A178242 Numerator of n*(5+n)/((n+1)*(n+4)).

0, 3, 7, 6, 9, 25, 33, 21, 26, 63, 75, 44, 51, 117, 133, 75, 84, 187, 207, 114, 125, 273, 297, 161, 174, 375, 403, 216, 231, 493, 525, 279, 296, 627, 663, 350, 369, 777, 817, 429, 450, 943, 987, 516, 539, 1125, 1173, 611, 636, 1323, 1375, 714, 741, 1537, 1593

Offset

0,2

Comments

Sequence of differences denominator(n) - numerator(n) = 1,2,2,1... = A014695(n).

Denominator: A160050(n+2).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Wikipedia, Quasi-polynomial.

Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Formula

a(n) = numerator(A176027(n)/A001793(n+1)).

a(n) = A060819(n)*A060819(n+5).

a(n) = +3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*a(n-6) +6*a(n-7) -3*a(n-8) +a(n-9).

a(n) = 3*a(n-4) -3*a(n-8) +a(n-12).

G.f.: x*(-3+2*x-3*x^2-3*x^3+x^7) / ( (x-1)^3*(x^2+1)^3 ).

a(n) = n*(n+5)*((-1)^((2*n-(-1)^n-3)/4)+3)/8 = n*(n+5)*(3-i^(n*(n+1)))/8, where i=sqrt(-1); also a(n) = a(n-4)*A028557(n)/A028557(n-4) for n>4. - Bruno Berselli, Dec 30 2010

From Peter Bala, Aug 07 2022: (Start)

a(n) = numerator of n*(n+5)/4.

a(n) is quasi-polynomial in n: a(4*n) = n*(4*n+5) = A343560(n+1); a(4*n+1) = (2*n+3)*(4*n+1); a(4*n+2) = (2*n+1)*(4*n+7); a(4*n+3) = (n+2)*(4*n+3) = A180863(n+2). (End)

Sum_{n>=1} 1/a(n) = 112/75 - Pi/10. - Amiram Eldar, Aug 16 2022

Example

The reduced fractions are 0, 3/5, 7/9, 6/7, 9/10, 25/27, 33/35, 21/22, 26/27, 63/65, 75/77, 44/45, ..

Maple

A178242 := proc(n) n*(5+n)/(n+1)/(n+4) ;  numer(%) ; end proc:
seq(A178242(n), n=0..80) ; # R. J. Mathar, Dec 20 2010

Mathematica

f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 5] &, 50, 0]
Table[Numerator[n*(5+n)/((n+1)*(n+4))], {n, 0, 50}] (* G. C. Greubel, Sep 21 2018 *)

Prog

(Magma) [Floor(n*(n+5)*((-1)^((2*n-(-1)^n-3)/4)+3)/8) : n in [0..50]]; // Vincenzo Librandi, Oct 08 2011
(PARI) vector(50, n, n--; numerator(n*(5+n)/((n+1)*(n+4)))) \\ G. C. Greubel, Sep 21 2018

Crossrefs

Cf. A001793, A014695, A028557, A060819, A160050, A176027.

Sequence in context: A371240 A117823 A338067 * A352168 A070882 A109635

Adjacent sequences: A178239 A178240 A178241 * A178243 A178244 A178245

Keyword

nonn,easy,frac

Author

Paul Curtz, Dec 20 2010

Status

approved