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Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=8/5.
+10
3
1, 3, 5, 9, 17, 30, 52, 90, 154, 262, 446, 758, 1286, 2182, 3702, 6278, 10646, 18054, 30614, 51910, 88022, 149254, 253078, 429126, 727638, 1233798, 2092054, 3547334, 6014934, 10199046, 17293718, 29323590, 49721686, 84309126, 142956310, 242399686, 411017942
COMMENTS
Conjecture: the sequence is strictly increasing.
FORMULA
G.f.: 1/(Sum_{k>=0} [(k+1)*r](-x)^k), where r = 8/5 and [ ] = floor.
G.f.: (1 + x)^2*(1 - x + x^2 - x^3 + x^4) / ((1 - x)*(1 - x - 2*x^3)).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 2*a(n-4) for n>3.
(End)
MATHEMATICA
r = 8/5;
u = 1000; (* # initial terms from given series *)
v = 100; (* # coefficients in reciprocal series *)
CoefficientList[Series[1/Sum[Floor[r*(k + 1)] (-x)^k, {k, 0, u}], {x, 0, v}], x]
LinearRecurrence[{2, -1, 2, -2}, {1, 3, 5, 9, 17, 30, 52}, 40] (* Harvey P. Dale, Oct 13 2023 *)
PROG
(PARI) Vec((1 + x)^2*(1 - x + x^2 - x^3 + x^4) / ((1 - x)*(1 - x - 2*x^3)) + O(x^50)) \\ Colin Barker, Jul 20 2017
Decimal expansion of the Hausdorff dimension of the Heighway-Harter dragon curve boundary.
+10
2
1, 5, 2, 3, 6, 2, 7, 0, 8, 6, 2, 0, 2, 4, 9, 2, 1, 0, 6, 2, 7, 7, 6, 8, 3, 9, 3, 5, 9, 5, 4, 2, 1, 6, 6, 2, 7, 2, 8, 4, 9, 3, 6, 3, 8, 3, 4, 0, 1, 1, 9, 3, 4, 7, 8, 1, 3, 8, 6, 9, 0, 9, 0, 9, 4, 5, 7, 9, 2, 1, 6, 6, 2, 8, 9, 5, 8, 8, 4, 1, 0, 6, 8, 9, 2, 6, 6, 4, 2, 2, 7, 4, 6, 4, 7, 1, 3, 9, 4, 2, 8, 1, 1, 2, 4
COMMENTS
The value for 'twindragon' is the same.
FORMULA
Equals log_2((1+(73+6*sqrt(87))^(1/3)+(73-6*sqrt(87))^(1/3))/3).
Equals 2*log( A289265)/log(2) [Chang and Zhang, equation 9]. Equals log( A289265)/log(sqrt(2)). (End)
EXAMPLE
1.5236270862024921062776839359542166272849363834011934781386909094...
MATHEMATICA
RealDigits[Log2[(1 + (73+6*Sqrt[87])^(1/3) + (73-6*Sqrt[87])^(1/3))/3], 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)
PROG
(PARI) log((1+(73+6*sqrt(87))^(1/3)+(73-6*sqrt(87))^(1/3))/3)/log(2)
Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 2 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.
+10
2
1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 8, 4, 1, 10, 12, 1, 12, 24, 1, 14, 40, 8, 1, 16, 60, 32, 1, 18, 84, 80, 1, 20, 112, 160, 16, 1, 22, 144, 280, 80, 1, 24, 180, 448, 240, 1, 26, 220, 672, 560, 32, 1, 28, 264, 960, 1120, 192, 1, 30, 312, 1320, 2016, 672, 1, 32, 364, 1760, 3360, 1792, 64
COMMENTS
The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-right in center-justified triangle given in A013609 ((1+2*x)^n) and along a "second layer" of skew diagonals pointing top-left in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.) The coefficients in the expansion of 1/(1-x-2x^3) are given by the sequence generated by the row sums.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.695620769559862... (see A289265), when n approaches infinity.
REFERENCES
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 358, 359
FORMULA
T(n,k) = 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).
EXAMPLE
Triangle begins:
1;
1;
1;
1, 2;
1, 4;
1, 6;
1, 8, 4;
1, 10, 12;
1, 12, 24;
1, 14, 40, 8;
1, 16, 60, 32;
1, 18, 84, 80;
1, 20, 112, 160, 16;
1, 22, 144, 280, 80;
1, 24, 180, 448, 240;
1, 26, 220, 672, 560, 32;
1, 28, 264, 960, 1120, 192;
1, 30, 312, 1320, 2016, 672;
1, 32, 364, 1760, 3360, 1792, 64;
MATHEMATICA
t[n_, k_] := t[n, k] = 2^k/((n - 3 k)! k!) (n - 2 k)!; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]} ] // Flatten.
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]}] // Flatten.
PROG
(GAP) Flat(List([0..20], n->List([0..Int(n/3)], k->2^k/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # Muniru A Asiru, Jul 31 2018
Decimal expansion of the real root of x^3 - x - 2.
+10
1
1, 5, 2, 1, 3, 7, 9, 7, 0, 6, 8, 0, 4, 5, 6, 7, 5, 6, 9, 6, 0, 4, 0, 8, 0, 8, 3, 2, 2, 5, 4, 4, 3, 8, 5, 1, 4, 4, 2, 8, 3, 8, 9, 8, 2, 8, 4, 2, 7, 9, 0, 3, 9, 0, 9, 0, 9, 0, 4, 9, 8, 0, 1, 5, 4, 2, 8, 1, 5, 6, 4, 0, 3, 4, 3, 0, 5, 8, 8, 2, 1, 6, 0, 4, 9, 1, 6, 3, 7, 9, 2, 6, 9, 6, 7, 3, 3, 8, 7, 7, 0, 5, 6, 7, 9
FORMULA
Equals ((27 + 3*sqrt(78))^(1/3) + 3/(27 + 3*sqrt(78))^(1/3))/3.
Equals (1 + sqrt(78)/9)^(1/3) + (1 - sqrt(78)/9)^(1/3).
EXAMPLE
1.5213797068045675696040808322544385144283898284279039090904980154281564...
MATHEMATICA
RealDigits[x /. FindRoot[x^3 - x - 2, {x, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Jun 18 2023 *)
Decimal expansion of the real root of x^3 - 2*x^2 - 1.
+10
1
2, 2, 0, 5, 5, 6, 9, 4, 3, 0, 4, 0, 0, 5, 9, 0, 3, 1, 1, 7, 0, 2, 0, 2, 8, 6, 1, 7, 7, 8, 3, 8, 2, 3, 4, 2, 6, 3, 7, 7, 1, 0, 8, 9, 1, 9, 5, 9, 7, 6, 9, 9, 4, 4, 0, 4, 7, 0, 5, 5, 2, 2, 0, 3, 5, 5, 1, 8, 3, 4, 7, 9, 0, 3, 5, 9, 1, 6, 7, 4, 6, 9, 1, 7, 6, 4, 1, 8
COMMENTS
This is the minimum number having the property that there are uncountably many permutation classes with the growth rate equal to that number. [Vatter] - Andrey Zabolotskiy, Dec 04 2024
FORMULA
Equals ((172 + 12*sqrt(177))^(1/3)+16/(172 + 12*sqrt(177))^(1/3) + 4)/6.
Equals ((172 + 12*sqrt(177))^(1/3) + (172 - 12*sqrt(177))^(1/3) + 4)/6.
Equals (((1/2)*(43 + 3*sqrt(3*59)))^(1/3) + ((1/2)*(43 - 3*sqrt(3*59)))^(1/3) + 2)/3.
Equals 2*(1 + 2*cosh(log((43 + 3*sqrt(177))/16)/3))/3. - Vaclav Kotesovec, Aug 19 2022 Equals y + 2/3 where y = 1.538902... is the real root of y^3 - (4/3)*y - 43/27.
EXAMPLE
2.2055694304005903117020286177838234263771089195976994404705522035518347903...
MATHEMATICA
First[RealDigits[N[Root[#1^3-2#1^2-1 &, 1, 0], 78]]] (* Stefano Spezia, Aug 19 2022 *)
PROG
(PARI) solve(x=2, 3, x^3 - 2*x^2 - 1) \\ Michel Marcus, Aug 19 2022
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