A135611 -id:A135611 - cp's OEIS frontend

A110117 a(n) = floor(n * (sqrt(2) + sqrt(3))).

3, 6, 9, 12, 15, 18, 22, 25, 28, 31, 34, 37, 40, 44, 47, 50, 53, 56, 59, 62, 66, 69, 72, 75, 78, 81, 84, 88, 91, 94, 97, 100, 103, 106, 110, 113, 116, 119, 122, 125, 128, 132, 135, 138, 141, 144, 147, 151, 154, 157, 160, 163, 166, 169, 173, 176, 179, 182, 185, 188

Offset: 1

Reinhard Zumkeller, Jul 13 2005

nonn

Beatty sequence for sqrt(2)+sqrt(3); complement of A110118;

sqrt(2)+sqrt(3) = 3.14626... = A135611, a weak but interesting Pi approximation.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Pi Approximations
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A110119, A135611.

Table[Floor[n*(Sqrt[2] + Sqrt[3])], {n, 1, 50}] (* G. C. Greubel, Jul 02 2017 *)

for(n=1,50, print1(floor(n*(sqrt(2) + sqrt(3))), ", ")) \\ G. C. Greubel, Jul 02 2017

Typo in Link section fixed by Reinhard Zumkeller, Feb 15 2010

A138281 a(n) = floor((sqrt(2) + sqrt(3))^n).

Original entry on oeis.org

1, 3, 9, 31, 97, 308, 969, 3051, 9601, 30210, 95049, 299052, 940897, 2960313, 9313929, 29304086, 92198401, 290080547, 912670089, 2871501385, 9034502497, 28424933309, 89432354889, 281377831710, 885289046401, 2785353383794, 8763458109129, 27572156006234

Offset: 0

Reinhard Zumkeller, Mar 12 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A089078, A110117, A121503, A135611, A135798.

[Floor((Sqrt(2) + Sqrt(3))^n): n in [0..50]]; // G. C. Greubel, Jan 27 2018

Table[Floor[(Sqrt[2] + Sqrt[3])^n], {n, 0, 50}] (* G. C. Greubel, Jan 27 2018 *)

for(n=0,50, print1(floor((sqrt(2) + sqrt(3))^n), ", ")) \\ G. C. Greubel, Jan 27 2018

a(2*n) = floor(A001079(n) + A001078(n)*sqrt(6));
(sqrt(2) + sqrt(3))^(2*n) = A001079(n) + A001078(n)*sqrt(6);
a(2*n+1) = floor(A054320(n)*sqrt(2) + A138288(n)*sqrt(3));
(sqrt(2)+sqrt(3))^(2*n+1) = A054320(n)*sqrt(2) + A138288(n)*sqrt(3).

Terms a(16) and a(18) corrected, terms a(19) onward added by G. C. Greubel, Jan 27 2018

A377343 Decimal expansion of the surface area of a truncated cuboctahedron (great rhombicuboctahedron) with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 26 2024

			61.7551724393036681079496207885868453461497255502...

Eric Weisstein's World of Mathematics, Great Rhombicuboctahedron.
Wikipedia, Truncated cuboctahedron.

Cf. A377344 (volume), A377345 (circumradius), A377346 (midradius).
Cf. A343199 (analogous for a cuboctahedron).
Cf. A135611.

First[RealDigits[12*(2 + Sqrt[2] + Sqrt[3]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedCuboctahedron", "SurfaceArea"], 10, 100]]

Equals 12*(2 + sqrt(2) + sqrt(3)) = 12*(2 + A135611).

A089078 Continued fraction for sqrt(2) + sqrt(3).

Original entry on oeis.org

3, 6, 1, 5, 7, 1, 1, 4, 1, 38, 43, 1, 3, 2, 1, 1, 1, 1, 2, 4, 1, 4, 5, 1, 5, 1, 7, 22, 2, 5, 1, 1, 2, 1, 1, 31, 2, 1, 1, 3, 1, 44, 1, 89, 1, 8, 5, 2, 5, 1, 1, 4, 2, 8, 1, 17, 1, 4, 3, 4, 3, 2, 1, 1, 4, 2, 1, 9, 1, 15, 13, 1, 39, 20, 2, 152, 3, 2, 4, 1, 30, 1, 3, 1, 2, 1, 2, 16, 3, 24, 1, 9, 1, 172, 3, 1, 1

Offset: 0

Jeppe Stig Nielsen, Dec 04 2003

This is the most natural example of the fact that the sum of two periodic continued fractions need not have a periodic continued fraction.

a(n) is the numbers of squares removed at stage n of the continued-fraction partitioning of a rectangle of length L and width W satisfying W=L*sqrt(8); see A188640. - Clark Kimberling, Apr 13 2011

G. C. Greubel, Table of n, a(n) for n = 0..5000
G. Xiao, Contfrac

Cf. A135611.

r = 8^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
ContinuedFraction[Sqrt[2]+Sqrt[3],100] (* Harvey P. Dale, Aug 17 2019 *)

contfrac(sqrt(2)+sqrt(3)) \\ Michel Marcus, Mar 12 2017

A229194 Integers nearest to (2^((n-3)/2) + 3^((n-3)/2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 35, 58, 97, 163, 275, 466, 793, 1353, 2315, 3969, 6817, 11726, 20195, 34816, 60073, 103724, 179195, 309724, 535537, 926275, 1602515, 2773034, 4799353, 8307516, 14381675, 24899377, 43112257, 74651790, 129271235, 223862687, 387682633, 671402698, 1162785755, 2013837368, 3487832977, 6040770648, 10462450355, 18120829034, 31385253913, 54359521280, 94151567435, 163072632198

Offset: 0

Vladimir Pletser, Sep 15 2013

nonn

This sequence illustrates the second law of small numbers because it is a coincidence that the terms for 1 <= n <= 8 are the same as the Fibonacci numbers F(n) (A000045): a(n) = F(n) for 1 <= n <= 8.
Furthermore, the following terms are the sum of two Fibonacci numbers: a(9) = F(9) + F(2), a(10) = F(10) + F(4), a(11) = F(11) + F(6), a(14) = F(14) + F(11); or the algebraic sum of three Fibonacci numbers: a(12) = F(12) + F(8) - F(3), a(13) = F(13) + F(10) - F(7), a(14) = F(14) + F(12) - F(10), a(18) = F(19) - F(13) - F(8), a(19) = F(20) + F(10) - F(4); or the algebraic sum of four Fibonacci numbers: a(15) = F(15) + F(12) + F(9) + F(5), a(16) = F(16) + F(14) - F(6) - F(4), a(17) = F(18) - F(13) - F(9) - F(3), a(18) = F(18) + F(16) + F(14) + F(8), a(19) = F(19) + F(18) + F(10) - F(3).
Note that, for following values of n, a(n) > F(n+1) for n >= 20.
Remark as well that (2^(1/2) + 3^(1/2)) = 3.14626437... ~=  Pi (see A135611).

T. Koshy, Fibonacci and Lucas Numbers with Applications, New York, Wiley-Interscience, 2001
I. Stewart, L'univers des nombres, Belin-Pour La Science, Paris 2000.

Vladimir Pletser, Table of n, a(n) for n = 0..1000
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
I. Stewart, Fibonacci Forgeries
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Strong Law of Small Numbers.

Cf. A000045, A005181.

[Round(2^((n-3)/2) + 3^((n-3)/2)): n in [0..50]]; // Vincenzo Librandi, Sep 20 2013

seq(round(2^((n-3)/2)+3^((n-3)/2)), n=0..50);

Table[Round[2^((n - 3)/2) + 3^((n - 3)/2)], {n, 0, 50}] (* Vincenzo Librandi, Sep 20 2013 *)

a(n) = round(2^((n-3)/2) + 3^((n-3)/2)).

A110119 Self-inverse integer permutation induced by Beatty sequences for x and (x+1)/(2*sqrt(2)) with x=sqrt(2)+sqrt(3).

Original entry on oeis.org

3, 6, 1, 9, 12, 2, 15, 18, 4, 22, 25, 5, 28, 31, 7, 34, 37, 8, 40, 44, 47, 10, 50, 53, 11, 56, 59, 13, 62, 66, 14, 69, 72, 16, 75, 78, 17, 81, 84, 19, 88, 91, 94, 20, 97, 100, 21, 103, 106, 23, 110, 113, 24, 116, 119, 26, 122, 125, 27, 128, 132, 29, 135, 138, 141, 30, 144

Offset: 1

Reinhard Zumkeller, Jul 13 2005

nonn

Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences
Index entries for sequences that are permutations of the natural numbers

Cf. A108591, A110117, A110118.

Cf. A135611 (sqrt(2)+sqrt(3)).

a(A110117(n)) = A110118(n) and a(A110118(n)) = A110117(n).

A135798 Decimal expansion of sqrt(2) + sqrt(3) - Pi.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Mar 06 2008

			0.0046717163521791038664916824360675613153077298122228802575...

G. C. Greubel, Table of n, a(n) for n = 0..2000
Index entries for transcendental numbers

Cf. A000796, A135611.

Digits:=100: evalf(sqrt(2)+sqrt(3)-Pi); # Wesley Ivan Hurt, Jan 24 2017

Join[{0,0},RealDigits[Sqrt[2] + Sqrt[3] - Pi, 10, 100][[1]]] (* G. C. Greubel, Nov 09 2016 *)

A297701 Decimal expansion of 1 + sqrt(2) + sqrt(3).

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Jan 03 2018

This is an algebraic integer of degree 4, with minimal polynomial x^4 - 4*x^3 - 4*x^2 + 16*x - 8.

			  1.0000000000000000000000000000...
+ 1.4142135623730950488016887242...
+ 1.7320508075688772935274463415...
= 4.1462643699419723423291350657...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 4

Essentially the same as A135611. Cf. A002193, A002194, A014176, A165663, A188582.

SetDefaultRealField(RealField(100)); 1 + Sqrt(2) + Sqrt(3); // G. C. Greubel, Nov 20 2018

RealDigits[1 + Sqrt[2] + Sqrt[3], 10, 100][[1]]

1+sqrt(2)+sqrt(3) \\ Felix Fröhlich, Jan 06 2018

numerical_approx(1+sqrt(2)+sqrt(3), digits=100) # G. C. Greubel, Nov 20 2018

1 + sqrt(2) + sqrt(3) = 1 + sqrt(5 + 2 sqrt(6)).

1 + A002193 + A002194 = A014176 + A002194 = A165663 + A188582. - Felix Fröhlich, Jan 06 2018

Terms a(52) onward corrected by G. C. Greubel, Nov 20 2018

A340616 Decimal expansion of sqrt(3)-sqrt(2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 14 2021

From Bernard Schott, Jan 16 2021: (Start)
Equals the smallest positive root of x^4 - 10*x^2 + 1 (minimal polynomial).
An approximation to 1/Pi (see corresponding comment in A135611). (End)

			0.317837245195782244725...

Cf. A002193 (sqrt(2)), A002194 (sqrt(3)), A135611 (sqrt(2) + sqrt(3)), A246723 (5-2*sqrt(6)).

Cf. A172264 (Beatty sequence).

Maple
```
sqrt(3)-sqrt(2) ; evalf(%) ;
```

RealDigits[Sqrt[3] - Sqrt[2], 10, 100][[1]] (* Wesley Ivan Hurt, Jan 14 2021 *)

sqrt(3)-sqrt(2) \\ Michel Marcus, Jan 15 2021

Equals A002194 - A002193.
Equals A135611 - A010466.
Equals 1/A135611. - Bernard Schott, Jan 15 2021
Equals sqrt(A246723). - Kevin Ryde, Jan 15 2021

cp's OEIS Frontend

A110117 a(n) = floor(n * (sqrt(2) + sqrt(3))).

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A138281 a(n) = floor((sqrt(2) + sqrt(3))^n).

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A377343 Decimal expansion of the surface area of a truncated cuboctahedron (great rhombicuboctahedron) with unit edge length.

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A089078 Continued fraction for sqrt(2) + sqrt(3).

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A229194 Integers nearest to (2^((n-3)/2) + 3^((n-3)/2)).

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A110119 Self-inverse integer permutation induced by Beatty sequences for x and (x+1)/(2*sqrt(2)) with x=sqrt(2)+sqrt(3).

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A135798 Decimal expansion of sqrt(2) + sqrt(3) - Pi.

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Maple

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A297701 Decimal expansion of 1 + sqrt(2) + sqrt(3).

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