A268539 Numbers k such that 48*k+25 is a perfect square.

0, 2, 3, 7, 17, 25, 28, 38, 58, 72, 77, 93, 123, 143, 150, 172, 212, 238, 247, 275, 325, 357, 368, 402, 462, 500, 513, 553, 623, 667, 682, 728, 808, 858, 875, 927, 1017, 1073, 1092, 1150, 1250, 1312, 1333, 1397, 1507, 1575, 1598, 1668, 1788, 1862, 1887, 1963, 2093, 2173

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

Links

Zak Seidov, Table of n, a(n) for n = 1..10000

M. Merca, The bisectional pentagonal number theorem, Journal of Number Theory, Volume 157, December 2015, Pages 223-232.

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)

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

Cf. A154293.

Subsequence of A011865.

Sequence in context: A245590 A110480 A338175 * A083822 A349665 A030086

Adjacent sequences: A268536 A268537 A268538 * A268540 A268541 A268542

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 24 2016

Extensions

More terms from Michel Marcus, Feb 25 2016

Status

approved