A235162 -id:A235162 - cp's OEIS frontend

A317026 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 8 * T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

1, 1, 1, 8, 1, 16, 1, 24, 64, 1, 32, 192, 1, 40, 384, 512, 1, 48, 640, 2048, 1, 56, 960, 5120, 4096, 1, 64, 1344, 10240, 20480, 1, 72, 1792, 17920, 61440, 32768, 1, 80, 2304, 28672, 143360, 196608, 1, 88, 2880, 43008, 286720, 688128, 262144, 1, 96, 3520, 61440, 516096, 1835008, 1835008

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 A013615 ((1+8*x)^n) and along skew diagonals pointing top-left in center-justified triangle given in A038279 ((8+x)^n).
The coefficients in the expansion of 1/(1-x-8*x^2) are given by the sequence generated by the row sums.
The row sums are Generalized Fibonacci numbers (see A015443).
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.3722813232690143..., when n approaches infinity; see A235162 (Decimal expansion of (sqrt(33)+1)/2).

			Triangle begins:
  1;
  1;
  1, 8;
  1, 16;
  1, 24, 64;
  1, 32, 192;
  1, 40, 384, 512;
  1, 48, 640, 2048;
  1, 56, 960, 5120, 4096;
  1, 64, 1344, 10240, 20480;
  1, 72, 1792, 17920, 61440, 32768;
  1, 80, 2304, 28672, 143360, 196608;
  1, 88, 2880, 43008, 286720, 688128, 262144;
  1, 96, 3520, 61440, 516096, 1835008, 1835008;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, Pages 70, 98

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 8x)^n
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (8 + x)^n

Row sums give A015443.

Cf. A013615, A038279, A235162.

Flat(List([0..13],n->List([0..Int(n/2)],k->8^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] + 8 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten
Table[8^k Binomial[n - k, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten

T(n,k) = 8^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).

A236290 Decimal expansion of (sqrt(33) - 1) / 2.

Original entry on oeis.org

2, 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 07 2014

Decimal expansion of sqrt(8 - sqrt(8 - sqrt(8 - sqrt(8 - ... )))).
The sequence with a(1) = 3 is decimal expansion of sqrt(8 + sqrt(8 + sqrt(8 + sqrt(8 + ... )))).
A quadratic integer with minimal polynomial x^2 + x - 8. - Charles R Greathouse IV, Apr 21 2016

			2.37228132326901432992530573410946465911013222899139618384993873528...

Index entries for algebraic numbers, degree 2

Cf. A010488, A235162.

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

(sqrt(33)-1)/2 \\ Charles R Greathouse IV, Apr 21 2016

A235162 - 1.

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

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A317026 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 8 * T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

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A236290 Decimal expansion of (sqrt(33) - 1) / 2.

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

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