A047329 Numbers that are congruent to {1, 3, 5, 6} mod 7.

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 26, 27, 29, 31, 33, 34, 36, 38, 40, 41, 43, 45, 47, 48, 50, 52, 54, 55, 57, 59, 61, 62, 64, 66, 68, 69, 71, 73, 75, 76, 78, 80, 82, 83, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 103, 104, 106, 108, 110, 111

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• Robert Fludd, Utriusque Cosmi ... Historia, Oppenheim, 1617-1619.

Links

• Robert Fludd, Page 158 of "Utriusque Cosmi" in Beinecke Rare Book and Manuscript Library Photonegatives Collection.
• Robert Fludd, Larger version of the same image
• Robert Fludd, Utriusque Cosmi, Maioris scilicet et Minoris, metaphysica, physica, atque technica Historia, available as ZIP or PDF download.
• Wikipedia, Robert Fludd
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A047280, A047383, A082977.

Programs

Haskell

a047329 n = a047329_list !! (n-1)
a047329_list = [1, 3, 5, 6] ++ map (+ 7) a047329_list
-- Reinhard Zumkeller, Jan 07 2014

Magma

[Floor((7*n-1)/4) : n in [1..100]]; // Wesley Ivan Hurt, May 21 2016

Maple

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

Mathematica

Table[Floor[(7*n-1)/4], {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
#+{1,3,5,6}&/@(7*Range[0,20])//Flatten (* Harvey P. Dale, Jan 07 2021 *)

Formula

a(n) = floor((7n-1)/4). - Gary Detlefs, Mar 07 2010

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

From Wesley Ivan Hurt, May 21 2016: (Start)

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

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

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

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

Extensions

Fludd reference from Brendan McKay, May 27 2003

More terms from Wesley Ivan Hurt, May 21 2016