A171983 -id:A171983 - cp's OEIS frontend

A010470 Decimal expansion of square root of 13.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 3 followed by {1, 1, 1, 1, 6} repeated. - Harry J. Smith, Jun 02 2009
The convergents to sqrt(13) are given in A041018/A041019. - Wolfdieter Lang, Nov 23 2017
The fundamental algebraic (integer) number in the field Q(sqrt(13)) is (1 + sqrt(13))/2  = A209927. - Wolfdieter Lang, Nov 21 2023

			3.605551275463989293119221267470495946251296573845246212710453056227166...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.31.4, p. 201.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2.

Cf. A010122 (continued fraction), A041018/A041019 (convergents), A248242 (Egyptian fraction), A171983 (Beatty sequence).

Cf. A020770 (reciprocal), A209927, A295330, A344069.

RealDigits[N[Sqrt[13],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 20080); x=sqrt(13); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010470.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

A198270 Ceiling(n*sqrt(13)).

Original entry on oeis.org

0, 4, 8, 11, 15, 19, 22, 26, 29, 33, 37, 40, 44, 47, 51, 55, 58, 62, 65, 69, 73, 76, 80, 83, 87, 91, 94, 98, 101, 105, 109, 112, 116, 119, 123, 127, 130, 134, 138, 141, 145, 148, 152, 156, 159, 163, 166, 170, 174, 177, 181, 184, 188, 192

Offset: 0

Vincenzo Librandi, Oct 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A171983.

Magma
```
[Ceiling(n*Sqrt(13)): n in [0..60]]
```

With[{c=Sqrt[13]},Ceiling[c*Range[0,60]]] (* Harvey P. Dale, Jul 31 2012 *)

A194116 a(n) = Sum_{j=1..n} floor(j*sqrt(13)); n-th partial sum of Beatty sequence for sqrt(13).

Original entry on oeis.org

3, 10, 20, 34, 52, 73, 98, 126, 158, 194, 233, 276, 322, 372, 426, 483, 544, 608, 676, 748, 823, 902, 984, 1070, 1160, 1253, 1350, 1450, 1554, 1662, 1773, 1888, 2006, 2128, 2254, 2383, 2516, 2653, 2793, 2937, 3084, 3235, 3390, 3548, 3710, 3875

Offset: 1

Clark Kimberling, Aug 16 2011

nonn

Cf. A171983 (Beatty sequence for sqrt(13)).

c[n_] := Sum[Floor[j*Sqrt[13]], {j, 1, n}];
c = Table[c[n], {n, 1, 90}]

from sympy import integer_nthroot
def A194116(n): return sum(integer_nthroot(13*j**2,2)[0] for j in range(1,n+1)) # Chai Wah Wu, Mar 17 2021

A198271 Round(n*sqrt(13)).

Original entry on oeis.org

0, 4, 7, 11, 14, 18, 22, 25, 29, 32, 36, 40, 43, 47, 50, 54, 58, 61, 65, 69, 72, 76, 79, 83, 87, 90, 94, 97, 101, 105, 108, 112, 115, 119, 123, 126, 130, 133, 137, 141, 144, 148, 151, 155, 159, 162, 166, 169, 173, 177, 180, 184, 187, 191

Offset: 0

Vincenzo Librandi, Oct 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A171983.

Magma
```
[Round(n*Sqrt(13)): n in [0..60]]
```

Round[Sqrt[13]Range[0,60]] (* Harvey P. Dale, Mar 25 2016 *)

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A010470 Decimal expansion of square root of 13.

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A198270 Ceiling(n*sqrt(13)).

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A194116 a(n) = Sum_{j=1..n} floor(j*sqrt(13)); n-th partial sum of Beatty sequence for sqrt(13).

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Python

A198271 Round(n*sqrt(13)).

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