A047251 - OEIS

A047251

Numbers that are congruent to {1, 3, 4, 5} (mod 6).

1, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 31, 33, 34, 35, 37, 39, 40, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 67, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 85, 87, 88, 89, 91, 93, 94, 95, 97, 99

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OFFSET

1,2

COMMENTS

"Polyrhythmic Sequence" P(2,3): numbers congruent to 1 (mod 2) and 1 (mod 3). (See A267027 for definition and description.) - Bob Selcoe, Jan 12 2016

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

FORMULA

From R. J. Mathar, Oct 08 2011: (Start)

a(n) = 3*n/2 - 1/2 - cos(Pi*n/2)/2.

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

a(n) = (-2 - (-i)^n - i^n + 6n)/4, with i=sqrt(-1). - Colin Barker, Oct 19 2015

From Wesley Ivan Hurt, May 31 2016: (Start)

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.

a(2k) = A047270(k), a(2k-1) = A016777(k-1) for n>0. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = 5*sqrt(3)*Pi/36 - log(2)/3 + log(3)/4. - Amiram Eldar, Dec 17 2021

MAPLE

A047251:=n->(-2-(-I)^n-I^n+6*n)/4: seq(A047251(n), n=1..100); # Wesley Ivan Hurt, May 31 2016

MATHEMATICA

Select[Range[0, 200], MemberQ[{1, 3, 4, 5}, Mod[#, 6]] &] (* Vincenzo Librandi, Jan 12 2016 *)

LinearRecurrence[{2, -2, 2, -1}, {1, 3, 4, 5}, 70] (* Harvey P. Dale, Feb 27 2024 *)

PROG

(PARI) a(n) = (-2-(-I)^n-I^n+6*n)/4 \\ Colin Barker, Oct 19 2015

(PARI) Vec(x*(x^3+x+1)/((x-1)^2*(x^2+1)) + O(x^100)) \\ Colin Barker, Oct 19 2015

(Magma) [n: n in [0..150]|n mod 6 in {1, 3, 4, 5}]; // Vincenzo Librandi, Jan 12 2016

CROSSREFS

Cf. A016777, A047270, A267027.

Sequence in context: A273670 A153329 A213637 * A258932 A183213 A183172

Adjacent sequences: A047248 A047249 A047250 * A047252 A047253 A047254

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved