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Search: a157258 -id:a157258
Displaying 1-7 of 7 results found.
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A090488
Decimal expansion of 2 + 2*sqrt(2).
+10
22
4, 8, 2, 8, 4, 2, 7, 1, 2, 4, 7, 4, 6, 1, 9, 0, 0, 9, 7, 6, 0, 3, 3, 7, 7, 4, 4, 8, 4, 1, 9, 3, 9, 6, 1, 5, 7, 1, 3, 9, 3, 4, 3, 7, 5, 0, 7, 5, 3, 8, 9, 6, 1, 4, 6, 3, 5, 3, 3, 5, 9, 4, 7, 5, 9, 8, 1, 4, 6, 4, 9, 5, 6, 9, 2, 4, 2, 1, 4, 0, 7, 7, 7, 0, 0, 7, 7, 5, 0, 6, 8, 6, 5, 5, 2, 8, 3, 1, 4, 5, 4
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Side length of smallest square containing five circles of radius 1. - Charles R Greathouse IV, Apr 05 2011
Equals n + n/(n +n/(n +n/(n +....))) for n = 4. See also A090388. - Stanislav Sykora, Jan 23 2014
Also the area of a regular octagon with unit edge length. - Stanislav Sykora, Apr 12 2015
The positive solution to x^2 - 4*x - 4 = 0. The negative solution is -1 * A163960 = -0.82842... . - Michal Paulovic, Dec 12 2023
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Octagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 2
FORMULA
Equals 1 + A086178 = 2*A014176. - R. J. Mathar, Sep 03 2007
From Michal Paulovic, Dec 12 2023: (Start)
Equals A010466 + 2.
Equals A156035 - 1.
Equals A157258 - 5.
Equals A163960 + 4.
Equals A365823 - 2.
Equals [4; 1, 4, ...] (periodic continued fraction expansion).
Equals sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * ...)))). (End)
EXAMPLE
4.828427124746190097603377448419396157139343750...
MATHEMATICA
RealDigits[2+2Sqrt[2], 10, 120][[1]] (* Harvey P. Dale, Mar 11 2015 *)
PROG
(PARI) 2*(1 + sqrt(2)) \\ G. C. Greubel, Jul 03 2017
CROSSREFS
Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014
Cf. A002193, A010466, A014176, A086178, A156035, A157258, A163960, A365823.
Cf. Areas of other regular polygons: A120011, A102771, A104956, A178817, A256853, A178816, A256854, A178809.
KEYWORD
easy,nonn,cons
AUTHOR
Felix Tubiana, Feb 05 2004
EXTENSIONS
Better definition from Rick L. Shepherd, Jul 02 2004
STATUS
approved
A129288
Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 41)^2 = y^2.
+10
12
0, 36, 39, 123, 319, 336, 820, 1960, 2059, 4879, 11523, 12100, 28536, 67260, 70623, 166419, 392119, 411720, 970060, 2285536, 2399779, 5654023, 13321179, 13987036, 32954160, 77641620, 81522519, 192071019, 452528623, 475148160, 1119472036
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Also values x of Pythagorean triples (x, x+41, y).
Corresponding values y of solutions (x, y) are in A157257.
lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (7 + 2*sqrt(2))/(7 - 2*sqrt(2)) for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3 + 2*sqrt(2))*(7 - 2*sqrt(2))^2/(7 + 2*sqrt(2))^2 for n mod 3 = 0.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).
FORMULA
a(n) = 6*a(n-3) - a(n-6) + 82 for n > 6; a(1)=0, a(2)=36, a(3)=39, a(4)=123, a(5)=319, a(6)=336.
G.f.: x*(36 + 3*x + 84*x^2 - 20*x^3 - x^4 - 20*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
a(3*k + 1) = 41*A001652(k) for k >= 0.
MATHEMATICA
LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 36, 39, 123, 319, 336, 820}, 40] (* Harvey P. Dale, Jan 18 2015 *)
PROG
(PARI) forstep(n=0, 1200000000, [3 , 1], if(issquare(2*n^2+82*n+1681), print1(n, ", ")))
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(36+3*x+84*x^2-20*x^3-x^4-20*x^5)/((1-x)*(1-6*x^3+ x^6)))); // G. C. Greubel, May 07 2018
CROSSREFS
Cf. A157257, A001652, A156035 (decimal expansion of 3 + 2*sqrt(2)), A157258 (decimal expansion of 7 + 2*sqrt(2)), A157259 (decimal expansion of 7 - 2*sqrt(2)), A157260 (decimal expansion of (7 + 2*sqrt(2))/(7 - 2*sqrt(2))).
KEYWORD
nonn,easy
AUTHOR
Mohamed Bouhamida, May 26 2007
EXTENSIONS
Edited and extended by Klaus Brockhaus, Feb 26 2009
STATUS
approved
A157259
Decimal expansion of 7 - 2*sqrt(2).
+10
9
4, 1, 7, 1, 5, 7, 2, 8, 7, 5, 2, 5, 3, 8, 0, 9, 9, 0, 2, 3, 9, 6, 6, 2, 2, 5, 5, 1, 5, 8, 0, 6, 0, 3, 8, 4, 2, 8, 6, 0, 6, 5, 6, 2, 4, 9, 2, 4, 6, 1, 0, 3, 8, 5, 3, 6, 4, 6, 6, 4, 0, 5, 2, 4, 0, 1, 8, 5, 3, 5, 0, 4, 3, 0, 7, 5, 7, 8, 5, 9, 2, 2, 2, 9, 9, 2, 2, 4, 9, 3, 1, 3, 4, 4, 7, 1, 6, 8, 5, 4, 5, 2, 9, 9, 7
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
lim_{n -> infinity} b(n)/b(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {1, 2}, b = A129288.
lim_{n -> infinity} b(n)/b(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {0, 2}, b = A157257.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
Pierre-Antoine Guihéneuf, Rotations Discrètes, Images des Mathématiques, CNRS, 2018. See 3-2*sqrt(2), the fractional part of this constant, about the loss of information when rotating an image.
Index entries for algebraic numbers, degree 2
FORMULA
Equals 3 + Sum_{k>=0} binomial(2*k,k)/((k+1) * 8^k). - Amiram Eldar, Aug 03 2020
Equals 4 + exp(-arccosh(3)). - Amiram Eldar, Jul 06 2023
Equals 4 + (2-sqrt(2))/(2+sqrt(2)). - Davide Rotondo, Jun 08 2024
EXAMPLE
7 - 2*sqrt(2) = 4.17157287525380990239...
MATHEMATICA
RealDigits[7-2*Sqrt[2], 10, 120][[1]] (* Harvey P. Dale, May 01 2012 *)
PROG
(PARI) 7 - 2*sqrt(2) \\ G. C. Greubel, Nov 28 2017
(Magma) [7 - 2*Sqrt(2)]; // G. C. Greubel, Nov 28 2017
CROSSREFS
Cf. A129288, A157257, A157258 (decimal expansion of 7+2*sqrt(2)), A157260 (decimal expansion of (7+2*sqrt(2))/(7-2*sqrt(2))).
KEYWORD
cons,nonn
AUTHOR
Klaus Brockhaus, Feb 26 2009
STATUS
approved
A163960
Decimal expansion of 2*(sqrt(2) - 1).
+10
7
8, 2, 8, 4, 2, 7, 1, 2, 4, 7, 4, 6, 1, 9, 0, 0, 9, 7, 6, 0, 3, 3, 7, 7, 4, 4, 8, 4, 1, 9, 3, 9, 6, 1, 5, 7, 1, 3, 9, 3, 4, 3, 7, 5, 0, 7, 5, 3, 8, 9, 6, 1, 4, 6, 3, 5, 3, 3, 5, 9, 4, 7, 5, 9, 8, 1, 4, 6, 4, 9, 5, 6, 9, 2, 4, 2, 1, 4, 0, 7, 7, 7, 0, 0, 7, 7, 5, 0, 6, 8, 6, 5, 5, 2, 8, 3, 1, 4, 5
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(1,1,sqrt(2)). (See A195284.) - Clark Kimberling, Sep 14 2011
REFERENCES
J. M. Steele, Probability Theory and Combinatorial Optimization, SIAM, 1997, p. 3.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
FORMULA
Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)/((k+1) * 4^k). - Amiram Eldar, May 06 2022
Equals Sum_{k>=1} (-1)^(k+1)/A084158(k). - Amiram Eldar, Dec 02 2024
EXAMPLE
0.82842712474619009760337744841939615713934375075389614635335...
MATHEMATICA
RealDigits[2(Sqrt[2]-1), 10, 120][[1]] (* Harvey P. Dale, May 27 2016 *)
PROG
(PARI) 2*(sqrt(2)-1) \\ G. C. Greubel, Aug 13 2017
CROSSREFS
Cf. A084158, A156035, A195284.
Essentially the same digit sequence as A010466, A086178, A090488 and A157258.
KEYWORD
nonn,cons,changed
AUTHOR
N. J. A. Sloane, Oct 02 2010
STATUS
approved
A157257
Positive numbers y such that y^2 is of the form x^2+(x+41)^2 with integer x.
+10
6
29, 41, 85, 89, 205, 481, 505, 1189, 2801, 2941, 6929, 16325, 17141, 40385, 95149, 99905, 235381, 554569, 582289, 1371901, 3232265, 3393829, 7996025, 18839021, 19780685, 46604249, 109801861, 115290281, 271629469, 639972145, 671961001
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
(-20, a(1)) and (A129288(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+41)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(7-2*sqrt(2))^2/(7+2*sqrt(2))^2 for n mod 3 = 1.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1001
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).
FORMULA
a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=29, a(2)=41, a(3)=85, a(4)=89, a(5)=205, a(6)=481.
G.f.: (1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 41*A001653(k) for k >= 1.
EXAMPLE
(-20, a(1)) = (-20, 29) is a solution: (-20)^2+(-20+41)^2 = 400+441 = 841 = 29^2.
(A129288(1), a(2)) = (0, 41) is a solution: 0^2+(0+41)^2 = 1681 = 41^2.
(A129288(3), a(4)) = (39, 89) is a solution: 39^2+(39+41)^2 = 1521+6400 = 7921 = 89^2.
MATHEMATICA
CoefficientList[Series[(1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6), {x, 0, 50}], x] (* G. C. Greubel, Feb 04 2018 *)
PROG
(PARI) {forstep(n=-20, 500000000, [3 , 1], if(issquare(n^2+(n+41)^2, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6)) \\ G. C. Greubel, Feb 04 2018
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6))) // G. C. Greubel, Feb 04 2018
CROSSREFS
Cf. A129288, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157258 (decimal expansion of 7+2*sqrt(2)), A157259 (decimal expansion of 7-2*sqrt(2)), A157260 (decimal expansion of (7+2*sqrt(2))/(7-2*sqrt(2))).
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Feb 26 2009
STATUS
approved
A157260
Decimal expansion of (7 + 2*sqrt(2))/(7 - 2*sqrt(2)).
+10
6
2, 3, 5, 6, 0, 4, 8, 2, 8, 6, 4, 9, 8, 6, 9, 9, 0, 5, 7, 7, 1, 8, 2, 2, 6, 4, 4, 5, 8, 0, 1, 7, 4, 5, 0, 2, 9, 2, 6, 7, 0, 9, 2, 9, 8, 8, 0, 6, 2, 3, 0, 6, 0, 0, 1, 1, 9, 3, 8, 3, 0, 0, 6, 4, 9, 6, 9, 2, 8, 0, 7, 1, 6, 9, 9, 8, 5, 1, 2, 1, 2, 4, 0, 9, 2, 9, 4, 7, 5, 8, 4, 4, 1, 8, 8, 7, 7, 1, 7, 1, 6, 2, 3, 9, 1
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129288.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A157257.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
FORMULA
Equals (57 + 28*sqrt(2))/41. - Klaus Brockhaus, May 01 2009
EXAMPLE
(7 +2*sqrt(2))/(7 -2*sqrt(2)) = 2.35604828649869905771...
MATHEMATICA
RealDigits[(7 + 2*Sqrt[2])/(7 - 2*Sqrt[2]), 10, 50][[1]] (* G. C. Greubel, Nov 28 2017 *)
PROG
(PARI) (7+2*sqrt(2))/(7-2*sqrt(2)) \\ G. C. Greubel, Nov 28 2017
(Magma) [(7+2*Sqrt(2))/(7-2*Sqrt(2))]; // G. C. Greubel, Nov 28 2017
CROSSREFS
Cf. A129288, A157257, A157258 (decimal expansion of 7+2*sqrt(2)), A157259 (decimal expansion of 7-2*sqrt(2)).
Cf. A157300 (decimal expansion of (1683+58*sqrt(2))/41^2). - Klaus Brockhaus, May 01 2009
KEYWORD
cons,nonn
AUTHOR
Klaus Brockhaus, Feb 26 2009
STATUS
approved
A365823
Decimal expansion of 2*(2 + sqrt(2)).
+10
3
6, 8, 2, 8, 4, 2, 7, 1, 2, 4, 7, 4, 6, 1, 9, 0, 0, 9, 7, 6, 0, 3, 3, 7, 7, 4, 4, 8, 4, 1, 9, 3, 9, 6, 1, 5, 7, 1, 3, 9, 3, 4, 3, 7, 5, 0, 7, 5, 3, 8, 9, 6, 1, 4, 6, 3, 5, 3, 3, 5, 9, 4, 7, 5, 9, 8, 1, 4, 6, 4, 9, 5, 6, 9, 2, 4, 2, 1, 4, 0, 7, 7
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The greater one of the solutions to x^2 - 8 * x + 8 = 0. The other solution is A157259 - 3 = 1.17157... . - Michal Paulovic, Nov 14 2023
LINKS
Table of n, a(n) for n=1..82.
FORMULA
Equals 2*sqrt(2)*(1 + sqrt(2)) = 2*(2 + sqrt(2)). This is an integer in the quadratic number field Q(sqrt(2)).
Equals lim_{n->oo} A057084(n + 1)/A057084(n).
Equals continued fraction with periodic term [[6], [1, 4]]. - Peter Luschny, Nov 13 2023
Equals -3+A157258 = 1+A156035 = 2+A090488 = 3+A086178 = 4+A010466 = 6+A163960. - Alois P. Heinz, Nov 15 2023
EXAMPLE
6.8284271247461900976033774484193961571393437507538961...
MAPLE
evalf(4+sqrt(8), 130); # Alois P. Heinz, Nov 13 2023
MATHEMATICA
First[RealDigits[2*(2 + Sqrt[2]), 10, 99]] (* Stefano Spezia, Nov 11 2023 *)
PROG
(PARI) \\ Works in v2.13 and higher; n = 100 decimal places
my(n=100); digits(floor(10^n*(4+quadgen(32)))) \\ Michal Paulovic, Nov 14 2023
CROSSREFS
Cf. A000129, A002193, A014176, A049310, A057084, A157259.
Cf. A156035, A163960.
Essentially the same as A157258, A090488, A086178 and A010466.
KEYWORD
nonn,cons,easy
AUTHOR
Wolfdieter Lang, Nov 13 2023
STATUS
approved
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