A151971 - OEIS

%I #37 Sep 08 2022 08:45:38

%S 0,1,7,15,21,22,28,36,42,43,49,57,63,64,70,78,84,85,91,99,105,106,112,

%T 120,126,127,133,141,147,148,154,162,168,169,175,183,189,190,196,204,

%U 210,211,217,225,231,232,238,246,252,253,259,267,273,274,280,288,294,295,301,309

%N Numbers n such that n^2 - n is divisible by 21.

%C Equivalently, numbers that are congruent to {0, 1, 7, 15} mod 21. - _Bruno Berselli_, Aug 06 2012

%H Vincenzo Librandi, <a href="/A151971/b151971.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

%F From _Bruno Berselli_, Aug 06 2012: (Start)

%F G.f.: x^2*(1+6*x+8*x^2+6*x^3)/((1+x)*(1-x)^2*(1+x^2)).

%F a(n) = (42*n +14*i^((n-1)*n) -3*(-1)^n -3)/8 -7, where i=sqrt(-1). (End)

%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - _Wesley Ivan Hurt_, Jun 07 2016

%F E.g.f.: (24 + (21*x - 31)*cosh(x) + 7*(sin(x) + cos(x) + (3*x - 4)*sinh(x)))/4. - _Ilya Gutkovskiy_, Jun 07 2016

%p A151971:=n->(42*n+14*I^((n-1)*n)-3*I^(2*n)-3)/8-7: seq(A151971(n), n=1..100); # _Wesley Ivan Hurt_, Jun 07 2016

%t Select[Range[0,400], Divisible[#^2-#,21]&] (* _Harvey P. Dale_, Jun 04 2012 *)

%o (Magma) [n: n in [0..309] | IsZero((n^2-n) mod 21)]; // _Bruno Berselli_, Aug 06 2012

%o (Maxima) makelist((42*n+14*%i^((n-1)*n)-3*(-1)^n-3)/8-7, n, 1, 60); /* _Bruno Berselli_, Aug 06 2012 */

%Y For m^2 == m (mod n), see: n=2: A001477; n=3: A032766; n=4: A042948; n=5: A008851; n=6: A032766; n=7: A047274; n=8: A047393; n=9: A090570; n=10: A008851; n=11: A112651; n=12: A112652; n=13:A112653; n=14: A047274; n=15: A151972; n=16: A151977; n=17: A151978; n=18: A090570; n=19: A151979; n=20: A151980; n=21: A151971; n=22: A112651; n=24: A151973; n=26: A112653; n=30: A151972; n=32: A151983; n=34: A151978; n=38: A151979; n=42: A151971; n=48: A151981; n=64: A151984.

%Y Cf. A215202.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_, Aug 23 2009