A293990 a(n) = (3*n + ((n-2) mod 4))/2.

1, 3, 3, 5, 7, 9, 9, 11, 13, 15, 15, 17, 19, 21, 21, 23, 25, 27, 27, 29, 31, 33, 33, 35, 37, 39, 39, 41, 43, 45, 45, 47, 49, 51, 51, 53, 55, 57, 57, 59, 61, 63, 63, 65, 67, 69, 69, 71, 73, 75, 75, 77, 79, 81, 81, 83, 85, 87, 87, 89, 91, 93, 93

Offset

0,2

Comments

The product (2/3) * (4/3) * (6/5) * (6/7) * (8/9) * (10/9) * (12/11) * (12/13) * ... = Pi/(2*sqrt(3)). The denominators are a(n) for n >= 1 and numerators are a(n-1) + A093148(n) for n >= 1 -> [2, 4, 6, 6, 8, 10, 12, 12, ...].

Let r(n) = (a(n)-1)/(a(n)+1)) if a(n) mod 4 = 1, (a(n)+1)/(a(n)-1)) otherwise; then Product_{n>=1} r(n) = (2/1) * (2/1) * (2/3) * (4/3) * (4/5) * (4/5) * (6/5) * (6/7) * ... = Pi*sqrt(3)/2 = 2.72069904635132677...

The odd numbers of partial sums this sequence, are identified with the A003215 sequence. Also the prime numbers that appear in partial sums in this sequence, are identified with the A002407 sequence.

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

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

Formula

Sum_{n>=0} 1/a(n)^2 = 5*Pi^2/36 = 1.3707783890401886970... = 10*A086729.

(a(n) - n) * (-1)^(n+1) = A134967(n) for n >= 0.

a(n) - n = A162330(n) for n >= 0.

a(n) - n = A285869(n+1) for n >= 0.

a(n) + a(n+1) = A157932(n+2) for n >= 0.

a(n) + (2*n+1) = A047298(n+1) for n >= 0.

From Colin Barker, Oct 21 2017: (Start)

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

a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.

(End)

a(n + 8) = a(n) + 12. - David A. Corneth, Oct 21 2017

a(4*k+4) * a(4*k+3) - a(4*k+2) * a(4*k+1) = 2*A063305(k+3) for k >= 0.

Sum_{n>=0} 1/(a(n) + a(n+2))^2 = (4*Pi^2 - 27) / 108 = (A214549 - 1) / 4.

Maple

A293990:=n->(3*n+((n-2) mod 4))/2: seq(A293990(n), n=0..100); # Wesley Ivan Hurt, Oct 29 2017

Mathematica

Table[(3*n + Mod[(n - 2), 4])/2, {n, 0, 100}] (* Wesley Ivan Hurt, Oct 29 2017 *)
f[n_] := (3n + Mod[n - 2, 4])/2; Array[f, 65, 0] (* or *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 3, 5, 7}, 65] (* or *)
CoefficientList[ Series[(x^4 + 2x^3 + 2x + 1)/((x - 1)^2 (x^3 + x^2 + x + 1)), {x, 0, 64}], x] (* Robert G. Wilson v, Nov 28 2017 *)

Prog

(PARI) a(n) = (3*n + (n-2)%4) / 2
(PARI) Vec(x*(1 + 2*x + 2*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^30)) \\ Colin Barker, Oct 21 2017
(PARI) first(n) = my(start=[1, 3, 3, 5, 7, 9, 9, 11]); if(n<=8, return(start)); my(res=vector(n)); for (i=1, 8, res[i] = start[i]); for(i = 1, n-8 , res[i+8] = res[i] + 12); res \\ David A. Corneth, Oct 21 2017
(Magma) [(3*n+((n-2) mod 4))/2 : n in [0..100]]; // Wesley Ivan Hurt, Oct 29 2017

Crossrefs

Cf. A007310, A047298, A063305, A093148, A134967, A157932, A162330, A168329, A214549, A285869, A003215, A002407.

Sequence in context: A168279 A171957 A278166 * A247130 A271974 A050824

Adjacent sequences: A293987 A293988 A293989 * A293991 A293992 A293993

Keyword

nonn,easy

Author

Dimitris Valianatos, Oct 21 2017

Status

approved