OFFSET
1,2
COMMENTS
Equivalently, integers of the form (h+5)*(h-5)/48, where h must be odd, h = 2*m+1, thus also integers of the form (m+3)*(m-2)/12, with m = 2, 5, 6, 9, 14, 17, 18, ... = {2, 5, 6, 9} + 12 N. - M. F. Hasler, Mar 02 2016
The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(8*n))*(1 + q^(8*n-1))*(1 + q^(8*n-7))/(1 + q^n) = Sum_{n >= 0} q^(2*n*(n+1)) * Product_{k >= 2*n+2} 1 - q^k = 1 - q^2 - q^3 + q^7 + q^17 - q^25 - q^28 + + - - ... (by the quintuple product identity and Mc Laughlin et al., S.38, p 16). - Peter Bala, Dec 30 2024
LINKS
Zak Seidov, Table of n, a(n) for n = 1..10000
J. Mc Laughlin, A. V. Sills and P. Zimmer, Rogers-Ramanujan-Slater Type Identities, Electronic J. Combinatorics, DS15, 1-59, May 31, 2008.
Mircea Merca, The bisectional pentagonal number theorem, Journal of Number Theory, Volume 157, December 2015, Pages 223-232, see Corollary 4.4.
Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).
FORMULA
For n>25, a(n) = 3*( a(n-8)-a(n-16) ) + a(n-24). - Zak Seidov, Feb 28 2016
From Robert Israel, Feb 29 2016: (Start)
Let L = [5, 11, 13, 19, 29, 35, 37, 43].
Then a(i + 8*j) = ( (L(i) + 48*j)^2 - 25 )/48 for i = 1..8, j >= 0. (End)
From Bruno Berselli, Feb 29 2016: (Start)
G.f.: x^2*(2 - 3*x + 8*x^2 - 3*x^3 + 2*x^4)/((1 - x)^3*(1 + x^2)^2).
a(n) = a(-n+1) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7) for n>6.
a(n) = (3*(n-1)*n + (2*n-1)*(-1)^((n-2)*(n-1)/2) - 1)/4. Therefore:
a(4*k) = k*(12*k -5),
a(4*k+1) = k*(12*k +5),
a(4*k+2) = k*(12*k+11)+2 = (3*k+2)*(4*k+1),
a(4*k+3) = k*(12*k+13)+3 = (3*k+1)*(4*k+3).
From the previous formulas follows that 2, 3, 7 and 17 are the only primes of the sequence. (End)
Sum_{n>=2} 1/a(n) = 12/25 + (4/sqrt(3)-1)*Pi/5. - Amiram Eldar, Jul 30 2024
MAPLE
L := [5, 11, 13, 19, 29, 35, 37, 43]:
seq(seq(((L[i]+48*j)^2-25)/48, i=1..8), j=0..10); # Robert Israel, Feb 29 2016
MATHEMATICA
Select[Range[0, 2500], IntegerQ[Sqrt[48 # + 25]] &] (* Vincenzo Librandi, Feb 25 2016 *)
Table[(3 (n - 1) n + (2 n - 1) (-1)^((n - 2) (n - 1)/2) - 1)/4, {n, 1, 60}] (* Bruno Berselli, Feb 29 2016 *)
LinearRecurrence[{3, -5, 7, -7, 5, -3, 1}, {0, 2, 3, 7, 17, 25, 28}, 48] (* Robert G. Wilson v, Mar 05 2016 *)
CoefficientList[ Series[ x*(2 - 3x + 8x^2 - 3x^3 + 2x^4)/((1 - x)^3*(1 + x^2)^2), {x, 0, 47}], x] (* Robert G. Wilson v, Mar 05 2016 *)
PROG
(PARI) isok(n) = issquare(48*n+25); \\ Michel Marcus, Feb 25 2016
(Magma) [n: n in [0..2200] | IsSquare(48*n+25)]; // Vincenzo Librandi, Feb 25 2016
(Sage) [n for n in (0..2200) if is_square(48*n+25)] # Bruno Berselli, Feb 29 2016
(PARI) A268539(n)={my(m=n\4*12+[-3, 2, 5, 6][n%4+1]); (3+m)*(m-2)/12} \\ M. F. Hasler, Mar 03 2016
(Python) from gmpy2 import is_square
[k for k in range(2200) if is_square(48*k+25)] # Bruno Berselli, Dec 05 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Feb 24 2016
EXTENSIONS
More terms from Michel Marcus, Feb 25 2016
STATUS
approved