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%I #46 Mar 15 2022 04:31:47
%S 0,1,8,13,17,24,45,56,64,77,112,129,141,160,209,232,248,273,336,365,
%T 385,416,493,528,552,589,680,721,749,792,897,944,976,1025,1144,1197,
%U 1233,1288,1421,1480,1520,1581,1728,1793,1837,1904,2065,2136,2184,2257,2432
%N Integers of the form (n^2 - 1) / 15.
%C Equivalently, numbers in increasing order of the form m(15m+2) or m(15m+8)+1, where m = 0,-1,1,-2,2,-3,3,... [_Bruno Berselli_, Nov 27 2012]
%C The sequence terms occur as exponents in the expansion of the identity Product_{n >= 0} (1 - x^(20*n+1))*(1 - x^(20*n+19))*(1 - x^(20*n+8))*(1 - x^(20*n+12))*(1 - x^(20*n+9))*(1 - x^(20*n+11))*(1 - x^(10*n+10)) = Sum_{n >= 0} x^(n^2+n)*Product_{k >= 2*n+1} 1 - x^k = 1 - x - x^8 + x^13 + x^17 - - + + .... See Andrews et al., p. 591, Exercise 6(c).
%D George E. Andrews, Richard Askey, and Ranjan Roy, Special Functions, Cambridge University Press, 1999.
%H Vincenzo Librandi, <a href="/A204221/b204221.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).
%F |A204220(n)| is the characteristic function of the numbers in this sequence.
%F a(-1 - n) = a(n).
%F G.f. -x*(x^2-x+1)*(x^4+8*x^3+12*x^2+8*x+1) / ( (1+x)^2*(x^2+1)^2*(x-1)^3 ). - _R. J. Mathar_, Jan 28 2012
%F a(n) = (30*n-10*i^(n(n-1))+3*(-1)^n+7)*(30*n-10*i^(n(n-1))+3*(-1)^n+23)/960, where i=sqrt(-1). - _Bruno Berselli_, Nov 28 2012
%F Sum_{n>=1} 1/a(n) = 15/4 - cot(2*Pi/15)*Pi/2 - Pi/(2*sqrt(3)) + sqrt(1+2/sqrt(5))*Pi/2. - _Amiram Eldar_, Mar 15 2022
%t Select[Range[0, 2500], IntegerQ[Sqrt[15 # + 1]] &] (* _Bruno Berselli_, Nov 23 2012 *)
%o (PARI) {a(n) = (15*n^2 + n*[8, 2, 28, 22][n%4 + 1] + 12) \ 16}
%o (Magma) [n: n in [0..2500] | IsSquare(15*n+1)]; // _Bruno Berselli_, Nov 23 2012
%o (Magma) /* By comment: */ s:=[0, 1] cat &cat[[t*(15*t+2), t*(15*t+8)+1]: t in [-n,n], n in [1..13]]; Sort(s); // _Bruno Berselli_, Nov 27 2012
%Y Cf. A204220, A204542 (square roots of 15*a(n)+1).
%Y Cf. similar sequences listed in A219257.
%K nonn,easy
%O 0,3
%A _Michael Somos_, Jan 13 2012
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