Paulo Romero Zanconato Pinto - cp's OEIS frontend

A277117 Decimal expansion of e^6/(Pi^5+Pi^4), where e = exp(1).

1, 0, 0, 0, 0, 0, 0, 0, 4, 3, 8, 0, 8, 1, 0, 7, 6, 2, 9, 9, 4, 7, 6, 3, 7, 4, 3, 2, 0, 1, 2, 2, 8, 9, 0, 0, 5, 8, 1, 9, 0, 4, 0, 9, 2, 1, 5, 3, 0, 6, 0, 3, 6, 9, 5, 9, 2, 3, 3, 5, 2, 0, 0, 4, 8, 6, 4, 3, 9, 0, 0, 0, 6, 1, 4, 3, 7, 1, 5, 8, 8, 9, 8, 2, 6, 5, 2, 5, 4, 8, 5, 3, 2, 4, 4, 1, 6, 8, 9, 9, 7, 5, 8, 3, 8, 6

Offset: 1

Paulo Romero Zanconato Pinto, Sep 30 2016

			1.0000000438081076299476374320122890058190409215306036959233520...

D. Wilson, pers. comm.

Eric Weisstein's World of Mathematics, Almost Integer
Wikipedia, Near-identities involving e and pi.

Cf. A000796, A001113, A092425, A092731, A092512, A239606.

RealDigits[E^6/(Pi^5+Pi^4),10,120][[1]] (* Harvey P. Dale, Apr 07 2019 *)

exp(6)/(Pi^5+Pi^4) \\ Michel Marcus, Oct 02 2016

Equals A092512 /(A092731 + A092425).

A277115 Decimal expansion of e*phi/Pi, where phi = (sqrt(5) + 1)/2.

Original entry on oeis.org

1, 4, 0, 0, 0, 1, 3, 5, 8, 3, 6, 9, 0, 4, 8, 4, 8, 5, 6, 2, 9, 8, 6, 1, 3, 5, 0, 2, 9, 9, 9, 7, 9, 0, 2, 6, 0, 3, 8, 1, 9, 8, 6, 6, 9, 0, 2, 5, 3, 1, 0, 6, 4, 2, 9, 9, 1, 7, 5, 9, 1, 7, 3, 0, 4, 4, 8, 1, 3, 8, 9, 5, 4, 6, 4, 0, 1, 5, 0, 6, 8, 2, 9, 0, 9, 8, 5, 8, 8, 1, 3, 8, 0, 1, 6, 1, 7, 5, 6, 5, 0, 8, 2, 6, 4

Offset: 1

Paulo Romero Zanconato Pinto, Sep 30 2016

An approximation to gamma(2) = 7/5. See A274981. - Omar E. Pol, Sep 30 2016

			1.400013583690484856298613502999790260381986690253106429917...

J. DePompeo (pers. comm., Mar. 29, 2004)

Eric Weisstein's World of Mathematics, Almost Integer

Cf. A000796, A001113, A001622, A094885, A274981.

First@ RealDigits[N[E GoldenRatio/Pi, 120]] (* Michael De Vlieger, Sep 30 2016 *)

PARI
```
exp(1)/Pi*(1+sqrt(5))/2;
```

Equals A001113 * A001622 / A000796.

A277113 a(n) = floor(n/(1-Pi/(sqrt(5)+1))).

Original entry on oeis.org

34, 68, 102, 137, 171, 205, 239, 274, 308, 342, 376, 411, 445, 479, 513, 548, 582, 616, 650, 685, 719, 753, 787, 822, 856, 890, 924, 959, 993, 1027, 1061, 1096, 1130, 1164, 1198, 1233, 1267, 1301, 1335, 1370, 1404, 1438, 1472, 1507, 1541, 1575, 1609, 1644, 1678

Offset: 1

Paulo Romero Zanconato Pinto, Sep 30 2016

nonn

The goal is to generate a ratio near 1 from two well-known constants.

			For n = 10 we have that floor(10/(1-Pi/(sqrt(5)+1))) = floor(10/0.02919448...) = floor(342.5304983...) so a(10) = 342.

Paulo Romero Zanconato Pinto, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Beatty sequences

Complement of A277112.

Cf. A000796, A001622, A094881, A094882.

A277113:=n->floor(n/(1-Pi/(sqrt(5)+1))): seq(A277113(n), n=1..100);

f[n_] := Floor[n/(1-Pi/(Sqrt[5]+1))]; Array[f, 100, 1]

a(n) = n\(1-Pi/(sqrt(5)+1)) \\ Michel Marcus, Oct 29 2016

a(n) = floor(n/(1-Pi/(sqrt(5)+1))).

More terms from Michel Marcus, Oct 29 2016

A277112 a(n) = floor(n*(1+sqrt(5))/Pi).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70

Offset: 0

Paulo Romero Zanconato Pinto, Sep 30 2016

			For n = 1000 we have that floor(1000*(1+sqrt(5))/Pi) = floor(1000*1.0300724296...) = floor(1030.0724296...) so a(1000) = 1030.

Index entries for sequences related to Beatty sequences

Complement of A277113.

Cf. A000796, A001622, A094881, A094882.

A277112:=n->floor(n*(1+sqrt(5))/Pi): seq(A277112(n), n=0..100); # Wesley Ivan Hurt, Oct 31 2016

f[n_] := Floor[n*(1+Sqrt[5])/Pi]; Array[f, 100, 0]

a(n) = floor(n*(1+sqrt(5))/Pi).

A277051 a(n) = floor(n/(1-3/Pi)).

Original entry on oeis.org

22, 44, 66, 88, 110, 133, 155, 177, 199, 221, 244, 266, 288, 310, 332, 355, 377, 399, 421, 443, 465, 488, 510, 532, 554, 576, 599, 621, 643, 665, 687, 710, 732, 754, 776, 798, 820, 843, 865, 887, 909, 931, 954, 976, 998, 1020, 1042, 1065, 1087, 1109, 1131, 1153, 1175

Offset: 1

Paulo Romero Zanconato Pinto, Sep 26 2016

			For n = 10 we have that floor(10/(1-3/Pi)) = floor(10/0.04719755...) = floor(211.8754045...) so a(10) = 221.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Beatty sequences

Complement of A274399.

Cf. A000796, A019670.

A277051:=n->floor(n/(1-3/Pi)): seq(A277051(n), n=1..100); # Wesley Ivan Hurt, Sep 26 2016

f[n_] := Floor[n/(1-3/Pi)]; Array[f, 100, 1]

makelist(floor(n / (1-3 / %pi )), n, 1, 100);

a(n)=n\(1-3/Pi) \\ Charles R Greathouse IV, Sep 26 2016

a(n) = floor(n/(1-3/Pi)).

A274399 a(n) = floor(n*Pi/3).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70

Offset: 0

Paulo Romero Zanconato Pinto, Sep 26 2016

			For n = 1000 we have that floor(1000*Pi/3) = floor(1000*1.04719755...) = floor(1047.19755...) so a(1000) = 1047.

Paulo Romero Zanconato Pinto, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Beatty sequences

Complement of A277051.

Cf. A000796, A019670.

A274399:=n->floor(n*Pi/3): seq(A274399(n), n=0..100); # Wesley Ivan Hurt, Sep 26 2016

f[n_] := Floor[n*Pi/Floor[Pi]]; Array[f, 100, 0]

Maxima
```
makelist(floor(n*(%pi/3)), n, 0, 100);
```

a(n) = n*Pi\3; \\ Michel Marcus, Sep 26 2016

a(n) = floor(n*Pi/floor(Pi)) = floor(n*Pi/3).

A277092 Decimal expansion of e^Pi/Pi^e.

Original entry on oeis.org

1, 0, 3, 0, 3, 4, 5, 5, 2, 4, 2, 1, 6, 2, 1, 0, 8, 3, 2, 4, 4, 1, 5, 5, 2, 4, 3, 7, 5, 4, 4, 1, 4, 2, 3, 9, 1, 3, 3, 1, 1, 6, 7, 4, 5, 3, 5, 4, 2, 6, 3, 5, 0, 4, 7, 7, 5, 2, 0, 6, 0, 3, 7, 6, 9, 4, 3, 6, 8, 5, 8, 3, 3, 3, 3, 6, 7, 0, 7, 8, 4, 6, 6, 5, 3, 6, 6

Offset: 1

Paulo Romero Zanconato Pinto, Sep 30 2016

			1.030345524216210832441552437544142391331167453542635047752...

Cf. A000796, A001113, A039661, A059850.

RealDigits[N[E^Pi/Pi^E, 120]] (* Michael De Vlieger, Sep 30 2016 *)

More digits from Jon E. Schoenfield, Mar 15 2018

A277052 a(n) = n+floor(n/(2/sqrt(Pi)-1)).

Original entry on oeis.org

8, 17, 26, 35, 43, 52, 61, 70, 79, 87, 96, 105, 114, 123, 131, 140, 149, 158, 166, 175, 184, 193, 202, 210, 219, 228, 237, 246, 254, 263, 272, 281, 290, 298, 307, 316, 325, 333, 342, 351, 360, 369, 377, 386, 395, 404, 413, 421, 430, 439, 448, 457, 465, 474

Offset: 1

Paulo Romero Zanconato Pinto, Sep 26 2016

Carauleanu Marc, Table of n, a(n) for n = 1..4242

Complement of A277050.

Cf. A190732, A037086.

A277052:=n->n+floor(n/(2/sqrt(Pi)-1)): seq(A277052(n), n=1..100); # Wesley Ivan Hurt, Sep 26 2016

f[n_] := n + Floor[n/(2/Sqrt[Pi]-1)]; Array[f, 100, 1]

PARI
```
a(n) = n + floor(n/(2/sqrt(Pi)-1));
```

a(n) = n + floor(n/(2/sqrt(Pi) - 1)).

A277050 a(n) = floor(2*n/sqrt(Pi)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75

Offset: 0

Paulo Romero Zanconato Pinto, Sep 26 2016

Paulo Romero Zanconato Pinto, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Beatty sequences

Cf. A037086, A190732, A277052 (complement).

A277050:=n->floor(2*n/sqrt(Pi)): seq(A277050(n), n=0..100); # Wesley Ivan Hurt, Sep 26 2016

Mathematica
```
Table[Floor[2 * n/Sqrt[Pi]], {n, 100}]
```
PARI
```
a(n) = floor(2*n/sqrt(Pi));
```

a(n) = floor(2*n/sqrt(Pi)).

cp's OEIS Frontend

User: Paulo Romero Zanconato Pinto

A277117 Decimal expansion of e^6/(Pi^5+Pi^4), where e = exp(1).

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A277115 Decimal expansion of e*phi/Pi, where phi = (sqrt(5) + 1)/2.

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A277113 a(n) = floor(n/(1-Pi/(sqrt(5)+1))).

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A277112 a(n) = floor(n*(1+sqrt(5))/Pi).

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A277051 a(n) = floor(n/(1-3/Pi)).

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A274399 a(n) = floor(n*Pi/3).

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A277092 Decimal expansion of e^Pi/Pi^e.

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