A219191 Numbers of the form k*(7*k+1), where k = 0,-1,1,-2,2,-3,3,...

0, 6, 8, 26, 30, 60, 66, 108, 116, 170, 180, 246, 258, 336, 350, 440, 456, 558, 576, 690, 710, 836, 858, 996, 1020, 1170, 1196, 1358, 1386, 1560, 1590, 1776, 1808, 2006, 2040, 2250, 2286, 2508, 2546, 2780, 2820, 3066, 3108, 3366, 3410, 3680, 3726, 4008

Offset: 1

Author

Bruno Berselli, Nov 14 2012

Keywords

• nonn
• easy

Comments

Equivalently, numbers m such that 28*m+1 is a square.

Also, integer values of h*(h+1)/7.

Let F(r) = Product_{n >= 1} 1 - q^(14*n-r). The sequence terms are the exponents in the expansion of F(0)*F(6)*F(8) = 1 - q^6 - q^8 + q^26 + q^30 - q^60 - q^66 + + - - ... (by the triple product identity).- Peter Bala, Dec 25 2024

Links

• Bruno Berselli, 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. numbers of the form k*(i*k+1) with k in A001057: i=0, A001057; i=1, A110660; i=2, A000217; i=3, A152749; i=4, A074378; i=5, A219190; i=6, A036498; i=7, this sequence; i=8, A154260.

Cf. A113801 (square roots of 28*a(n)+1, see the comment).

Cf. similar sequences listed in A219257.

Subsequence of A011860.

Programs

Magma

k:=7; f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..25]];

Magma

I:=[0,6,8,26,30]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Maple

A := proc (q) local n; for n from 0 to q do if type(sqrt(28*n+1), integer) then print(n) fi; od; end: A(4100); # Peter Bala, Dec 25 2024

Mathematica

Rest[Flatten[{# (7 # - 1), # (7 # + 1)} & /@ Range[0, 25]]]
CoefficientList[Series[2 x (3 + x + 3 x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,6,8,26,30},50] (* Harvey P. Dale, Sep 14 2022 *)

Formula

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

a(n) = a(-n+1) = (14*n*(n-1)+5*(-1)^n*(2*n-1)+5)/8.

a(n) = 2*A057570(n) = (1/7)*A047335(n)*A047274(n+1).

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