A266958 - OEIS

%I #31 Sep 08 2022 08:46:15

%S -1,4,12,27,43,68,92,127,159,204,244,299,347,412,468,543,607,692,764,

%T 859,939,1044,1132,1247,1343,1468,1572,1707,1819,1964,2084,2239,2367,

%U 2532,2668,2843,2987,3172,3324,3519,3679,3884,4052,4267,4443,4668,4852,5087,5279,5524

%N Numbers m such that 9*m+13 is a square.

%C Equivalently, numbers of the form h*(9*h+4)-1, where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ...

%C Also, integer values of k*(k+4)/9 minus 1.

%C Is A063289 (after -1) the list of the square roots of 9*a(n)+13?

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F G.f.: x*(-1 + 5*x + 10*x^2 + 5*x^3 - x^4)/((1 + x)^2*(1 - x)^3).

%F a(n) = a(-n+1) = (18*(n-1)*n + (2*n-1)*(-1)^n - 7)/8.

%F a(n) = A185039(n) + 1.

%t Select[Range[-1, 6000], IntegerQ[Sqrt[9 # + 13]] &]

%t Table[(18 (n-1) n + (2 n - 1) (-1)^n - 7)/8, {n, 1, 50}]

%t LinearRecurrence[{1,2,-2,-1,1},{-1,4,12,27,43},50] (* _Harvey P. Dale_, Jan 20 2020 *)

%o (Sage) [n for n in range(-1,6000) if is_square(9*n+13)]

%o (Sage) [(18*(n-1)*n+(2*n-1)*(-1)^n-7)/8 for n in range(1,50)]

%o (PARI) for(n=-1, 6000, if(issquare(9*n+13), print1(n, ", ")))

%o (PARI) vector(50, n, n; (18*(n-1)*n+(2*n-1)*(-1)^n-7)/8)

%o (Python) from gmpy2 import is_square

%o [n for n in range(-1,6000) if is_square(9*n+13)]

%o (Python) [(18*(n-1)*n+(2*n-1)*(-1)**n-7)/8 for n in range(1,60)]

%o (Magma) [n: n in [-1..6000] | IsSquare(9*n+13)];

%o (Magma) [(18*(n-1)*n+(2*n-1)*(-1)^n-7)/8: n in [1..50]];

%Y Cf. A185039.

%Y Cf. similar sequences listed in A266956.

%K sign,easy

%O 1,2

%A _Bruno Berselli_, Jan 07 2016