A265228 Interleave the even numbers with the numbers that are congruent to {1, 3, 7} mod 8.

0, 1, 2, 3, 4, 7, 6, 9, 8, 11, 10, 15, 12, 17, 14, 19, 16, 23, 18, 25, 20, 27, 22, 31, 24, 33, 26, 35, 28, 39, 30, 41, 32, 43, 34, 47, 36, 49, 38, 51, 40, 55, 42, 57, 44, 59, 46, 63, 48, 65, 50, 67, 52, 71, 54, 73, 56, 75, 58, 79, 60, 81, 62, 83, 64, 87, 66

Offset: 0

Author

Paul Curtz, Dec 06 2015

Keywords

• nonn
• easy

Comments

b(n) denotes the sequence:

0, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 1, 2, -2, 2, -2, 2, -2, 2, 3, -3, 3, -3, 3, -3, 3, 4, -4, ..., and

c(n) = n + b(n) = n + floor((n+1)/7)*(-1)^((n+1) mod 7) provides:

0, 1, 2, 3, 4, 5, 7, 6, 9, 8, 11, 10, 13, 15, 12, 17, 14, 19, 16, 21, 23, 18, 25, 20, 27, 22, 29, ..., which is a permutation of A001477.

a(n) differs from c(n) because c(n) contains the terms of the form 8*k+5.

Links

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

Crossrefs

Cf. A001477, A004770, A005843, A047529, A079979, A260160, A260699, A260708.

Programs

Mathematica

lim = 11; Riffle[Range[0, 6 lim, 2], Select[Range[8 lim], MemberQ[{1, 3, 7}, Mod[#, 8]] &]] (* Michael De Vlieger, Dec 06 2015 *)

PARI

concat(0, Vec(x*(1+2*x+2*x^2+2*x^3+4*x^4+2*x^5+x^6)/((1-x)^2 *(1+x)^2*(1-x+x^2)*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Dec 06 2015

PARI

vector(100, n, n--; n+(1-(-1)^n)*floor(n/6+1/3)) \\ Altug Alkan, Dec 09 2015

Formula

a(n) = n + 2*A260160(n) = n + (1-(-1)^n)*floor(n/6+1/3). Therefore, for odd n, a(n) = A047529((n+1)/2); otherwise, a(n) = n.

a(n) = a(n-6) - (-1)^n + 7.

a(n) = A260708(n) - A260699(n-1) - A079979(n+3), with A260699(-1) = 0.

From Colin Barker, Dec 06 2015: (Start)

a(n) = a(n-2) + a(n-6) - a(n-8) for n > 7.

G.f.: x*(1+2*x+2*x^2+2*x^3+4*x^4+2*x^5+x^6) / ((1-x)^2*(1+x)^2*(1-x+x^2)*(1+x+x^2)). (End)