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

Author

Michael Somos, Jan 13 2012

Keywords

• nonn
• easy

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).

Crossrefs

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

Cf. similar sequences listed in A219257.

Programs

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

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 *)

PARI

{a(n) = (15*n^2 + n*[8, 2, 28, 22][n%4 + 1] + 12) \ 16}

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)