A047203 Numbers that are congruent to {0, 2, 3, 4} mod 5.
0, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 72, 73, 74, 75, 77, 78, 79, 80, 82, 83, 84, 85, 87, 88, 89
Offset: 1
Links
- Melvyn B. Nathanson, On the fractional parts of roots of positive real numbers, Amer. Math. Monthly, Vol. 120, No. 5 (2013), pp. 409-429 [see p. 417].
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Programs
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Magma
[Floor((5*n-2)/4) : n in [1..100]]; // Wesley Ivan Hurt, May 14 2016
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Maple
seq(floor(5*n-2)/4), n=1..72); # Gary Detlefs, Mar 06 2010 seq(floor((15*n-5)/12), n=1..72); # Gary Detlefs, Mar 07 2010
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Mathematica
Flatten[Table[5*n + {0, 2, 3, 4}, {n, 0, 20}]] (* T. D. Noe, Nov 12 2013 *) -
PARI
a(n)=(5*n-2)\4 \\ Charles R Greathouse IV, Jun 11 2015
Formula
A027445(a(n)) mod 10 = 0. - Reinhard Zumkeller, Oct 23 2006
a(n) = floor((5n-2)/4). - Gary Detlefs, Mar 06 2010
a(n) = floor((15n-5)/12). - Gary Detlefs, Mar 07 2010
G.f.: x^2*(2+x+x^2+x^3)/((1+x)*(1+x^2)*(x-1)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 14 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (10*n-7+(-1)^n+2*(-1)^((2*n+3+(-1)^n)/4))/8.
a(n) = A047207(n+1) - 1.
a(n+2) = n + 2 + A002265(n) for n>0.
a(n+3)-a(n+2) = A177704(n) for n>0.
a(1-n) = - A001068(n). (End)
Sum_{n>=2} (-1)^n/a(n) = log(5)/4 + sqrt(5)*log(phi)/10 - sqrt(5-2*sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021
Extensions
More terms from Reinhard Zumkeller, Oct 23 2006
Comments