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Numbers that are congruent to {1, 3, 5, 6} mod 7.
2

%I #27 Sep 08 2022 08:44:57

%S 1,3,5,6,8,10,12,13,15,17,19,20,22,24,26,27,29,31,33,34,36,38,40,41,

%T 43,45,47,48,50,52,54,55,57,59,61,62,64,66,68,69,71,73,75,76,78,80,82,

%U 83,85,87,89,90,92,94,96,97,99,101,103,104,106,108,110,111

%N Numbers that are congruent to {1, 3, 5, 6} mod 7.

%D Robert Fludd, Utriusque Cosmi ... Historia, Oppenheim, 1617-1619.

%H Robert Fludd, <a href="http://beinecke.library.yale.edu/dl_crosscollex/photoneg/printoneITEM.asp?pid=39002036242312&amp;iid=3624231">Page 158 of "Utriusque Cosmi"</a> in Beinecke Rare Book and Manuscript Library Photonegatives Collection.

%H Robert Fludd, <a href="http://highway55.library.yale.edu/PHOTONEGIMG/zoom/Z362/z3624231.jpg">Larger version of the same image</a>

%H Robert Fludd, <a href="http://billheidrick.com/Orpd/RFludd/">Utriusque Cosmi, Maioris scilicet et Minoris, metaphysica, physica, atque technica Historia</a>, available as ZIP or PDF download.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Robert_Fludd">Robert Fludd</a>

%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) = floor((7n-1)/4). - _Gary Detlefs_, Mar 07 2010

%F G.f.: (x*(1+2*x+2*x^2+x^3+x^4)) / ((1+x)*(x^2+1)*(x-1)^2). - _R. J. Mathar_, Oct 08 2011

%F From _Wesley Ivan Hurt_, May 21 2016: (Start)

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

%F a(n) = (14n-5-i^(2n)-(1+i)*i^(-n)-(1-i)*i^n)/8 where i=sqrt(-1).

%F a(2n) = A047280(n), a(2n-1) = A047383(n). (End)

%F E.g.f.: (4 - sin(x) - cos(x) + (7*x - 2)*sinh(x) + (7*x - 3)*cosh(x))/4. - _Ilya Gutkovskiy_, May 21 2016

%p A047329:=n->floor((7*n-1)/4): seq(A047329(n), n=1..100); # _Wesley Ivan Hurt_, May 21 2016

%t Table[Floor[(7*n-1)/4], {n, 80}] (* _Wesley Ivan Hurt_, May 21 2016 *)

%t #+{1,3,5,6}&/@(7*Range[0,20])//Flatten (* _Harvey P. Dale_, Jan 07 2021 *)

%o (Haskell)

%o a047329 n = a047329_list !! (n-1)

%o a047329_list = [1, 3, 5, 6] ++ map (+ 7) a047329_list

%o -- _Reinhard Zumkeller_, Jan 07 2014

%o (Magma) [Floor((7*n-1)/4) : n in [1..100]]; // _Wesley Ivan Hurt_, May 21 2016

%Y Cf. A047280, A047383, A082977.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E Fludd reference from _Brendan McKay_, May 27 2003

%E More terms from _Wesley Ivan Hurt_, May 21 2016