A144181 -id:A144181 - cp's OEIS frontend

A144182 Eigentriangle, row sums = A144181.

1, 0, 1, 2, 0, 2, 4, 2, 0, 3, -4, 4, 2, 0, 9, 0, -4, 4, 6, 0, 11, -8, 0, -4, 12, 18, 0, 17, -16, -8, 0, -12, 36, 22, 0, 35, 16, -16, -8, 0, -36, 44, 34, 0, 57, 0, 16, -16, -24, 0, -44, 68, 70, 0, 91, 32, 0, 16, -48, -72, 0, -68, 140, 114, 0, 161

Offset: 0

Gary W. Adamson, Sep 13 2008

Row sums = A144181: (1, 1, 3, 9, 11, 17, 35,...).
Left border = A118434: (1, 0, 2, 4, -4, 0, -8,...); (i.e. row sums of the self-inverse triangle A118433).
Triangle A144183 = partial sums starting from the right of A144182.
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle are:
1;
0, 1;
2, 0, 1;
4, 2, 0, 3;
-4, 4, 2, 0, 9;
0, -4, 4, 6, 0, 11;
-8, 0, -4, 12, 18, 0, 17;
-16, -8, 0, -12, 36, 22, 0, 35;
...
row 3 = (4, 2, 0, 3) = termwise products of (4, 2, 0, 1) and (1, 1, 1, 3) = (4*1, 2*1, 0*1, 1*3).

A144181, Cf. A118434, A144183

Triangle read by rows, T(n,k) = A118434(n-k)*A144181(k-1); where A144181(k-1) = A144181 shifted to (1, 1, 1, 3, 9, 11, 17, 35, 57, 91, 161,...).

A289265 Decimal expansion of the real root of x^3 - x^2 - 2 = 0.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 14 2017

			1.6956207695598620574163671001175353426181793882085077...

D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves, unpublished, 1976, end of section 2. See links in A003229.

Angel Chang and Tianrong Zhang, The Fractal Geometry of the Boundary of Dragon Curves, Journal of Recreational Mathematics, volume 30, number 1, 1999-2000, pages 9-22.
Index entries for algebraic numbers, degree 3.

Sequences growing as this power: A003229, A003476, A003479, A052537, A077949, A144181, A164395, A164399, A164410, A164414, A164471, A203175, A227036, A289260, A292764.

Cf. A078140 (includes guide to constants similar to A289260).

z = 2000; r = 8/5;
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A289260 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]] (* A289265 *)

solve(x=1, 2, x^3 - x^2 - 2) \\ Michel Marcus, Oct 26 2019

r = D^(1/3) + (1/9)*D^(-1/3) + 1/3 where D = 28/27 + (1/9)*sqrt(29*3) [Chang and Zhang] from the usual cubic solution formula. Or similarly r = (1/3)*(1 + C + 1/C) where C = (28 + sqrt(29*27))^(1/3). - Kevin Ryde, Oct 25 2019

A144183 Triangle read by rows, A144182 * A000012.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 9, 5, 3, 3, 11, 15, 11, 9, 9, 17, 17, 21, 17, 11, 11, 35, 43, 43, 47, 35, 17, 17, 57, 73, 81, 81, 93, 57, 35, 35, 91, 75, 91, 99, 99, 135, 91, 57, 57, 161, 161, 145, 161, 185, 185, 229, 161, 91, 91, 275, 243, 243, 227, 275, 347, 347, 415, 275, 161, 161

Offset: 0

Gary W. Adamson, Sep 13 2008

Left border = A144181: (1, 1, 3, 9, 11, 17, 35,...) = INVERT transform of A118434. Right border = A144181 shifted.

			First few rows of the triangle =
1;
1, 1;
3, 1, 1;
9, 5, 3, 3;
11, 15, 11, 9, 9;
17, 17, 21, 17, 11, 11;
35, 43, 43, 47, 35, 17, 17;
57, 73, 81, 81, 93, 57, 35, 35;
91, 75, 91, 99, 99, 135, 91, 57, 57;
...
Row 3 = (9, 5, 3, 3) = partial sums from the right of row 3, triangle A144182: (4, 2, 0, 3).

A144182, Cf. A144181, A118434

Triangle read by rows, A144182 * A000012; (equivalent to taking partial row sums

of A144182 starting from the right). A000012 = an infinite lower triangular matrix with all 1's and the rest zeros.

cp's OEIS Frontend

A144182 Eigentriangle, row sums = A144181.

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A289265 Decimal expansion of the real root of x^3 - x^2 - 2 = 0.

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A144183 Triangle read by rows, A144182 * A000012.

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