A108369 -id:A108369 - cp's OEIS frontend

A108368 Coefficients of x/(1-3x-3x^2-x^3).

0, 1, 3, 12, 46, 177, 681, 2620, 10080, 38781, 149203, 574032, 2208486, 8496757, 32689761, 125768040, 483870160, 1861604361, 7162191603, 27555258052, 106013953326, 407869825737, 1569206595241, 6037243216260, 23227219260240, 89362594024741, 343806683071203

Offset: 0

Michael Somos, Jun 01 2005

From Enrique Navarrete, Jul 09 2024: (Start)
a(n+1) is the number of generalized compositions of n using parts of size at most 3 where there are binomial(3,i) types of i.
For example, the following table gives the type of composition, the number of such compositions, and the total number of compositions of n = 5 using parts of size at most 3 where there are binomial(3,i) types of i (ie. 3 types of 1, 3 types of 2 and 1 type of 3):
    Type                     Number              Total
    3+2                        2                    6
    3+1+1                      3                   27
    2+2+1                      3                   81
    2+1+1+1                    4                  324
    1+1+1+1+1                  1                  243,
adding to a(6) = 681.
The coefficients of 1/(1 - C(k,1)*x - C(k,2)*x^2 - C(k,3)*x^3 - ... - C(k,k)*x^k) give the number of generalized compositions of n using parts of size at most k where there are binomial(k,i) types of i. (End).
For n>0, the expansion of (4^(1/3) + 2^(1/3) + 1)^n is a(n)*4^(1/3) + (a(n) + a(n-1))*2^(1/3) + (a(n) + 2*a(n-1) + a(n-2)). - Greg Dresden, Aug 14 2024

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 562.

Michael De Vlieger, Table of n, a(n) for n = 0..1710
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (3,3,1).

Cf. A000012, A000129, A374454, A374455.

Cf. A108369.

CoefficientList[Series[x/(1-3*x-3*x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,3,1},{0,1,3},40] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)

a(n)=if(n>=0, polcoeff(x/(1-3*x-3*x^2-x^3)+x*O(x^n),n), n=-1-n; polcoeff(x/(1+3*x+3*x^2-x^3)+x*O(x^n),n))

x=a(n), z=a(-n), y=a(n)+a(n-1), t=a(-n)+a(-n-1) is a solution to 2(x^3+z^3)=y^3+t^3.
G.f.: x/(1-3*x-3*x^2-x^3).
a(n) = 3*a(n-1)+3*a(n-2)+a(n-3).
a(-1-n) = A108369(n).
a(n+1) = Sum_{k>=0} (1/2)^(k+1) * binomial(3*k,n). - Seiichi Manyama, Aug 03 2024

A317969 Decimal expansion of (2^(1/3)-1)^(1/3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 27 2018

(2^(1/3)-1)^(1/3) = (1/9)^(1/3) - (2/9)^(1/3) + (4/9)^(1/3) is a famous and remarkable identity of Ramanujan's.

Ramanujan's question 1076 (ii), see Berndt and Rankin in References: Show that (4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8) = (4/9)^(1/3)-(2/9)^(1/3)+(1/9)^(1/3). - Hugo Pfoertner, Aug 28 2018

			0.638185820860644153015503659444067701265157543977997683421...

B. C. Berndt and R. A. Rankin, Ramanujan: Essays and Surveys, American Mathematical Society, 2001, ISBN 0-8218-2624-7, page 222 (JIMS 11, page 199).
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.1.2, p. 4.
S. Ramanujan, Coll. Papers, Chelsea, 1962, page 331, Question 682; page 334 Question 1076.

Susan Landau, Simplification of nested radicals, SIAM Journal on Computing 21.1 (1992): 85-110. See page 85. [Do not confuse this paper with the short FOCS conference paper with the same title, which is only a few pages long.]
S. Ramanujan, Question 682, Journal of the Indian Mathematical Society, VII, p. 160.
S. Ramanujan, Question 1076, Journal of the Indian Mathematical Society, XI, p. 199.
Vincent Thill, Radicaux et Ramanujan, April 2021, see k.

Cf. A318526.

Cf. A002580, A005480, A108369.

evalf((4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8)); # Muniru A Asiru, Aug 28 2018

RealDigits[N[Power[Power[2, (3)^-1] - 1, (3)^-1], 100]] (* Peter Cullen Burbery, Apr 09 2022 *)

(4*(2/3)^(1/3)-5*(1/3)^(1/3))^(1/8) /* Hugo Pfoertner Aug 28 2018 */

sqrtn(1/9, 3) - sqrtn(2/9, 3) + sqrtn(4/9, 3) \\ Michel Marcus, Jan 07 2022

From Michel Marcus, Jan 08 2022: (Start)
Equals (A002580-1)^(1/3).
k^(3*n) = x(n) + A002580*y(n) + A005480*z(n) where k is this constant z(n) = A108369(n-1), y(n) = z(n)+z(n+1), x(n) = y(n)+y(n+1); A002580 and A005480 are the cube root of 2 and 4. (End)
Minimal polynomial: 1 - 3*x^3 - 3*x^6 - x^9. - Stefano Spezia, Oct 15 2024

A373463 Expansion of 1/((1 + x)^5 - 2*x^5).

Original entry on oeis.org

1, -5, 15, -35, 70, -124, 190, -220, 55, 715, -2999, 8585, -20580, 43520, -81940, 134376, -176195, 118435, 279235, -1572395, 4900626, -12339900, 27139450, -53163300, 91745475, -131888749, 125584845, 66464465, -781173960, 2736565920, -7295547624, 16717081040

Offset: 0

Seiichi Manyama, Aug 04 2024

Index entries for linear recurrences with constant coefficients, signature (-5,-10,-10,-5,1).

Cf. A108369, A375163.

Cf. A000750.

CoefficientList[Series[1/((1+x)^5-2x^5),{x,0,40}],x] (* or *) LinearRecurrence[{-5,-10,-10,-5,1},{1,-5,15,-35,70},40] (* Harvey P. Dale, May 25 2025 *)

my(N=40, x='x+O('x^N)); Vec(1/((1+x)^5-2*x^5))

a(n) = -5*a(n-1) - 10*a(n-2) - 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = (-1)^n * Sum_{k=0..floor(n/5)} (-2)^k * binomial(n+4,5*k+4).

A375163 Expansion of 1/((1 + x)^7 - 2*x^7).

Original entry on oeis.org

1, -7, 28, -84, 210, -462, 924, -1714, 2975, -4795, 6888, -7616, 1428, 27132, -116276, 352632, -919303, 2180101, -4815188, 10010980, -19650050, 36268478, -62075384, 95196038, -118778345, 73332329, 207719344, -1138579904, 3670243032, -9852434200, 23858720872

Offset: 0

Seiichi Manyama, Aug 05 2024

Index entries for linear recurrences with constant coefficients, signature (-7,-21,-35,-35,-21,-7,1).

Cf. A108369, A373463.

Cf. A049018.

my(N=40, x='x+O('x^N)); Vec(1/((1+x)^7-2*x^7))

a(n) = -7*a(n-1) - 21*a(n-2) - 35*a(n-3) - 35*a(n-4) - 21*a(n-5) - 7*a(n-6) + a(n-7).

a(n) = (-1)^n * Sum_{k=0..floor(n/7)} (-2)^k * binomial(n+6,7*k+6).

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A108368 Coefficients of x/(1-3*x-3*x^2-x^3).

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A317969 Decimal expansion of (2^(1/3)-1)^(1/3).

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A373463 Expansion of 1/((1 + x)^5 - 2*x^5).

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A375163 Expansion of 1/((1 + x)^7 - 2*x^7).

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A108368 Coefficients of x/(1-3x-3x^2-x^3).