A223140 -id:A223140 - cp's OEIS frontend

A367454 Decimal expansion of (-1 + sqrt(29))/14 = 1/A223140.

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

Offset: 0

Wolfdieter Lang, Jan 05 2024

c^n = 7*A(-(n+1)) + A(-n)*phi29, for n >= 0, where phi29 = A223140, and A(-n) = A015442(-n) = sqrt(-7)^(-(n+1))*S(-(n+1), 1/sqrt(-7)) = -(i/sqrt(7))^(n+1)*S(n-1, i/sqrt(7)), with i = sqrt(-1) and the S-Chebyshev polynomials (see A049310), where S(-n, x) = -S(n-2, x), for n >= 1, and S(n, -x) = (-1)^n*S(n, x).

			c = 0.3132260576524645736607650351100235397353657258317706312628...

Index entries for sequences related to Chebyshev polynomials.

Cf. A010484, A015442, A049310, A223140.

Flatten[First[RealDigits[(-1 + Sqrt[29])/14,10,96]]] (* Stefano Spezia, Jan 05 2024 *)

c = 1/phi29 = (-1 + phi(29))/7, with phi29 = A223140.

A010484 Decimal expansion of square root of 29.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 5 followed by {2, 1, 1, 2, 10} repeated. - Harry J. Smith, Jun 04 2009

The fundamental algebraic (integer) number in the field Q(sqrt(29)) is (1 + sqrt(29))/2 = A223140. - Wolfdieter Lang, Nov 21 2023

			5.385164807134504031250710491540329556295120161644788837680388670016645....

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

Cf. A010128 (continued fraction). - Harry J. Smith, Jun 04 2009

Cf. A010128.

RealDigits[N[Sqrt[29], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)

default(realprecision, 20080); x=sqrt(29); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010484.txt", n, " ", d));  \\ Harry J. Smith, Jun 04 2009

A235162 Decimal expansion of (sqrt(33) + 1) / 2.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Feb 06 2014

Solution of y^2 - y - 8 = 0.
Decimal expansion of sqrt(8 + sqrt(8 + sqrt(8 + sqrt(8 + ... )))).
The sequence with a(1) = 2 is decimal expansion of sqrt(8 - sqrt(8 - sqrt(8 - sqrt(8 - ... )))).
A basis for the integers of the real quadratic number field K(sqrt(33)) is
  <1, omega(33)>, where omega(33) = (1 + sqrt(33))/2. - Wolfdieter Lang, Feb 11 2020

			3.37228132326901432992530573410946465911013222899139618384993873528...

Christopher M. Conrey, Table of n, a(n) for n = 1..10000

Cf. A001622, A010488, A209927, A222132, A222134, A223140.

val = vpa((sqrt(sym(33))+1)/2,10001); list = char(val)-'0'; list = list([1,3:end-1]); % Christopher M. Conrey, Jan 26 2022

Mathematica
```
RealDigits[(1 + Sqrt[33])/2, 10, 130]
```

A223141 Decimal expansion of (sqrt(29) - 1)/2.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Apr 02 2013

Decimal expansion of sqrt(7 - sqrt(7 - sqrt(7 - sqrt(7 - ... )))).

Sequence with a(1) = 3 is decimal expansion of sqrt(7 + sqrt(7 + sqrt(7 + sqrt(7 + ... )))) - A223140.

			2.1925824035672520156253552457701...

Cf. A098318, A085551, A223140.

RealDigits[(Sqrt[29] - 1)/2, 10, 130][[1]]

Closed form: (sqrt(29) - 1)/2 = A098318-3 = 10*A085551+2 = A223140-1.

sqrt(7 - sqrt(7 - sqrt(7 - sqrt(7 - ... )))) + 1 = sqrt(7 + sqrt(7 + sqrt(7 + sqrt(7 + ... )))). See A223140.

A317016 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 7 * T(n-2,k-1) for k = 0..floor(n/2). T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 1, 7, 1, 14, 1, 21, 49, 1, 28, 147, 1, 35, 294, 343, 1, 42, 490, 1372, 1, 49, 735, 3430, 2401, 1, 56, 1029, 6860, 12005, 1, 63, 1372, 12005, 36015, 16807, 1, 70, 1764, 19208, 84035, 100842, 1, 77, 2205, 28812, 168070, 352947, 117649, 1, 84, 2695, 41160, 302526, 941192, 823543

Offset: 0

Zagros Lalo, Jul 19 2018

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013614 ((1+7*x)^n) and along skew diagonals pointing top-left in center-justified triangle given in A027466 ((7+x)^n).
The coefficients in the expansion of 1/(1-x-7*x^2) are given by the sequence generated by the row sums.
The row sums are Generalized Fibonacci numbers (see A015442).
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.192582403567252..., when n approaches infinity (see A223140).

			Triangle begins:
  1;
  1;
  1,  7;
  1, 14;
  1, 21,   49;
  1, 28,  147;
  1, 35,  294,   343;
  1, 42,  490,  1372;
  1, 49,  735,  3430,   2401;
  1, 56, 1029,  6860,  12005;
  1, 63, 1372, 12005,  36015,  16807;
  1, 70, 1764, 19208,  84035, 100842;
  1, 77, 2205, 28812, 168070, 352947, 117649;
  1, 84, 2695, 41160, 302526, 941192, 823543;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pages 70, 96.

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 7x)^n
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (7 + x)^n

Row sums give A015442.

Cf. A013614, A027466.

Flat(List([0..13],n->List([0..Int(n/2)],k->7^k*Binomial(n-k,k)))); # Muniru A Asiru, Jul 19 2018

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 7 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten
Table[7^k Binomial[n - k, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten

T(n,k) = 7^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).

A320029 Decimal expansion of sqrt(9 + sqrt(9 + sqrt(9 + sqrt(9 + ...)))) = (sqrt(37) + 1)/2.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Oct 03 2018

For x >= 0, sqrt(x + sqrt(x + sqrt(x + sqrt(x + ...)))) = (sqrt(4*x+1) + 1)/2. This is an integer for each x such that 2*x is a term in A000217.

			3.541381265149109844499842122601033531042485047393205593209576523243166362659...

Index entries for algebraic numbers, degree 2

Cf. A000007, A001622, A000038, A209927, A222132, A222134, A223140, A235162.

evalf((sqrt(37)+1)/2,120); # Muniru A Asiru, Oct 07 2018

RealDigits[ Fold[ Sqrt[#1 + #2] &, 0, Table[9, {135}]], 10, 111][[1]] (* or *)
RealDigits[(Sqrt[37] + 1)/2, 10, 111][[1]]

(sqrt(37)+1)/2 \\ Altug Alkan, Oct 03 2018

Minimal polynomial: x^2 - x - 9. - Stefano Spezia, Jul 02 2025

A367456 Expansion of (1 - x)/(1 - x - 7*x^2).

Original entry on oeis.org

1, 0, 7, 7, 56, 105, 497, 1232, 4711, 13335, 46312, 139657, 463841, 1441440, 4688327, 14778407, 47596696, 151045545, 484222417, 1541541232, 4931098151, 15721886775, 50239573832, 160292781257, 511969798081, 1634019266880, 5217807853447, 16655942721607, 53180597695736, 169772196746985

Offset: 0

Wolfdieter Lang, Jan 16 2024

a(n) appears in the formula for powers of the fundamental algebraic number c = (1 + sqrt(29))/2 = A223140 of the quadratic number field Q(sqrt(29)): c^n = a(n) + A015442(n), for n >= 0. The formulas given below and in A015442 in terms of S-Chebyshev polynomials are valid also for c^(-n), for n >= 0, with 1/c = (-1 + sqrt(29))/14 = A367454.

a(n) is the number of compositions (ordered partitions) of n into parts >= 2 and there are 7 sorts of each part. - Joerg Arndt, Jan 16 2024

Cf.: A010484, A015442 (partial sums), A049310, A223140, A367454.

LinearRecurrence[{1,7},{1,0},30] (* James C. McMahon, Jan 16 2024 *)

a(n) = a(n-1) + 7*a(n-2), with a(0) = 1, a(1) = 0.
G.f.: (1 - x)/(1 - x - 7*x^2).
a(n) = 7*A015442(n-1), with A015442(-1) = 1/7.
a(n) = 7*(-i*sqrt(7))^(n-2)*S(n-2, i/sqrt(7)), with i = sqrt(-1) and the S-Chebyshev polynomial (see A049310). S(-2, x) = -1 and S(-1, x) = 0. The Fibonacci polynomials are F(n, x) = (-i)^(n-1)*S(n-1, i*x).

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A367454 Decimal expansion of (-1 + sqrt(29))/14 = 1/A223140.

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A010484 Decimal expansion of square root of 29.

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A235162 Decimal expansion of (sqrt(33) + 1) / 2.

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A223141 Decimal expansion of (sqrt(29) - 1)/2.

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A317016 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 7 * T(n-2,k-1) for k = 0..floor(n/2). T(n,k)=0 for n or k < 0.

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A320029 Decimal expansion of sqrt(9 + sqrt(9 + sqrt(9 + sqrt(9 + ...)))) = (sqrt(37) + 1)/2.

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A367456 Expansion of (1 - x)/(1 - x - 7*x^2).

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