 28547, 46253, 75013, 121479, 196841, 318669, 516087, 835333, 1352361, 2188635, 3542541, 5732721, 9277775, 15013009, 24294887, 39311999, 63613547, 102932207
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] + a[Floor[(n-1)/2]];
Table[a[n], {n, 0, 30}]  (* A298352 *)

A298353 a(n) = a(n-1) + a(n-2) + a([(n-1)/2]), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 7, 12, 22, 37, 66, 110, 188, 310, 520, 852, 1409, 2298, 3773, 6137, 10020, 16267, 26475, 42930, 69715, 112955, 183190, 296665, 480707, 778224, 1260340, 2039973, 3302611, 5344882, 8651266, 13999921, 22657324, 36663382, 59330726, 96004128, 155351121
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] + a[Floor[(n-1)/2]];
Table[a[n], {n, 0, 30}]  (* A298353 *)

A298354 a(n) = a(n-1) + a(n-2) + 2 a([(n-1)/2]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 4, 7, 13, 22, 43, 73, 130, 217, 373, 616, 1033, 1693, 2812, 4591, 7549, 12286, 20095, 32641, 53170, 86245, 140161, 227152, 368545, 596929, 967540, 1566535, 2537461, 4107382, 6650467, 10763473, 17423122, 28195777, 45633997, 73844872, 119503441
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] + 2 a[Floor[(n-1)/2]];
Table[a[n], {n, 0, 30}]  (* A298354 *)

A298355 a(n) = a(n-1) + a(n-2) + 2 a([(n-1)/2]), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 9, 16, 31, 53, 102, 173, 307, 512, 881, 1455, 2442, 4003, 6649, 10856, 17851, 29053, 47518, 77185, 125727, 203936, 331425, 537123, 871458, 1411491, 2287833, 3704208, 6000047, 9712261, 15725606, 25451165, 41198483, 66671360, 107905545, 174612607
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] + 2 a[Floor[(n-1)/2]];
Table[a[n], {n, 0, 30}]  (* A298355 *)

A298356 a(n) = a(n-1) + a(n-2) + a([n/2]) + a([n/3]) + ... + a([n/n]), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 4, 8, 16, 32, 57, 103, 178, 308, 514, 874, 1441, 2394, 3926, 6462, 10531, 17231, 28001, 45614, 74026, 120258, 194903, 316210, 512171, 830007, 1343883, 2176578, 3523150, 5704107, 9231637, 14942711, 24181525, 39135483, 63328289, 102482212, 165828942
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[a[Floor[n/k]], {k, 2, n}];
Table[a[n], {n, 0, 30}]  (* A298356 *)
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A298356(n):
    if n <= 2:
        return 1
    c, j = A298356(n-1)+A298356(n-2), 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A298356(k1)
        j, k1 = j2, n//j2
    return c+n-j+1 # Chai Wah Wu, Mar 31 2021
Previous Showing 21-30 of 42 results. Next

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