A269819 - OEIS

%I #30 Sep 08 2022 08:46:16

%S 5,11,13,19,29,35,37,43,53,59,61,67,77,83,85,91,101,107,109,115,125,

%T 131,133,139,149,155,157,163,173,179,181,187,197,203,205,211,221,227,

%U 229,235,245,251,253,259,269,275,277,283,293,299,301,307,317

%N Numbers that are congruent to {5, 11, 13, 19} mod 24.

%C No terms are multiples of 3.

%C Numbers such that (j+5)*(j-5)/48 are positive integers. Equivalent to positive integers (m+3)*(m-2)/12, with m == {2,5,6,9} mod 12 (observation made in A268539 by _M. F. Hasler_, Mar 02 2016).

%H Colin Barker, <a href="/A269819/b269819.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).

%F a(n) = a(n-4) + 24.

%F a(n) = sqrt(48*A268539(n) + 25).

%F G.f.: x*(1+x)*(5-4*x+5*x^2) / ((1-x)^2*(1+x^2)). - _Colin Barker_, Mar 06 2016

%F From _Wesley Ivan Hurt_, Jun 04 2016: (Start)

%F a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.

%F a(n) = 6*n-3-(1-i)*i^(-n)-(1+i)*i^n for i=sqrt(-1). (End)

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (2-sqrt(2))*Pi/12. - _Amiram Eldar_, Dec 31 2021

%p A269819:=n->6*n-3-(1-I)*I^(-n)-(1+I)*I^n: seq(A269819(n), n=1..80); # _Wesley Ivan Hurt_, Jun 04 2016

%t Table[24 n + {5, 11, 13, 19}, {n, 0, 12}] // Flatten (* _Michael De Vlieger_, Mar 07 2016 *)

%t Table[6n-3-(1-I)*I^(-n)-(1+I)*I^n, {n, 80}] (* _Wesley Ivan Hurt_, Jun 04 2016 *)

%t LinearRecurrence[{2,-2,2,-1},{5,11,13,19},60] (* _Harvey P. Dale_, Nov 17 2017 *)

%o (Magma) I:=[5,11,13,19]; [n le 4 select I[n] else Self(n-4) + 24 : n in [1..60]]; // _Vincenzo Librandi_, Mar 06 2016

%o (PARI) Vec(x*(1+x)*(5-4*x+5*x^2)/((1-x)^2*(1+x^2)) + O(x^100)) \\ _Colin Barker_, Mar 06 2016

%o (Magma) [n : n in [0..400] | n mod 24 in [5, 11, 13, 19]]; // _Wesley Ivan Hurt_, Jun 04 2016

%Y Subsequence of A001651.

%Y Cf. A268539.

%K nonn,easy

%O 1,1

%A _Bob Selcoe_, Mar 05 2016

%E Incorrect term 252 replaced by two missing terms 251 and 253 by _Colin Barker_, Mar 06 2016