A131717 Natural numbers A000027 with 6n+4 and 6n+5 terms swapped.

1, 2, 3, 5, 4, 6, 7, 8, 9, 11, 10, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 23, 22, 24, 25, 26, 27, 29, 28, 30, 31, 32, 33, 35, 34, 36, 37, 38, 39, 41, 40, 42, 43, 44, 45, 47, 46, 48, 49, 50, 51, 53, 52, 54, 55, 56, 57, 59, 58, 60, 61, 62, 63, 65, 64, 66, 67, 68, 69, 71, 70, 72

Offset: 1

Author

Paul Curtz, Sep 15 2007

Keywords

• nonn
• easy

Comments

Hexaperiodic differences: 1, 1, 2, -1, 2, 1; 0, 1, -3, 3, -1, 0 (even palindromic signed); 1,-4, 6, -4, 1, 0.

Links

• Frieder Mittmann, Table of n, a(n) for n = 1..1002
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Crossrefs

Cf. A131042.

Programs

Maple

seq(seq(6*i+s,s=[1,2,3,5,4,6]),i=0..100); # Robert Israel, Nov 11 2014

Mathematica

Drop[CoefficientList[Series[x (2x^5 - x^4 + 2x^3 + x^2 + x + 1)/((x - 1)^2 (x + 1) (x^2 - x + 1) (x^2 + x + 1)), {x, 0, 100}], x], 1] (* Indranil Ghosh, Apr 18 2017 *)
Table[Sum[(7 #1 - 13 #2 + 17 #3 - 3 #4 + 2 #5 + 2 #6)/30 & @@ Mod[k + Range[0, 5], 6], {k, 0, n}], {n, 0, 71}] (* Michael De Vlieger, Apr 22 2017 *)

PARI

Vec(x*(2*x^5-x^4+2*x^3+x^2+x+1)/((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Nov 11 2014

Formula

a(n) = A008585(n/3) if n is congruent to 0 mod 3. - Frieder Mittmann, Nov 11 2014

a(n) = A007310((n-1)/3) if n is congruent to 1 mod 3. - Frieder Mittmann, Nov 11 2014

a(n) = A047235((n-2)/3) if n is congruent to 2 mod 3. - Frieder Mittmann, Nov 11 2014

G.f.: x*(2*x^5-x^4+2*x^3+x^2+x+1) / ((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Nov 11 2014

a(n) = (24*floor(n/6)-3*(n^2-3*n-2)-9*floor(n/3)*(3*floor(n/3)-2*n+3)+(-1)^floor(n/3)*(3*n^2-5*n-6+3*floor(n/3)*(9*floor(n/3)-6*n+5)))/4. - Luce ETIENNE, Apr 18 2017