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

%I A268539 #80 May 11 2025 01:04:53
%S A268539 0,2,3,7,17,25,28,38,58,72,77,93,123,143,150,172,212,238,247,275,325,
%T A268539 357,368,402,462,500,513,553,623,667,682,728,808,858,875,927,1017,
%U A268539 1073,1092,1150,1250,1312,1333,1397,1507,1575,1598,1668,1788,1862,1887,1963,2093,2173
%N A268539 Numbers k such that 48*k + 25 is a perfect square.
%C A268539 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
%C A268539 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
%C A268539 Conjecture: the sequence terms are also the exponents in the expansion of Sum_{n >= 0} q^n/(Product_{k = 1..2*n+1} 1 + q^k) = 1 + q^2 - q^3 - q^7 + q^17 + q^25 - - + + .... - _Peter Bala_, Jan 15 2025
%H A268539 Zak Seidov, <a href="/A268539/b268539.txt">Table of n, a(n) for n = 1..10000</a>
%H A268539 J. Mc Laughlin, A. V. Sills and P. Zimmer, <a href="https://doi.org/10.37236/36">Rogers-Ramanujan-Slater Type Identities</a>, Electronic J. Combinatorics, DS15, 1-59, May 31, 2008.
%H A268539 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.05.008">The bisectional pentagonal number theorem</a>, Journal of Number Theory, Volume 157, December 2015, Pages 223-232, see Corollary 4.4.
%H A268539 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-5,7,-7,5,-3,1).
%F A268539 For n>25, a(n) = 3*( a(n-8)-a(n-16) ) + a(n-24). - _Zak Seidov_, Feb 28 2016
%F A268539 From _Robert Israel_, Feb 29 2016: (Start)
%F A268539 Let L = [5, 11, 13, 19, 29, 35, 37, 43].
%F A268539 Then a(i + 8*j) = ( (L(i) + 48*j)^2 - 25 )/48 for i = 1..8, j >= 0. (End)
%F A268539 From _Bruno Berselli_, Feb 29 2016: (Start)
%F A268539 G.f.: x^2*(2 - 3*x + 8*x^2 - 3*x^3 + 2*x^4)/((1 - x)^3*(1 + x^2)^2).
%F A268539 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.
%F A268539 a(n) = (3*(n-1)*n + (2*n-1)*(-1)^((n-2)*(n-1)/2) - 1)/4. Therefore:
%F A268539 a(4*k)   = k*(12*k -5),
%F A268539 a(4*k+1) = k*(12*k +5),
%F A268539 a(4*k+2) = k*(12*k+11)+2 = (3*k+2)*(4*k+1),
%F A268539 a(4*k+3) = k*(12*k+13)+3 = (3*k+1)*(4*k+3).
%F A268539 From the previous formulas follows that 2, 3, 7 and 17 are the only primes of the sequence. (End)
%F A268539 Sum_{n>=2} 1/a(n) = 12/25 + (4/sqrt(3)-1)*Pi/5. - _Amiram Eldar_, Jul 30 2024
%p A268539 L := [5, 11, 13, 19, 29, 35, 37, 43]:
%p A268539 seq(seq(((L[i]+48*j)^2-25)/48, i=1..8), j=0..10); # _Robert Israel_, Feb 29 2016
%t A268539 Select[Range[0, 2500], IntegerQ[Sqrt[48 # + 25]] &] (* _Vincenzo Librandi_, Feb 25 2016 *)
%t A268539 Table[(3 (n - 1) n + (2 n - 1) (-1)^((n - 2) (n - 1)/2) - 1)/4, {n, 1, 60}] (* _Bruno Berselli_, Feb 29 2016 *)
%t A268539 LinearRecurrence[{3, -5, 7, -7, 5, -3, 1}, {0, 2, 3, 7, 17, 25, 28}, 48] (* _Robert G. Wilson v_, Mar 05 2016 *)
%t A268539 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 *)
%o A268539 (PARI) isok(n) = issquare(48*n+25); \\ _Michel Marcus_, Feb 25 2016
%o A268539 (Magma) [n: n in [0..2200] | IsSquare(48*n+25)]; // _Vincenzo Librandi_, Feb 25 2016
%o A268539 (Sage) [n for n in (0..2200) if is_square(48*n+25)] # _Bruno Berselli_, Feb 29 2016
%o A268539 (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
%o A268539 (Python) from gmpy2 import is_square
%o A268539 [k for k in range(2200) if is_square(48*k+25)] # _Bruno Berselli_, Dec 05 2016
%Y A268539 Cf. A078405, A154293.
%Y A268539 Subsequence of A011865.
%K A268539 nonn,easy
%O A268539 1,2
%A A268539 _N. J. A. Sloane_, Feb 24 2016
%E A268539 More terms from _Michel Marcus_, Feb 25 2016