A204221 Integers of the form (N^2 - 1) / 15.

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

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). - Peter Bala, Feb 22 2021.

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

John Greene and James A. Sellers, Extending recent parity results of Nyirenda and Mugwangwavari for partitions with initial repetitions, Integers (2025), Vol. 25, Art. No. A32. See p. 8.

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*(1+x^2)^2*(1-x)^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

From Peter Bala, Dec 17 2024: (Start)

a(n) is quasi-polynomial in n: for n >= 0,

a(4*n+1) = 15*n^2 + 8*n + 1; a(4*n+2) = 15*n^2 + 22*n + 8;

a(4*n+3) = 15*n^2 + 28*n + 13; a(4*n+4) = 15*n^2 + 32*n + 17.

For 1 <= k <= 4, a(4*n+k) = (N_k(n)^2 - 1)/15, where N_1(n) = 15*n + 4, N_2(n) = 15*n + 11, N_3(n) = 15*n + 14 and N_4(n) = 15*n + 16. (End)

Maple

A204221 := proc(q) local n;
for n from 0 to q do
 if type(sqrt(15*n+1), integer) then print(n);
fi; od; end:
A204221(2500); # Peter Bala, Dec 18 2024

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. A204220, A204542 (square roots of 15*a(n)+1), A379210.

Cf. similar sequences listed in A219257.

Sequence in context: A070113 A178968 A006613 * A348277 A337308 A014134

Adjacent sequences: A204218 A204219 A204220 * A204222 A204223 A204224

Keyword

nonn,easy

Author

Michael Somos, Jan 13 2012

Status

approved