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A212161
Numbers congruent to 6 or 10 mod 17.
5
6, 10, 23, 27, 40, 44, 57, 61, 74, 78, 91, 95, 108, 112, 125, 129, 142, 146, 159, 163, 176, 180, 193, 197, 210, 214, 227, 231, 244, 248, 261, 265, 278, 282, 295, 299, 312, 316, 329, 333, 346, 350
OFFSET
0,1
COMMENTS
A001844(N) = N^2 + (N+1)^2 = 4*A000217(N) + 1 is divisible by 17 if and only if N = a(n), n >= 0. For the proof it suffices to show that only N=6 and N=10 from {0,1,...,16} satisfy A001844(N) == 0 (mod 17). Note that only primes of the form p = 4*k+1 (A002144) can be divisors of A001844 (see a Wolfdieter Lang comment there giving the reference). Note also that if N^2 + (N+1)^2 == 0 (mod p), with any prime p (necessarily from A002144), then also p-1-N satisfies this congruence. This explains why 10 = 17-1-6 is the (incongruent) companion of 6.
Partial sums of the sequence 6,4,13,4,13,4,13,4,13,4,13,... (see the o.g.f., and subtract 6 to see the remaining 4, 13=17-4 periodicity).
LINKS
FORMULA
Bisection: a(2*n) = 17*n + 6, a(2*n+1) = 17*n + 10, n >= 0.
O.g.f.: (6 + 4*x + 7*x^2)/((1-x)*(1-x^2)).
E.g.f.: ((34*x + 15)*exp(x) + 9*exp(-x))/4. - David Lovler, Aug 09 2022
EXAMPLE
Divisibility of A001844 by 17:
n=0: A001844(6) = 85 = 5*17 == 0 (mod 17).
n=2: A001844(23) = 1105 = 5*13*17 == 0 (mod 17).
However, 8^2 + 9^2 = 145 == 9 (mod 17) is not divisible by 17 because 8 is not a term of the present sequence.
MATHEMATICA
Table[1/4*(34*n+9*(-1)^n+15), {n, 0, 60}] (* Vincenzo Librandi, May 24 2012 *)
PROG
(Magma) [1/4*(34*n+9*(-1)^n+15): n in [0..60]]; // Vincenzo Librandi, May 24 2012
(PARI) a(n) = (34*n + 9*(-1)^n + 15)/4 \\ David Lovler, Aug 09 2022
CROSSREFS
Cf. A047219 (p=5), A212160 (p=13).
Sequence in context: A166305 A047651 A079426 * A132994 A199885 A024499
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, May 09 2012
STATUS
approved