A276670 Numerator of (n-1)*n*(n+1)/4.

0, 0, 3, 6, 15, 30, 105, 84, 126, 180, 495, 330, 429, 546, 1365, 840, 1020, 1224, 2907, 1710, 1995, 2310, 5313, 3036, 3450, 3900, 8775, 4914, 5481, 6090, 13485, 7440, 8184, 8976, 19635, 10710, 11655, 12654, 27417, 14820, 15990, 17220, 37023, 19866, 21285

Offset: 0

Author

Paul Curtz, Oct 05 2016

Keywords

• nonn
• easy

Comments

Consider the sequence [2/(n+1), autosequence of the second kind] (see A003506), and its successive differences:

2, 1, 2/3, 1/2, 2/5, 1/3, 2/7, 1/4, 2/9, ... (see A026741)

-1, -1/3, -1/6, -1/10, -1/15, -1/21, -1/28, -1/36, -1/45, ... (see A000217)

2/3, 1/6, 1/15, 1/30, 2/105, 1/84, 1/126, 1/180, 2/495, ...

...

Each fraction in the third row is essentially the reciprocal of (n-1)*n*(n+1)/4 (3/2, 6, 15, 30, 105/2, ... ).

The numbers (= 3*A138190) are divisible by

1) -1, 1, 1, 1, 3, 2, 5, 3, 7, ... hence f(n) = 0, 0, 3, 6, 5, 15, 21, 28, 18, ...

2) 1, 1, 3, 3, 5, 5, 7, 7, 9, ... hence g(n) = 0, 0, 1, 2, 3, 6, 15, 12, 14, ...

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,-6,0,0,0,4,0,0,0,-1).

Crossrefs

Cf. A000217, A003506, A007531, A027641, A138190, A109613, A236398 (denominators).

Programs

Maple

seq(numer((n^3-n)/4), n=0..100); # Robert Israel, Oct 05 2016

Mathematica

f[n_] := Numerator[(n - 1) n (n + 1)/4]; Array[f, 40, 0] (* Robert G. Wilson v, Oct 05 2016 *)

PARI

concat(vector(2), Vec(3*x^2*(1 +2*x +5*x^2 +10*x^3 +31*x^4 +20*x^5 +22*x^6 +20*x^7 +31*x^8 +10*x^9 +5*x^10 +2*x^11 +x^12) / ((1 -x)^4*(1 +x)^4*(1 +x^2)^4) + O(x^30))) \\ Colin Barker, Oct 09 2016

Formula

a(n) = 3*A138190(n), for n>=1.

a(n) = 4*a(n-4) - 6*a(n-8) + 4*a(n-12) - a(n-16).

a(n) = A007531(n+1)/2 if n == 2 (mod 4), otherwise a(n) = A007531(n+1)/4. - Robert Israel, Oct 05 2016

G.f.: 3*x^2*(1 +2*x +5*x^2 +10*x^3 +31*x^4 +20*x^5 +22*x^6 +20*x^7 +31*x^8 +10*x^9 +5*x^10 +2*x^11 +x^12) / ((1 -x)^4*(1 +x)^4*(1 +x^2)^4). - Colin Barker, Oct 09 2016

Sum_{n>=2} 1/a(n) = 1 - log(2)/2. - Amiram Eldar, Aug 13 2022

Extensions

More terms from Robert G. Wilson v, Oct 05 2016