A269819 Numbers that are congruent to {5, 11, 13, 19} mod 24.

5, 11, 13, 19, 29, 35, 37, 43, 53, 59, 61, 67, 77, 83, 85, 91, 101, 107, 109, 115, 125, 131, 133, 139, 149, 155, 157, 163, 173, 179, 181, 187, 197, 203, 205, 211, 221, 227, 229, 235, 245, 251, 253, 259, 269, 275, 277, 283, 293, 299, 301, 307, 317

Offset: 1

Author

Bob Selcoe, Mar 05 2016

Keywords

• nonn
• easy

Comments

No terms are multiples of 3.

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).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Subsequence of A001651.

Cf. A268539.

Programs

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

Magma

[n : n in [0..400] | n mod 24 in [5, 11, 13, 19]]; // Wesley Ivan Hurt, Jun 04 2016

Maple

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

Mathematica

Table[24 n + {5, 11, 13, 19}, {n, 0, 12}] // Flatten (* Michael De Vlieger, Mar 07 2016 *)
Table[6n-3-(1-I)*I^(-n)-(1+I)*I^n, {n, 80}] (* Wesley Ivan Hurt, Jun 04 2016 *)
LinearRecurrence[{2,-2,2,-1},{5,11,13,19},60] (* Harvey P. Dale, Nov 17 2017 *)

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

Formula

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

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

G.f.: x*(1+x)*(5-4*x+5*x^2) / ((1-x)^2*(1+x^2)). - Colin Barker, Mar 06 2016

From Wesley Ivan Hurt, Jun 04 2016: (Start)

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

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

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

Extensions

Incorrect term 252 replaced by two missing terms 251 and 253 by Colin Barker, Mar 06 2016