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A266957 Numbers m such that 9*m+10 is a square.

%I A266957 #27 Sep 08 2022 08:46:15
%S A266957 -1,6,10,31,39,74,86,135,151,214,234,311,335,426,454,559,591,710,746,
%T A266957 879,919,1066,1110,1271,1319,1494,1546,1735,1791,1994,2054,2271,2335,
%U A266957 2566,2634,2879,2951,3210,3286,3559,3639,3926,4010,4311,4399,4714,4806,5135,5231,5574
%N A266957 Numbers m such that 9*m+10 is a square.
%C A266957 Equivalently, numbers of the form h*(9*h+2)-1, where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ...
%C A266957 Also, integer values of k*(k+2)/9 minus 1.
%H A266957 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A266957 G.f.: x*(-1 + 7*x + 6*x^2 + 7*x^3 - x^4)/((1 + x)^2*(1 - x)^3).
%F A266957 a(n) = a(-n+1) = (18*(n-1)*n + 5*(2*n-1)*(-1)^n - 3)/8.
%F A266957 a(n) = A132355(n) + 1.
%t A266957 Select[Range[-1, 6000], IntegerQ[Sqrt[9 # + 10]] &]
%t A266957 Table[(18 (n - 1) n + 5 (2 n - 1) (-1)^n - 3)/8, {n, 1, 50}]
%o A266957 (Sage) [n for n in (-1..6000) if is_square(9*n+10)]
%o A266957 (Sage) [(18*(n-1)*n+5*(2*n-1)*(-1)^n-3)/8 for n in (1..50)]
%o A266957 (PARI) for(n=-1, 6000, if(issquare(9*n+10), print1(n, ", ")))
%o A266957 (PARI) vector(50, n, n; (18*(n-1)*n+5*(2*n-1)*(-1)^n-3)/8)
%o A266957 (Python) from gmpy2 import is_square
%o A266957 [n for n in range(-1,6000) if is_square(9*n+10)]
%o A266957 (Python) [(18*(n-1)*n+5*(2*n-1)*(-1)**n-3)/8 for n in range(1, 60)]
%o A266957 (Magma) [n: n in [-1..6000] | IsSquare(9*n+10)];
%o A266957 (Magma) [(18*(n-1)*n+5*(2*n-1)*(-1)^n-3)/8: n in [1..50]];
%Y A266957 Cf. A132355.
%Y A266957 Cf. similar sequences listed in A266956.
%Y A266957 Cf. A056020: square roots of 9*a(n)+10.
%K A266957 sign,easy
%O A266957 1,2
%A A266957 _Bruno Berselli_, Jan 07 2016