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A204454 Odd numbers not divisible by 11.

%I A204454 #38 Oct 20 2020 21:31:21
%S A204454 1,3,5,7,9,13,15,17,19,21,23,25,27,29,31,35,37,39,41,43,45,47,49,51,
%T A204454 53,57,59,61,63,65,67,69,71,73,75,79,81,83,85,87,89,91,93,95,97,101,
%U A204454 103,105,107,109,111,113,115,117,119,123,125,127,129,131
%N A204454 Odd numbers not divisible by 11.
%C A204454 Up to a(45) this sequence coincides with A029740, but 101 is not in A029740.
%C A204454 This sequence is the fourth member of the family of sequences of odd numbers not divisible by a given odd prime p. For p = 3, 5, and 7 these sequences are A007310, A045572, and A162699, respectively.  The formula is
%C A204454   a(p;n) = 2*n+1 + 2*floor((n-(p+1)/2)/(p-1)), n>=1, p an odd prime. If one puts a(p;0):=0, the o.g.f. is
%C A204454   G(p;x) = (x/((1-x^(p-1))*(1-x)))*(1 + 2*sum(x^k,k=1..(p-3)/2) + 4*x^((p-1)/2) + 2*sum(x^((p-1)/2+k),k=1..(p-3)/2) + x^(p-1)).
%C A204454 See the array A204456 with the coefficients of the numerator polynomials of these o.g.f.s.
%C A204454 This sequence gives also the numbers relatively prime to 2 and 11.
%C A204454 Another formula is a(p;n) = 2*n-1 + 2*floor(( n-(p-3)/2)/(p-1)), n>=1. From the rows of the array A204456 for the o.g.f. one can show first: a(p;n) = n + sum(floor((n+p-3-k)/(p-1)),k=1..(p-3)/2) + 3*floor((n+(p-3)/2)/(p-1)) + sum(floor((n+(p-3)/2-k)/(p-1)),k=1..(p-1)/2), p an odd prime, n>=1. - _Wolfdieter Lang_, Jan 26 2012
%C A204454 Recurrences for odd p: a(p;n) = a(p;n-(p-1)) + 2*p. For first differences: a(p;n) = a(p;n-1) + a(p;n-p+1) - a(p;n-p), n>=p, and inputs a(p;0):=-1 (here not 0) and a(p;k) for k=1,...,p-1. See the formula sections of the A-numbers for the instances p = 3, 5, and 7 for the contributions from _Zak Seidov_ and _R. J. Mathar_. From this recurrence follows the o.g.f. (starting with x^1) directly. Above it has been found from the formula for a(p;n). Here the coefficients of the numerator polynomial of the o.g.f. (besides the 1s for x^1 and x^p) arise as first differences of the input members of the {a(p;n)} sequence. - _Wolfdieter Lang_, Jan 27 2012
%C A204454 Numbers coprime to 22. The asymptotic density of this sequence is 5/11. - _Amiram Eldar_, Oct 20 2020
%H A204454 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,1,-1).
%F A204454 a(n) = 2*n+1 + 2*floor((n-6)/10), n>=1. Note that this is -1 for n=0, but the following o.g.f. uses a(0)=0.
%F A204454 O.g.f: x*(1+2*x+2*x^2+2*x^3+2*x^4+4*x^5+2*x^6+2*x^7+2*x^8+2*x^9+x^10)/((1-x^10)*(1-x)). See the comment above for p=11.
%F A204454 a(n) = n + sum(floor((n+9-k)/10),k=1..4) + 3*floor((n+4)/10) + sum(floor((n+4-k)/10),k=1..5) = n + (n-1) + 2*floor((n+4)/10), n>=1. See the line m=5, p=11 of the array A204456, and the general formula given in a comment above. - _Wolfdieter Lang_, Jan 26 2012
%F A204454 Recurrences: a(n) = a(n-10) + 2*11. First differences: a(n) = a(n-1) + a(n-10) - a(n-11), n>=11, and inputs a(p;0):=-1 ( here not 0) and a(p;k) for k=1,...,10. See the general comment above. - _Wolfdieter Lang_, Jan 27 2012
%e A204454 2*floor((n-6)/10), n>=0, is the sequence (the exponent of a number indicates how many times this number appears consecutively): (-2)^6 0^10 2^10 4^10 ... By adding these numbers to 2*n+1, n>=0, one obtains -1 for n=0 and a(n) for n>=1. The o.g.f is computed from this sum, but adjusted such that one obtains a vanishing a(0).
%e A204454 Recurrences: 31 = a(15) = a(5) + 2*11 = 9 + 22. a(15) = a(14) + a(5) - a(4) = 29 + 9 - 7 = 31. - _Wolfdieter Lang_, Jan 27 2012
%t A204454 Complement[Range[1, 131, 2], 11Range[11]] (* _Alonso del Arte_, Jan 24 2012 *)
%o A204454 (PARI) a(n)=2*n+(n-6)\10*2+1 \\ _Charles R Greathouse IV_, Jan 24 2012
%Y A204454 Cf. A007310, A045572, A162699, A204456, A204457, A204458.
%K A204454 nonn,easy
%O A204454 1,2
%A A204454 _Wolfdieter Lang_, Jan 24 2012