A218723 a(n) = (256^n - 1)/255.

0, 1, 257, 65793, 16843009, 4311810305, 1103823438081, 282578800148737, 72340172838076673, 18519084246547628289, 4740885567116192841985, 1213666705181745367548161, 310698676526526814092329217, 79538861190790864407636279553, 20361948464842461288354887565569

Offset: 0

Author

M. F. Hasler, Nov 04 2012

Keywords

• nonn
• easy

Comments

Partial sums of powers of 256 (A133752), q-integers for q=256.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. Cites this sequence.
• Index entries related to partial sums.
• Index entries related to q-numbers.
• Index entries for linear recurrences with constant coefficients, signature (257,-256).

Crossrefs

Cf. A133752.

Cf. similar sequences of the form (k^n-1)/(k-1) listed in A269025.

Programs

Magma

[n le 2 select n-1 else 257*Self(n-1) - 256*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

Mathematica

LinearRecurrence[{257, -256}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

PARI

A218723(n)=256^n\255

Python

def A218723(n): return (1<<(n<<3))//255 # Chai Wah Wu, Nov 10 2022

Formula

a(n) = floor(256^n/255).

From Vincenzo Librandi, Nov 07 2012: (Start)

G.f.: x/((1 - x)*(1 - 256*x)).

a(n) = 257*a(n-1) - 256*a(n-2). (End)

E.g.f.: exp(x)*(exp(255*x) - 1)/255. - Stefano Spezia, Mar 23 2023