A267755 - OEIS

%I #32 Sep 06 2023 13:24:34

%S 1,3,4,5,9,12,14,15,16,20,23,25,26,27,31,34,36,37,38,42,45,47,48,49,

%T 53,56,58,59,60,64,67,69,70,71,75,78,80,81,82,86,89,91,92,93,97,100,

%U 102,103,104,108,111,113,114,115,119,122,124,125,126,130,133,135,136,137

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

%C (m^k-1)/11 is a nonnegative integer when

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

%C . m is a member of A017401 and k is odd but not multiple of 5 (A045572),

%C . m is a member of A175885 and k is even but not multiple of 5 (A217562),

%C . m is a member of A160542 and k is a positive multiple of 10 (A008592),

%C apart from the trivial case in which k=0.

%C Also, numbers that are congruent to {1, 3, 4, 5, 9} mod 11. Therefore, the product of two terms belongs to the sequence.

%C Union of this sequence and A267541 is A160542.

%C a(n) is prime for n = 1, 3, 10, 14, 17, 21, 24, 27, 30, 33, 40, 44, 47, ...

%H Bruno Berselli, <a href="/A267755/b267755.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).

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

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

%F a(-n) = -A267541(n-1).

%F a(n) = n + 1 + 2*floor(n/5) + 3*floor((n+1)/5) + floor((n+4)/5). - _Ridouane Oudra_, Sep 06 2023

%e From the linear recurrence:

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

%p gf := (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6): deg := 64: series(gf, x, deg): seq(coeff(%, x, n), n=0..deg-1); # _Peter Luschny_, Jan 21 2016

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

%t LinearRecurrence[{1, 0, 0, 0, 1, -1}, {1, 3, 4, 5, 9, 12}, 70]

%t Select[Range[140], MemberQ[{1, 3, 4, 5, 9}, Mod[#, 11]]&]

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

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

%o (Sage)

%o gf = (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6)

%o print(taylor(gf, x, 0, 63).list()) # _Peter Luschny_, Jan 21 2016

%o (Magma) I:=[1,3,4,5,9,12]; [n le 6 select I[n] else Self(n-1)+Self(n-5)-Self(n-6): n in [1..70]]; // _Vincenzo Librandi_, Jan 21 2016

%Y Cf. A160542, A017329, A267541.

%Y Related sequences (see the first comment): A017401, A160542, A175885.

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Jan 20 2016