A112653 - OEIS

A112653

a(n) squared is congruent to a(n) (mod 13).

0, 1, 13, 14, 26, 27, 39, 40, 52, 53, 65, 66, 78, 79, 91, 92, 104, 105, 117, 118, 130, 131, 143, 144, 156, 157, 169, 170, 182, 183, 195, 196, 208, 209, 221, 222, 234, 235, 247, 248, 260, 261, 273, 274, 286, 287, 299, 300, 312, 313, 325, 326, 338, 339, 351 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	COMMENTS	`Numbers that are congruent to {0,1} mod 13. - Philippe Deléham, Oct 17 2001`
	LINKS	`Vincenzo Librandi, Table of n, a(n) for n = 0..1000` `Index entries for linear recurrences with constant coefficients, signature (1,1,-1).`
	FORMULA	`a(n) = Sum_{k>=0} A030308(n,k) * A005029(k-1) with A005029(-1) = 1. - Philippe Deléham, Oct 17 2011` `From Colin Barker, May 14 2012: (Start)` `a(n) = (11(-1+(-1)^n)+26n)/4.` `a(n) = a(n-1) + a(n-2) - a(n-3) for n > 2.` `G.f.: x(1+12x) / ((1-x)^2*(1+x)). (End)`
	EXAMPLE	`a(3) = 14 because 14*14 = 196 = 1 (mod 13) and 14 = 1 (mod 13).`
	MAPLE	`m:= 13; for n from 0 to 300 do if n^2 mod m = n mod m then print(n) fi od;`
	MATHEMATICA	`Select[Range[0, 400], MemberQ[{0, 1}, Mod[#, 13]]&] (* Vincenzo Librandi, May 17 2012 )` `Select[Range[0, 400], Mod[#, 13]==PowerMod[#, 2, 13]&] ( or ) LinearRecurrence[ {1, 1, -1}, {0, 1, 13}, 60] ( Harvey P. Dale, Feb 07 2023 *)`
	PROG	`(Magma) I:=[0, 1, 13]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..70]]; // Vincenzo Librandi, May 17 2012` `(PARI) a(n)=(11(-1+(-1)^n)+26n)/4 \\ Charles R Greathouse IV, Oct 16 2015`
	CROSSREFS	`Cf. A005029, A030308.` `Sequence in context: A296795 A079831 A022803 * A301660 A022103 A224224` `Adjacent sequences: A112650 A112651 A112652 * A112654 A112655 A112656`
	KEYWORD	`easy,nonn`
	AUTHOR	`Jeremy Gardiner, Dec 28 2005`
	STATUS	`approved`