|
%I #25 Sep 08 2022 08:46:05
%S 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 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 6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6,4,6
%N Period 2: repeat [6, 4].
%C A two number repeating sequence for constructing a summation sequence from negative to positive infinity containing all primes except 2 and 5.
%C Essentially the same as A168428, A101432 and A010711.
%C 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 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 s(n+) includes all numbers with last digits of 1 and 7.
%C s(n-) includes all numbers with last digits of 3 and 9.
%C Therefore, s(n) includes all primes (except 2 and 5) without duplication.
%C 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,1).
%F a(n) = 5+(-1)^n.
%F To generate the summation sequence s(n), start with s(0) = 1, and a(0) = 6.
%F For positive values of s(n): s(n+1) = s(n) + a(n)
%F For negative values of s(n): s(n-1) = s(n) - a(n-1). n is negative here.
%F See example values for s(n) below, for both positive and negative indices.
%F G.f.: ( 6+4*x ) / ( (1-x)*(1+x) ). - _R. J. Mathar_, Jun 12 2013
%F a(n) = a(n-2) for n>1. - _Wesley Ivan Hurt_, Jul 18 2016
%e s(1) = 7, s(2) = 11, s(3) = 17, s(4) = 21, s(5) = 27, s(6) = 31;
%e s(-1) = -3, s(-2) = -9, s(-3) = -13, s(-4) = -19, s(-5) = -23, s(-6) = -29;
%p A226294:=n->5+(-1)^n: seq(A226294(n), n=0..100); # _Wesley Ivan Hurt_, Mar 24 2014
%t Table[5+(-1)^n, {n,0,100}] (* _Wesley Ivan Hurt_, Mar 24 2014 *)
%o (PARI) a(n)=6-n%2*2 \\ _Charles R Greathouse IV_, Jul 17 2016
%o (Magma) &cat [[6, 4]^^50]; // _Wesley Ivan Hurt_, Jul 18 2016
%Y Cf. A010711, A101432, A168428, A226276.
%K nonn,easy,less
%O 0,1
%A _Richard R. Forberg_, Jun 02 2013
|
