A047290 -id:A047290 - cp's OEIS frontend

A225002 Duplicate of A047290.

1, 4, 6, 8, 11, 13, 15, 18, 20, 22, 25, 27, 29, 32, 34, 36, 39, 41, 43, 46, 48, 50, 53, 55, 57, 60, 62, 64, 67, 69, 71, 74, 76, 78, 81, 83, 85, 88, 90, 92, 95, 97, 99, 102, 104, 106, 109, 111, 113, 116, 118, 120, 123, 125, 127, 130, 132, 134, 137, 139, 141

Offset: 1

T. D. Noe, Apr 26 2013

dead

th = E^(3/7); t = Table[Floor[1/FractionalPart[th^(1/n)]], {n, 100}]

A224996 a(n) = floor(1/f(x^(1/n))) for x = 2, where f computes the fractional part.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 11, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 39, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 88, 90, 91, 93, 94, 96, 97

Offset: 2

T. D. Noe, Apr 26 2013

First denominator of continued fraction representing 2^(1/n): [1,a(n),....] so that 1+1/a(n) is first convergent for 2^(1/n). - Carmine Suriano, Apr 29 2014

a(n) is the largest integer y that satisfies (y+1)^n - y^n >= y^n, or equivalently (y+1)^n >= 2*y^n. - Charles Kusniec, Jan 19 2025

Melvyn B. Nathanson, On the fractional parts of roots of positive real numbers, Amer. Math. Monthly, 120 (2013), 409-429.

Cf. A224995, A224997, A224998, A001651, A047211, A047203, A047290, A047332.

Cf. A078607 (the smallest integer y that satisfies (y+1)^n - y^n < y^n).

th = 2; t = Table[Floor[1/FractionalPart[th^(1/n)]], {n, 2, 100}]

a(n) = floor(n/log(2)-1/2). - Andrey Zabolotskiy, Dec 01 2017

A224995 Floor(1/f(x^(1/n))) for x = 3/2, where f computes the fractional part.

Original entry on oeis.org

2, 4, 6, 9, 11, 14, 16, 19, 21, 24, 26, 29, 31, 34, 36, 38, 41, 43, 46, 48, 51, 53, 56, 58, 61, 63, 66, 68, 71, 73, 75, 78, 80, 83, 85, 88, 90, 93, 95, 98, 100, 103, 105, 108, 110, 112, 115, 117, 120, 122, 125, 127, 130, 132, 135, 137, 140, 142, 145, 147, 149

Offset: 1

T. D. Noe, Apr 26 2013

Melvyn B. Nathanson, On the fractional parts of roots of positive real numbers, Amer. Math. Monthly, 120 (2013), 409-429.

Cf. A224996, A224997, A224998, A001651, A047211, A047203, A047290, A047332.

th = 3/2; t = Table[Floor[1/FractionalPart[th^(1/n)]], {n, 100}]

a(n) = floor(n/log(3/2)-1/2) for n>1. - Andrey Zabolotskiy, Dec 01 2017

A224997 Floor(1/f(x^(1/n))) for x = 17, where f computes the fractional part.

Original entry on oeis.org

8, 1, 32, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25

Offset: 2

T. D. Noe, Apr 26 2013

Melvyn B. Nathanson, On the fractional parts of roots of positive real numbers, Amer. Math. Monthly, 120 (2013), 409-429.

Cf. A224995, A224996, A224998, A001651, A047211, A047203, A047290, A047332.

th = 17; t = Table[Floor[1/FractionalPart[th^(1/n)]], {n, 2, 100}]

a(n) = floor(n/log(17)-1/2) for n>7. - Andrey Zabolotskiy, Dec 01 2017

A224998 Floor(1/f(x^(1/n))) for x = Pi, where f computes the fractional part.

Original entry on oeis.org

7, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 58

Offset: 1

T. D. Noe, Apr 26 2013

Melvyn B. Nathanson, On the fractional parts of roots of positive real numbers, Amer. Math. Monthly, 120 (2013), 409-429.

Cf. A224995, A224996, A224997, A001651, A047211, A047203, A047290, A047332.

th = Pi; t = Table[Floor[1/FractionalPart[th^(1/n)]], {n, 100}]

a(n) = floor(n/log(Pi)-1/2) for n>4. - Andrey Zabolotskiy, Dec 01 2017

A220753 Expansion of (1+4x+5x^2-x^3)/((1-x)(1+x)(1-2*x^2)).

Original entry on oeis.org

1, 4, 8, 11, 22, 25, 50, 53, 106, 109, 218, 221, 442, 445, 890, 893, 1786, 1789, 3578, 3581, 7162, 7165, 14330, 14333, 28666, 28669, 57338, 57341, 114682, 114685, 229370, 229373, 458746, 458749, 917498, 917501, 1835002, 1835005, 3670010, 3670013

Offset: 0

Philippe Deléham, Apr 13 2013

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A005009, A048489, A063757.

m:=41; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+4*x+5*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)))); // Bruno Berselli, Apr 13 2013

Table[7 2^Floor[n/2] - (3/2) (3 + (-1)^n), {n, 0, 40}] (* Bruno Berselli, Apr 13 2013 *)
LinearRecurrence[{0, 3, 0, -2}, {1, 4, 8, 11}, 40] (* T. D. Noe, Apr 17 2013 *)

G.f.: (1+4*x+5*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)).
a(2n) = 7*2^n - 6 = A048489(n) = A063757(2n) = A005009(n)-6.
a(2n+1) = 7*2^n - 3 = A048489(n) + 3 = A063757(2n+1) - 3*A000225(n) = A005009(n)-3.
a(n) = a(n-1)*2 if n even.
a(n) = a(n-1)+3 if n odd.
a(n) = 3*a(n-2) - 2*a(n-4) with a(0)=1, a(1)=4, a(2)=8, a(3)=11.
a(n) = 7*2^floor(n/2) - (3/2)*(3+(-1)^n).
a(n) = A047290(A083416(n+1)). [Bruno Berselli, Apr 13 2013]

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A225002 Duplicate of A047290.

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A224996 a(n) = floor(1/f(x^(1/n))) for x = 2, where f computes the fractional part.

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A224995 Floor(1/f(x^(1/n))) for x = 3/2, where f computes the fractional part.

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A224997 Floor(1/f(x^(1/n))) for x = 17, where f computes the fractional part.

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A224998 Floor(1/f(x^(1/n))) for x = Pi, where f computes the fractional part.

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A220753 Expansion of (1+4*x+5*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)).

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Magma

Mathematica

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A220753 Expansion of (1+4x+5x^2-x^3)/((1-x)(1+x)(1-2*x^2)).