A057570 Numbers of the form n*(7n+-1)/2.

0, 3, 4, 13, 15, 30, 33, 54, 58, 85, 90, 123, 129, 168, 175, 220, 228, 279, 288, 345, 355, 418, 429, 498, 510, 585, 598, 679, 693, 780, 795, 888, 904, 1003, 1020, 1125, 1143, 1254, 1273, 1390, 1410, 1533, 1554, 1683, 1705, 1840, 1863, 2004

Offset: 1

Author

N. J. A. Sloane, Oct 04 2000

Keywords

• nonn
• easy

Comments

Also integers of the form Sum_{k = 1..n} k/7. - Alonso del Arte, Jan 20 2012

Sequence provides all integers m such that 56*m + 1 is a square. [Bruno Berselli, Oct 07 2015]

The sequence terms occur as the exponents in the expansion of Product_{n >= 1} (1 - x^(7*n)) * (1 + x^(7*n-3)) * (1 + x^(7*n-4)) = 1 + x^3 + x^4 + x^13 + x^15 + x^30 + x^33 + .... Cf. A363801. - Peter Bala, Nov 21 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A074378, A001318, A057569, A154260, A154292, A154293, A047341, A274830, A363801.

Programs

Mathematica

Select[Table[Plus@@Range[n]/7, {n, 0, 199}], IntegerQ] (* Alonso del Arte, Jan 20 2012 *)
CoefficientList[Series[-x (3 + x + 3 x^2) / ((1 + x)^2 (x - 1)^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,3,4,13,15},50] (* Harvey P. Dale, Sep 17 2023 *)

PARI

a(n)=(14*n*(n-1)+5*(2*n-1)*(-1)^n+5)/16 \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: -x^2*(3+x+3*x^2) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Jan 25 2011

a(n) = +1*a(n-1)+2*a(n-2)-2*a(n-3)-1*a(n-4)+1*a(n-5). - Joerg Arndt, Jan 25 2011

a(n) = (14*n*(n-1)+5*(2*n-1)*(-1)^n+5)/16. - Bruno Berselli, Jan 25 2011

a(n)-a(n-2) = A047341(n-1) for n>2. - Bruno Berselli, Jan 25 2011

Sum_{n>=2} 1/a(n) = 14 - 2*cot(Pi/7)*Pi. - Amiram Eldar, Mar 17 2022