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A268539 Numbers k such that 48*k+25 is a perfect square. 2
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 (list; graph; refs; listen; history; text; internal format)
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 xrange(2200) if is_square(48*k+25)] # Bruno Berselli, Dec 05 2016

CROSSREFS

Cf. A154293.

Subsequence of A011865.

Sequence in context: A289379 A245590 A110480 * A083822 A030086 A078721

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

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Last modified November 19 08:44 EST 2019. Contains 329318 sequences. (Running on oeis4.)