A271324 a(n) = n + floor(n/4) + (n mod 4).

0, 2, 4, 6, 5, 7, 9, 11, 10, 12, 14, 16, 15, 17, 19, 21, 20, 22, 24, 26, 25, 27, 29, 31, 30, 32, 34, 36, 35, 37, 39, 41, 40, 42, 44, 46, 45, 47, 49, 51, 50, 52, 54, 56, 55, 57, 59, 61, 60, 62, 64, 66, 65, 67, 69, 71, 70, 72, 74, 76, 75, 77, 79, 81, 80, 82, 84, 86, 85, 87, 89

Offset: 0

Author

Bruno Berselli, Apr 04 2016

Keywords

• nonn
• easy

Comments

Sort the terms in increasing order and add 1 to get sequence A032769.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A032769.

Cf. numbers of the form m + floor(m/k) + (m mod k): A028242 (k=-2), A000004 (k=-1), A005843 (k=1), A007494 (k=2), A063224 (k=3).

Programs

Magma

[n + Floor(n/4) + (n mod 4): n in [0..80]];

Mathematica

Table[n + Floor[n/4] + Mod[n, 4], {n, 0, 80}]

Maxima

makelist(n + floor(n/4) + mod(n, 4), n, 0, 80);

PARI

vector(80, n, n--; n + floor(n/4) + n%4)

Python

def A271324(n): return n+(n>>2)+(n&3) # Chai Wah Wu, Jan 29 2023

Sage

[n + floor(n/4) + n%4 for n in (0..80)]

Formula

O.g.f.: x*(2 + 2*x + 2*x^2 - x^3)/((1 - x)^2*(1 + x + x^2 + x^3)).

E.g.f.: ((6 + 5*x)*sinh(x) + (3 + 5*x)*cosh(x) - 3*(sin(x) + cos(x)))/4.

a(n) = 1 + (10*n - 6*(-1)^((n-1)*n/2) - 3*(-1)^n + 1)/8.

a(4*k + r) = 5*k + 2*r, with r = 0, 1, 2 or 3.

a(n + 4*k) = a(n) + 5*k.