A367454 -id:A367454 - cp's OEIS frontend

A223140 Decimal expansion of (sqrt(29) + 1)/2.

3, 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) = 2 is decimal expansion of sqrt(7 - sqrt(7 - sqrt(7 - sqrt(7 - ... )))) - A223141.
From Wolfdieter Lang, Jan 05 2024: (Start)
This number phi29 = (1 + sqrt(29))/2 is the fundamental algebraic integer in the quadratic number field Q(sqrt(29)) with minimal polynomial x^2 - x - 7. The other root is -A223141.
phi29^n = 7*A(n-1) + A(n)*phi29, where A(n) = A015442(n) with A(-1) = 1/7, for n >= 0. For negative powers n see A367454 = 1/phi29. (End)

			3.1925824035672520156253552457701...

Index entries for algebraic numbers, degree 2.

Cf. A223141, A367454.

Essentially the same as A098318 and A085551.

Mathematica
```
RealDigits[(1 + Sqrt[29])/2, 10, 130]
```

(sqrt(29)+1)/2 \\ Charles R Greathouse IV, Feb 10 2025

Closed form: (sqrt(29) + 1)/2 = A098318-2 = 10*A085551+3 = A223141+1.

sqrt(7 + sqrt(7 + sqrt(7 + sqrt(7 + ... )))) - 1 = sqrt(7 - sqrt(7 - sqrt(7 - sqrt(7 - ... )))). See A223141.

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).

cp's OEIS Frontend

A223140 Decimal expansion of (sqrt(29) + 1)/2.

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