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

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

Offset: 0

Author

Arkadiusz Wesolowski, Aug 09 2013

Keywords

• easy
• nonn

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).

Examples

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

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).

Crossrefs

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

Programs

Magma

[64^n+1 : n in [0..15]];

Mathematica

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

PARI

for(n=0, 15, print1(64^n+1, ", "))

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).