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A226294 Period 2: repeat [6, 4].

%I A226294 #27 Jun 29 2025 08:49:28
%S A226294 6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,
%T A226294 6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,
%U A226294 6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6
%N A226294 Period 2: repeat [6, 4].
%C A226294 A two number repeating sequence for constructing a summation sequence from negative to positive infinity containing all primes except 2 and 5.
%C A226294 Essentially the same as A168428, A101432 and A010711.
%C A226294 NOTE: This sequence has a shift in the starting value at index 0 relative to A010711. It is used here for the purpose stated with positive and negative indices making the formula in A010711 non-applicable.
%C A226294 This infinitely repeating sequence, a(n), of two numbers (6,4) starting with a(0) = 6, allows for the creation of an infinite summation sequence, s(n), extending from negative to positive infinity, using the formula below in parallel with how the same is done in A226276 using a different repeating sequence. Letting "s(n+)" be the set positive s(n) values, and "s(n-)" be the absolute value of the set of negative s(n) values, the following applies:
%C A226294 s(n+) includes all numbers with last digits of 1 and 7.
%C A226294 s(n-) includes all numbers with last digits of 3 and 9.
%C A226294 Therefore, s(n) includes all primes (except 2 and 5) without duplication.
%C A226294 This is one of only two such repeating patterns that accomplish this goal relative to the primes, while excluding all numbers with a last digit of 5. The other is (8,4,4,4) but with a different split between which primes occur as positive vs. negative numbers. See A226276 for details. Both patterns have the same density of primes relative to all s(n), and both, presumably, have the same average density of primes as positive vs. negative values of s(n).
%H A226294 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,1).
%F A226294 a(n) = 5+(-1)^n = 2*A176059(n).
%F A226294 To generate the summation sequence s(n), start with s(0) = 1, and a(0) = 6.
%F A226294 For positive values of s(n):  s(n+1) = s(n) + a(n)
%F A226294 For negative values of s(n):  s(n-1) = s(n) - a(n-1). n is negative here.
%F A226294 See example values for s(n) below, for both positive and negative indices.
%F A226294 G.f.: ( 6+4*x ) / ( (1-x)*(1+x) ). - _R. J. Mathar_, Jun 12 2013
%F A226294 a(n) = a(n-2) for n>1. - _Wesley Ivan Hurt_, Jul 18 2016
%e A226294 s(1) = 7, s(2) = 11, s(3) = 17, s(4) = 21, s(5) = 27, s(6) = 31;
%e A226294 s(-1) = -3, s(-2) = -9, s(-3) = -13, s(-4) = -19, s(-5) = -23, s(-6) = -29;
%p A226294 A226294:=n->5+(-1)^n: seq(A226294(n), n=0..100); # _Wesley Ivan Hurt_, Mar 24 2014
%t A226294 Table[5+(-1)^n, {n,0,100}] (* _Wesley Ivan Hurt_, Mar 24 2014 *)
%o A226294 (PARI) a(n)=6-n%2*2 \\ _Charles R Greathouse IV_, Jul 17 2016
%o A226294 (Magma) &cat [[6, 4]^^50]; // _Wesley Ivan Hurt_, Jul 18 2016
%Y A226294 Cf. A010711, A101432, A168428, A226276.
%K A226294 nonn,easy,less
%O A226294 0,1
%A A226294 _Richard R. Forberg_, Jun 02 2013