A228081 a(n) = 64^n + 1.

2, 65, 4097, 262145, 16777217, 1073741825, 68719476737, 4398046511105, 281474976710657, 18014398509481985, 1152921504606846977, 73786976294838206465, 4722366482869645213697, 302231454903657293676545, 19342813113834066795298817, 1237940039285380274899124225

Offset

0,1

Comments

These numbers can be written as the sum of two relatively prime squares and also as the sum of two relatively prime cubes (i.e., 2^(6*n) + 1 = (2^(3*n))^2 + 1^2 = (2^(2*n))^3 + 1^3).

Links

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..100

Index entries for sequences related to sums of cubes

Index entries for sequences related to sums of squares

Index entries for linear recurrences with constant coefficients, signature (65,-64).

Formula

a(n) = 64*a(n-1) - 63.

a(n) = A089357(n) + 1 = 2^A008588(n) + 1.

G.f.: (2 - 65*x)/((1 - x)*(1 - 64*x)).

E.g.f.: e^x + e^(64*x).

Example

a(2) = 64^2 + 1 = 4097.

Mathematica

Table[64^n + 1, {n, 0, 15}]
LinearRecurrence[{65, -64}, {2, 65}, 20] (* Harvey P. Dale, Jul 17 2020 *)

Prog

(Magma) [64^n+1 : n in [0..15]]
(PARI) for(n=0, 15, print1(64^n+1, ", "))

Crossrefs

Cf. A000051 (2^n + 1), A052539 (4^n + 1), A062395 (8^n + 1).

Sequence in context: A199145 A198665 A185029 * A214366 A220596 A326253

Adjacent sequences: A228078 A228079 A228080 * A228082 A228083 A228084

Keyword

easy,nonn

Author

Arkadiusz Wesolowski, Aug 09 2013

Status

approved