|
| |
|
|
A204221
|
|
Integers of the form (n^2 - 1) / 15.
|
|
6
|
|
|
|
0, 1, 8, 13, 17, 24, 45, 56, 64, 77, 112, 129, 141, 160, 209, 232, 248, 273, 336, 365, 385, 416, 493, 528, 552, 589, 680, 721, 749, 792, 897, 944, 976, 1025, 1144, 1197, 1233, 1288, 1421, 1480, 1520, 1581, 1728, 1793, 1837, 1904, 2065, 2136, 2184, 2257, 2432
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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
|
Cf. similar sequences listed in A219257.
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
