A298338 -id:A298338 - cp's OEIS frontend

A363503 a(n+1) = 2*a(n) + A298338(n-1), with a(1) = 1.

1, 3, 7, 15, 33, 71, 151, 319, 667, 1385, 2855, 5855, 11949, 24299, 49255, 99597, 200967, 404845, 814425, 1636581, 3285713, 6591853, 13216829, 26487447, 53062045, 106265431, 212759755, 425890401, 852381243, 1705734905, 3413043757, 6828635653, 13661395165

Offset: 1

Evangelos G. Filothodoros, Jun 06 2023

nonn

a(n) is an odd number that appears in m* = log(a(n)*phi^2) for the fermion condensate mass in odd dimensional and large N Gross-Neveu model at imaginary chemical potential and finite temperature.

			a(1) = 1;
a(2) = 2*a(1) + 1 = 2*1 + 1 = 3;
etc.

Evangelos G. Filothodoros, Strongly coupled fermions in odd dimensions and the running cut-off Lambda_d, arXiv:2306.14652 [hep-th], 2023.

Cf. A104457 (phi^2), A298338.

b(n) = if (n<=2, 1, b(n-1) + b(n-2) + b(n\2)); \\ A298338
a(n) = if (n==1, 1, 2*a(n-1) + b(n-2)); \\ Michel Marcus, Jun 06 2023

first(n) = {n = max(3, n); my(A298338 = vector(n+1), res = vector(n)); res[1] = 1; for(i = 1, 3, A298338[i] = 1); for(i = 4, n+1, A298338[i] = A298338[i-1] + A298338[i-2] + A298338[(i-1)\2 + 1]); for(i = 2, n, res[i] = 2*res[i-1] + A298338[i-1]); res} \\ David A. Corneth, Jun 08 2023

A298339 a(n) = a(n-1) + a(n-2) + a([n/2]), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 7, 13, 23, 43, 73, 129, 215, 367, 605, 1015, 1663, 2751, 4487, 7367, 11983, 19565, 31763, 51695, 83825, 136125, 220555, 357695, 579265, 938623, 1519551, 2460925, 3983227, 6448639, 10436353, 16892359, 27336079, 44240421, 71588483, 115848469

Offset: 0

Clark Kimberling, Feb 09 2018

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.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A000045, A298338.

a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + a[Floor[n/2]];
Table[a[n], {n, 0, 30}]  (* A298339 *)

A298400 a(n) = a(n-1) + a(n-2) - n*a(floor(n/2)), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, -1, -4, -10, -8, -11, 13, 38, 151, 299, 546, 949, 1649, 2763, 4204, 6746, 10266, 16290, 23536, 36655, 53613, 83391, 123900, 193641, 292867, 460885, 707580, 1120644, 1745334, 2780325, 4391131, 7032724, 11194491, 17991105, 28816020, 46427283, 74624283

Offset: 0

Clark Kimberling, Feb 10 2018

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.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A000045, A298338, A298401.

a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] - n*a[Floor[n/2]];
Table[a[n], {n, 0, 30}]  (* A298400 *)

A298401 a(n) = a(n-1) + a(n-2) - n*a(floor(n/2)), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, -1, -10, -26, -30, -49, 1, 42, 303, 631, 1294, 2315, 4295, 7345, 11624, 18952, 29820, 47974, 71734, 113345, 171197, 270029, 410170, 647849, 997829, 1583173, 2460742, 3919360, 6159752, 9851417, 15639201, 25107026, 40101859, 64545565, 103573904

Offset: 0

Clark Kimberling, Feb 10 2018

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.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A000045, A298338, A298400.

a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] - n*a[Floor[n/2]];
Table[a[n], {n, 0, 30}]  (* A298401 *)

A298402 a(n) = 2*a(n-1) - a(n-3) + a(floor(n/2)), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 30, 56, 100, 178, 308, 532, 902, 1526, 2550, 4254, 7038, 11626, 19098, 31336, 51224, 83658, 136288, 221884, 360642, 585898, 950814, 1542512, 2500652, 4053040, 6566118, 10635838, 17222890, 27886700, 45144600, 73077936, 118280798

Offset: 0

Clark Kimberling, Feb 10 2018

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.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A000045, A298338, A298403.

a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + a[Floor[n/2]];
Table[a[n], {n, 0, 90}]  (* A298402 *)

A298403 a(n) = 2*a(n-1) - a(n-3) + a(floor(n/2)), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 7, 15, 30, 60, 112, 209, 373, 664, 1149, 1985, 3366, 5695, 9517, 15877, 26268, 43392, 71280, 116956, 191184, 312237, 508667, 828135, 1346018, 2186735, 3548701, 5757079, 9333118, 15127052, 24506542, 39695843, 64280511, 104080748, 168491921, 272746723

Offset: 0

Clark Kimberling, Feb 10 2018

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.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A000045, A298338, A298402.

a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + a[Floor[n/2]];
Table[a[n], {n, 0, 90}]  (* A298403 *)

A298404 a(n) = 2*a(n-1) - a(n-3) + a(ceiling(n/2)), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 18, 36, 67, 125, 223, 397, 687, 1187, 2013, 3406, 5692, 9496, 15711, 25953, 42633, 69952, 114348, 186750, 304235, 495309, 805055, 1307888, 2122480, 3443311, 5582140, 9047492, 14657365, 23742086, 38446176, 62250698, 100775021, 163129819

Offset: 0

Clark Kimberling, Feb 10 2018

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.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A000045, A298338, A298405.

a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + a[Ceiling[n/2]];
Table[a[n], {n, 0, 90}]  (* A298404  *)

A298405 a(n) = 2*a(n-1) - a(n-3) + a(ceiling(n/2)), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 8, 17, 39, 78, 156, 290, 541, 965, 1718, 2973, 5137, 8712, 14741, 24635, 41099, 67998, 112326, 184518, 302756, 494904, 808263, 1316743, 2143719, 3484312, 5660593, 9186179, 14902787, 24159722, 39157900, 63437648, 102756673, 166396545, 269423440

Offset: 0

Clark Kimberling, Feb 10 2018

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.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A000045, A298338, A298404.

a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + a[Ceiling[n/2]];
Table[a[n], {n, 0, 90}]  (* A298405  *)

A298406 a(n) = 2*a(n-1) - a(n-3) + a(floor(n/2)) + a(floor(n/3)) + ... + a(floor(n/n)), where a(0) = 1, a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 3, 8, 19, 42, 84, 163, 301, 547, 961, 1682, 2879, 4902, 8241, 13807, 22917, 37962, 62487, 102690, 168096, 274798, 448000, 729829, 1186797, 1928729, 3130905, 5080360, 8237339, 13352743, 21634097, 35045477, 56753250, 91896553, 148771833, 240830555

Offset: 0

Clark Kimberling, Feb 10 2018

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.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A000045, A298338, A298407.

a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + Sum[a[Floor[n/k]], {k, 2, n}];
Table[a[n], {n, 0, 90}]  (* A298406  *)

from functools import lru_cache
@lru_cache(maxsize=None)
def A298406(n):
    if n <= 2:
        return 1
    c, j = 2*A298406(n-1)-A298406(n-3), 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A298406(k1)
        j, k1 = j2, n//j2
    return c+n-j+1 # Chai Wah Wu, Mar 31 2021

A298407 a(n) = 2*a(n-1) - a(n-3) + a(floor(n/2)) + a(floor(n/3)) + ... + a(floor(n/n)), where a(0) = 1, a(1) = 2, a(2) = 3.

Original entry on oeis.org

1, 2, 3, 9, 23, 52, 113, 223, 431, 794, 1442, 2532, 4433, 7589, 12924, 21730, 36411, 60440, 100125, 164816, 270863, 443390, 724846, 1181713, 1925113, 3130488, 5087530, 8258585, 13400782, 21728136, 35221342, 57065559, 92441545, 149701409, 242400952, 392424193

Offset: 0

Clark Kimberling, Feb 10 2018

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.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A000045, A298338, A298406.

a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + Sum[a[Floor[n/k]], {k, 2, n}];
Table[a[n], {n, 0, 90}]  (* A298407  *)

from functools import lru_cache
@lru_cache(maxsize=None)
def A298407(n):
    if n <= 2:
        return n+1
    c, j = 2*A298407(n-1)-A298407(n-3), 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A298407(k1)
        j, k1 = j2, n//j2
    return c+2*(n-j+1) # Chai Wah Wu, Mar 31 2021

cp's OEIS Frontend

A363503 a(n+1) = 2*a(n) + A298338(n-1), with a(1) = 1.

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A298339 a(n) = a(n-1) + a(n-2) + a([n/2]), where a(0) = 1, a(1) = 2, a(2) = 3.

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A298400 a(n) = a(n-1) + a(n-2) - n*a(floor(n/2)), where a(0) = 1, a(1) = 1, a(2) = 1.

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A298401 a(n) = a(n-1) + a(n-2) - n*a(floor(n/2)), where a(0) = 1, a(1) = 2, a(2) = 3.

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A298402 a(n) = 2*a(n-1) - a(n-3) + a(floor(n/2)), where a(0) = 1, a(1) = 1, a(2) = 1.

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Mathematica

A298403 a(n) = 2*a(n-1) - a(n-3) + a(floor(n/2)), where a(0) = 1, a(1) = 2, a(2) = 3.

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A298404 a(n) = 2*a(n-1) - a(n-3) + a(ceiling(n/2)), where a(0) = 1, a(1) = 1, a(2) = 1.

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Mathematica

A298405 a(n) = 2*a(n-1) - a(n-3) + a(ceiling(n/2)), where a(0) = 1, a(1) = 2, a(2) = 3.

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Mathematica

A298406 a(n) = 2*a(n-1) - a(n-3) + a(floor(n/2)) + a(floor(n/3)) + ... + a(floor(n/n)), where a(0) = 1, a(1) = 1, a(2) = 1.

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