A165232 - OEIS

A165232

a(0)=1, a(1)=9, a(n) = 18*a(n-1) - 9*a(n-2) for n>1.

1, 9, 153, 2673, 46737, 817209, 14289129, 249849441, 4368687777, 76387735017, 1335661040313, 23354409110481, 408358414625841, 7140261781270809, 124849486331241993, 2183028397930918593, 38170865785775356737, 667428328562578153929, 11670172122054428560089, 204056243239916510696241

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OFFSET

0,2

COMMENTS

The values of this sequence arise as a special case of the powers of the sum of two square roots, when considering the equation (sqrt(x) + sqrt(y))^(2*n) = Sum_{k=0..n} binomial(2*n, 2*k)*x^(n-k)*y^k + (Sum_{k=0..n} binomial(2*n, 2*k+1)*x^(n-k-1)*y^k)*sqrt(x*y). For x = 3, y = 6, the sequence corresponds to the terms of the first sum in the representation above. The associated recurrence relation for general description is a(n) = 2*(x+y)*a(n-1) - (x-y)^2*a(n-2), with initial values a(0) = 1 and a(1) = x+y. Additionally, the values of the second sum can be generated if the initial values are set to a(0) = 0 and a(1) = 2. - Martin Renner, May 26 2026

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (18,-9)

FORMULA

G.f.: (1-9*x)/(1-18*x+9*x^2).

MAPLE

a:= n-> (<<0|1>, <-9|18>>^n.<<1, 9>>)[1, 1]:

seq(a(n), n=0..19); # Alois P. Heinz, May 26 2026

MATHEMATICA

LinearRecurrence[{18, -9}, {1, 9}, 25] (* Paolo Xausa, Jan 04 2024 *)

(* Alternative: *)

nxt[{a_, b_}]:={b, 18b-9a}; NestList[nxt, {1, 9}, 20][[;; , 1]] (* Harvey P. Dale, Oct 02 2025 *)