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Numbers that are congruent to {0, 1, 2, 3} mod 7.
2

%I #34 Sep 08 2022 08:44:57

%S 0,1,2,3,7,8,9,10,14,15,16,17,21,22,23,24,28,29,30,31,35,36,37,38,42,

%T 43,44,45,49,50,51,52,56,57,58,59,63,64,65,66,70,71,72,73,77,78,79,80,

%U 84,85,86,87,91,92,93,94,98,99,100,101,105,106,107,108,112

%N Numbers that are congruent to {0, 1, 2, 3} mod 7.

%C Nonnegative m for which floor(2*m/7) = 2*floor(m/7). [_Bruno Berselli_, Dec 03 2015]

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

%F a(n) = 7*floor(n/4) + (n mod 4), with offset 0 and a(0) = 0. - _Gary Detlefs_, Mar 09 2010

%F G.f.: x^2*(1+x+x^2+4*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - _R. J. Mathar_, Dec 04 2011

%F From _Wesley Ivan Hurt_, May 23 2016: (Start)

%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.

%F a(n) = (14*n-23-3*i^(2*n)-(3-3*i)*i^(-n)-(3+3*i)*i^n)/8, where i=sqrt(-1).

%F a(2k) = A047356(k), a(2k-1) = A047352(k). (End)

%F E.g.f.: (16 + 3*(sin(x) - cos(x)) + (7*x - 10)*sinh(x) + (7*x - 13)*cosh(x))/4. - _Ilya Gutkovskiy_, May 24 2016

%p A047361:=n->(14*n-23-3*I^(2*n)-(3-3*I)*I^(-n)-(3+3*I)*I^n)/8: seq(A047361(n), n=1..100); # _Wesley Ivan Hurt_, May 23 2016

%t Flatten[#+{0,1,2,3}&/@(7*Range[0,20])] (* _Harvey P. Dale_, Jan 17 2013 *)

%o (PARI) concat(0, Vec(x^2*(1+x+x^2+4*x^3)/((1+x)*(x^2+1)*(x-1)^2) + O(x^100))) \\ _Altug Alkan_, Dec 09 2015

%o (Magma) [n : n in [0..150] | n mod 7 in [0..3]]; // _Wesley Ivan Hurt_, May 23 2016

%Y Cf. A047352, A047356.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_

%E More terms from _Wesley Ivan Hurt_, May 23 2016