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

%I A219191 #41 Dec 26 2024 10:15:05
%S A219191 0,6,8,26,30,60,66,108,116,170,180,246,258,336,350,440,456,558,576,
%T A219191 690,710,836,858,996,1020,1170,1196,1358,1386,1560,1590,1776,1808,
%U A219191 2006,2040,2250,2286,2508,2546,2780,2820,3066,3108,3366,3410,3680,3726,4008
%N A219191 Numbers of the form k*(7*k+1), where k = 0,-1,1,-2,2,-3,3,...
%C A219191 Equivalently, numbers m such that 28*m+1 is a square.
%C A219191 Also, integer values of h*(h+1)/7.
%C A219191 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
%H A219191 Bruno Berselli, <a href="/A219191/b219191.txt">Table of n, a(n) for n = 1..1000</a>
%H A219191 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A219191 G.f.: 2*x^2*(3+x+3*x^2)/((1+x)^2*(1-x)^3).
%F A219191 a(n) = a(-n+1) = (14*n*(n-1)+5*(-1)^n*(2*n-1)+5)/8.
%F A219191 a(n) = 2*A057570(n) = (1/7)*A047335(n)*A047274(n+1).
%F A219191 Sum_{n>=2} 1/a(n) = 7 - cot(Pi/7)*Pi. - _Amiram Eldar_, Mar 17 2022
%p A219191 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
%t A219191 Rest[Flatten[{# (7 # - 1), # (7 # + 1)} & /@ Range[0, 25]]]
%t A219191 CoefficientList[Series[2 x (3 + x + 3 x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%t A219191 LinearRecurrence[{1,2,-2,-1,1},{0,6,8,26,30},50] (* _Harvey P. Dale_, Sep 14 2022 *)
%o A219191 (Magma) k:=7; f:=func<n | n*(k*n+1)>; [0] cat [f(n*m): m in [-1,1], n in [1..25]];
%o A219191 (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
%Y A219191 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.
%Y A219191 Cf. A113801 (square roots of 28*a(n)+1, see the comment).
%Y A219191 Cf. similar sequences listed in A219257.
%Y A219191 Subsequence of A011860.
%K A219191 nonn,easy
%O A219191 1,2
%A A219191 _Bruno Berselli_, Nov 14 2012