A144454 First trisection of A061039.

0, 1, 8, 5, 8, 35, 16, 7, 80, 11, 40, 143, 56, 65, 224, 85, 32, 323, 40, 133, 440, 161, 176, 575, 208, 75, 728, 87, 280, 899, 320, 341, 1088, 385, 136, 1295, 152, 481, 1520, 533, 560, 1763, 616, 215, 2024, 235, 736, 2303, 800, 833, 2600, 901, 312, 2915, 336, 1045

Offset: 1

Author

Paul Curtz, Oct 07 2008

Keywords

• nonn
• frac
• easy

Comments

Numerator of (n^2-1)/(9n^2). Denominator is A147650(n).

Terms alternate between even and odd. The sequence modulo 9 reads (0, 1, 8, 5, 8, 8, 7, 7, 8, 2, 4, 8, 2, 2, 8, 4, 5, ...) (Is there a meaning to the interpretation as the constant 0.1858877824822845...?) The first appearance of 3 (mod 9) is at a(26)=75, the second at a(55)=336. The first appearance of 6 (mod 9) is at a(28)=87, the second at a(53)=312.

a(n) also gives the numerator of (n^2 - 1)/(3*((2*n)^2 - 1)) =: r(n-1), with denominators A300295(n-1), for n >= 1. For the proof see a comment in A300295; also for details on r(n) with the Jolley reference. - Wolfdieter Lang, Mar 15 2018

a(n) is also the numerator of Sum_{k=0..n} (1/((2*k-3)(2*k-1)*(2*k+1))). This summation is an offset adjusted form of formula 209 in Jolley's "Summation of Series". The closed form is given in the Lang comment above. - Gary Detlefs, Mar 15 2018

Examples

			The rationals (n^2 - 1)/(9*n^2) begin: 0/1, 1/12, 8/81, 5/48, 8/75, 35/324, 16/147, 7/64, 80/729, 11/100, 40/363, 143/1296, 56/507, 65/588, ... - _Wolfdieter Lang_, Mar 15 2018
		

References

• L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, pp. 40, 41.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,-3,0,0,0,0,0,0,0,0,1).

Crossrefs

Cf. A061039, A147650, A300295.

Programs

Maple

P:=n-> sum(1/((2*k-3)*(2*k-1)*(2*k+1)), k = 0 n); seq(numer(P(i))i,=1..50) # Gary Detlefs, Mar 15 2018

Mathematica

Numerator[Table[((n-1)(n+1))/(9n^2),{n,60}]] (* or *) LinearRecurrence[ {0,0,0,0, 0,0,0,0,3,0,0,0,0,0,0,0,0,-3,0,0,0,0,0,0,0,0,1}, {0,1,8,5,8,35,16,7,80,11,40, 143,56,65,224,85,32,323,40,133,440,161,176,575,208,75,728}, 60] (* Harvey P. Dale, Jan 16 2013 *)

PARI

concat(0, Vec(x^2*(1 + 8*x + 5*x^2 + 8*x^3 + 35*x^4 + 16*x^5 + 7*x^6 + 80*x^7 + 11*x^8 + 37*x^9 + 119*x^10 + 41*x^11 + 41*x^12 + 119*x^13 + 37*x^14 + 11*x^15 + 83*x^16 + 7*x^17 + 16*x^18 + 35*x^19 + 8*x^20 + 5*x^21 + 8*x^22 + x^23 - x^25) / ((1 - x)^3*(1 + x + x^2)^3*(1 + x^3 + x^6)^3) + O(x^60))) \\ Colin Barker, Mar 15 2018

PARI

a(n) = numerator(1/9-1/(3*n)^2); \\ Altug Alkan, Mar 15 2018

Sage

[numerator((1 - 1/n^2)/9) for n in (1..100)] # G. C. Greubel, Mar 07 2022

Formula

a(n) = A061039(3*n).

For n > 27, a(n) = 3*a(n-9) - 3*a(n-18) + a(n-27). - Harvey P. Dale, Jan 16 2013

a(n) = (n^2 - 1)/9 if n == 1 (mod 9) or == 8 (mod 9). For other n: a(n) = (n^2 - 1)/3 if n == 1 (mod 3) or == 2(mod 3), and a(n) = n^2 - 1 if n == 0 (mod 3). The proof uses the first comment and gcd(n^2-1, n^2) = 1. - Wolfdieter Lang, Mar 15 2018

G.f.: x^2*(1 + 8*x + 5*x^2 + 8*x^3 + 35*x^4 + 16*x^5 + 7*x^6 + 80*x^7 + 11*x^8 + 37*x^9 + 119*x^10 + 41*x^11 + 41*x^12 + 119*x^13 + 37*x^14 + 11*x^15 + 83*x^16 + 7*x^17 + 16*x^18 + 35*x^19 + 8*x^20 + 5*x^21 + 8*x^22 + x^23 - x^25) / ((1 - x)^3*(1 + x + x^2)^3*(1 + x^3 + x^6)^3). - Colin Barker, Mar 15 2018

Extensions

Edited and extended by R. J. Mathar, Oct 24 2008