A267541 - OEIS

A267541

Expansion of (2 + 4*x + x^2 + x^3 + 2*x^4 + x^5)/(1 - x - x^5 + x^6).

2, 6, 7, 8, 10, 13, 17, 18, 19, 21, 24, 28, 29, 30, 32, 35, 39, 40, 41, 43, 46, 50, 51, 52, 54, 57, 61, 62, 63, 65, 68, 72, 73, 74, 76, 79, 83, 84, 85, 87, 90, 94, 95, 96, 98, 101, 105, 106, 107, 109, 112, 116, 117, 118, 120, 123, 127, 128, 129, 131, 134, 138, 139, 140

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Also, numbers that are congruent to {2, 6, 7, 8, 10} mod 11.

(m^k+1)/11 is a nonnegative integer when

. m is a member of this sequence and k is an odd multiple of 5 (A017329),

. m is a member of A017509 and k is odd but not multiple of 5 (A045572).

If k is even, (m^k+1)/11 is never an integer.

The product of two terms does not belong to the sequence.

LINKS

Bruno Berselli, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

FORMULA

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

a(n) = a(n-1) + a(n-5) - a(n-6).

a(-n) = -A267755(n-1).

EXAMPLE

From the linear recurrence:

(-A267755) ..., -12, -9, -5, -4, -3, -1, 2, 6, 7, 8, 10, 13, ... (A267541)

MAPLE

gf := (2+4*x+x^2+x^3+2*x^4+x^5)/((1-x)^2*(1+x+x^2+x^3+ x^4)): deg := 64: series(gf, x, deg): seq(coeff(%, x, n), n=0..deg-1); # Peter Luschny, Jan 19 2016

MATHEMATICA

CoefficientList[Series[(2 + 4 x + x^2 + x^3 + 2 x^4 + x^5)/(1 - x - x^5 + x^6), {x, 0, 70}], x]

LinearRecurrence[{1, 0, 0, 0, 1, -1}, {2, 6, 7, 8, 10, 13}, 70]

Select[Range[150], MemberQ[{2, 6, 7, 8, 10}, Mod[#, 11]]&]

PROG

(PARI) Vec((2+4*x+x^2+x^3+2*x^4+x^5)/(1-x-x^5+x^6)+O(x^70))

(Magma) m:=70; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((2+4*x+x^2+x^3+2*x^4+x^5)/(1-x-x^5+x^6)));

(Sage)

gf = (2+4*x+x^2+x^3+2*x^4+x^5)/((1-x)^2*(1+x+x^2+x^3+ x^4))

print(taylor(gf, x, 0, 63).list()) # Peter Luschny, Jan 19 2016

CROSSREFS

Cf. A002266, A017329, A017509, A267755.