OFFSET
0,3
COMMENTS
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]
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).
REFERENCES
George E. Andrews, Richard Askey, and Ranjan Roy, Special Functions, Cambridge University Press, 1999.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).
FORMULA
|A204220(n)| is the characteristic function of the numbers in this sequence.
a(-1 - n) = a(n).
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
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
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
MATHEMATICA
Select[Range[0, 2500], IntegerQ[Sqrt[15 # + 1]] &] (* Bruno Berselli, Nov 23 2012 *)
PROG
(PARI) {a(n) = (15*n^2 + n*[8, 2, 28, 22][n%4 + 1] + 12) \ 16}
(Magma) [n: n in [0..2500] | IsSquare(15*n+1)]; // Bruno Berselli, Nov 23 2012
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Jan 13 2012
STATUS
approved