A001027 -id:A001027 - cp's OEIS frontend

A373939 Expansion of 1/(2 - 1/(1 - 9*x)^(2/3)).

1, 6, 81, 1116, 15471, 214974, 2990115, 41609376, 579152592, 8062039188, 112233642606, 1562482974384, 21752802921528, 302844316749948, 4216246872393294, 58699445770011312, 817227085779238851, 11377633930294786182, 158402278494943048161, 2205317108305048202244

Offset: 0

Seiichi Manyama, Aug 04 2024

Cf. A001027, A373818.

my(N=20, x='x+O('x^N)); Vec(1/(2-1/(1-9*x)^(2/3)))

a(n) = 9^n * Sum_{k>=0} (1/2)^(k+1) * binomial(n-1+2*k/3,n).

A334991 a(n) = 4^n + 3 * 18^n.

Original entry on oeis.org

4, 58, 988, 17560, 315184, 5669728, 102040768, 1836676480, 33059947264, 595078133248, 10711402728448, 192805234432000, 3470494161055744, 62468894664122368, 1124440103014678528, 20239921850506117120, 364318593294077722624, 6557734679233269465088, 118039224225958332203008

Offset: 0

Bernard Schott, May 18 2020

This sequence is a variation of the sequence A333385, variation proposed by Tony Gardiner in his book in reference.

Proposition: a(n) is a perfect square iff n = 0; in this case, a(0) = 4.

			a(4) = 4^4 + 3 * 18^4 = 315184 = 2^4 * 19699 is not a perfect square.

A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, page 115 (1991).

Index entries for linear recurrences with constant coefficients, signature (22,-72).

Cf. A000302, A001027, A333385.

Maple
```
S:=seq(4^n+3*18^n, n=0..20);
```

a(n) = A000302(n) + 3 * A001027(n).
a(n) = 22*a(n-1) - 72*a(n-2) for n>1.
G.f.: (4 - 30*x)/((1 - 4*x)*(1 - 18*x)). - Alejandro J. Becerra Jr., Jun 01 2020

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A373939 Expansion of 1/(2 - 1/(1 - 9*x)^(2/3)).

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A334991 a(n) = 4^n + 3 * 18^n.

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