A102899 -id:A102899 - cp's OEIS frontend

A353314 If n is of the form 3k, then a(n) = n, and if n is of the form 3k+r, with r = 1 or 2, then a(n) = 5*k + 3 + r.

0, 4, 5, 3, 9, 10, 6, 14, 15, 9, 19, 20, 12, 24, 25, 15, 29, 30, 18, 34, 35, 21, 39, 40, 24, 44, 45, 27, 49, 50, 30, 54, 55, 33, 59, 60, 36, 64, 65, 39, 69, 70, 42, 74, 75, 45, 79, 80, 48, 84, 85, 51, 89, 90, 54, 94, 95, 57, 99, 100, 60, 104, 105, 63, 109, 110, 66, 114, 115, 69, 119, 120, 72, 124, 125, 75, 129, 130

Offset: 0

Antti Karttunen, Apr 14 2022

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Nicholas Drozd, A Busy Beaver Champion Derived from Scratch
Nicholas Drozd, Feedback to Doron Zeilberger's opinion #155, Jan. 4, 2022.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A353313 (variant), A349876 (the first multiple of 3 reached when iterating this sequence), A349877 (number of iterations to reach the first multiple of 3), A353327 (A102899).

Array[If[#2 == 0, #1, 5 #1 + 3 + #2 & @@ QuotientRemainder[#1, 3]] & @@ {#, Mod[#, 3]} &, 78, 0] (* Michael De Vlieger, Apr 14 2022 *)

A353314(n) = { my(r=(n%3)); if(!r,n,((5*((n-r)/3)) + r + 3)); };

a(n) = n + A353327(n) = n + A102899(3+n).
From Chai Wah Wu, Jul 27 2022: (Start)
a(n) = 2*a(n-3) - a(n-6) for n > 5.
G.f.: x*(x^3 + 3*x^2 + 5*x + 4)/(x^6 - 2*x^3 + 1). (End)

A120691 First differences of coefficients in the continued fraction for e.

Original entry on oeis.org

2, -1, 1, -1, 0, 3, -3, 0, 5, -5, 0, 7, -7, 0, 9, -9, 0, 11, -11, 0, 13, -13, 0, 15, -15, 0, 17, -17, 0, 19, -19, 0, 21, -21, 0, 23, -23, 0, 25, -25, 0, 27, -27, 0, 29, -29, 0, 31, -31, 0, 33, -33, 0, 35, -35, 0, 37, -37, 0, 39, -39

Offset: 0

Paul Barry, Jun 27 2006

First differences of A003417.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,-1,1,1,1).

Cf. A003417, A102899.

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1-x)*(2+x+2*x^2-3*x^3-x^4+x^6)/(1-x^3)^2 )); // G. C. Greubel, Dec 28 2022

Join[{2},Differences[ContinuedFraction[E,120]]] (* or *) LinearRecurrence[{-1,-1,1,1,1},{2,-1,1,-1,0,3,-3},120] (* Harvey P. Dale, Jun 08 2016 *)

A120691(n)={n<2 && return(2-3*n); n=divrem(n-1,3); if(n[2],-(1+n[1]*2)*(-1)^n[2])} \\ - M. F. Hasler, May 01 2013

def b(n):
    if (n%3==1): return 0
    elif (n%3==2): return (2*n-1)/3
    else: return (3-2*n)/3
def A120691(n): return b(n) + (-1)^n*int(n<2)
[A120691(n) for n in range(71)] # G. C. Greubel, Dec 28 2022

G.f.: (1-x)*(2+x+2*x^2-3*x^3-x^4+x^6)/(1-2*x^3+x^6).
a(n) = 2*C(0,n) -C(1,n) +2*sin(2*Pi*(n-1)/3)*floor((2*n-1)/3)/sqrt(3). [Sign corrected by M. F. Hasler, May 01 2013]
a(0)=2, a(1)=-1, for n>0: a(3*n-1) = 2*n-1, a(3*n) = 1-2*n, a(3*n+1) = 0. - M. F. Hasler, May 01 2013
a(n) = - a(n-1) - a(n-2) + a(n-3) + a(n-4) + a(n-5) for n > 6. - Chai Wah Wu, Jul 27 2022
a(n) = 0 if n mod 3 = 1, a(n) = (2*n-1)/3 if n mod 3 = 2, a(n) = (3-2*n)/3 otherwise, with a(0) = 2, and a(1) = -1. - G. C. Greubel, Dec 28 2022

A353327 If n is a multiple of 3, then a(n) = 0, and if n is of the form 3k+r, with r = 1 or 2, then a(n) = 2*k + 3.

Original entry on oeis.org

0, 3, 3, 0, 5, 5, 0, 7, 7, 0, 9, 9, 0, 11, 11, 0, 13, 13, 0, 15, 15, 0, 17, 17, 0, 19, 19, 0, 21, 21, 0, 23, 23, 0, 25, 25, 0, 27, 27, 0, 29, 29, 0, 31, 31, 0, 33, 33, 0, 35, 35, 0, 37, 37, 0, 39, 39, 0, 41, 41, 0, 43, 43, 0, 45, 45, 0, 47, 47, 0, 49, 49, 0, 51, 51, 0, 53, 53, 0, 55, 55, 0, 57, 57, 0, 59, 59, 0

Offset: 0

Antti Karttunen, Apr 14 2022

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A010872, A102899, A120691, A353314.

A353327(n) = { my(r=(n%3)); if(!r,0,((2*((n-r)/3)) + 3)); };

a(n) = A353314(n) - n.
a(n) = A102899(3+n).
From Chai Wah Wu, Jul 27 2022: (Start)
a(n) = 2*a(n-3) - a(n-6) for n > 5.
G.f.: x*(-x^4 - x^3 + 3*x + 3)/(x^6 - 2*x^3 + 1). (End)

cp's OEIS Frontend

A353314 If n is of the form 3k, then a(n) = n, and if n is of the form 3k+r, with r = 1 or 2, then a(n) = 5*k + 3 + r.

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A120691 First differences of coefficients in the continued fraction for e.

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A353327 If n is a multiple of 3, then a(n) = 0, and if n is of the form 3k+r, with r = 1 or 2, then a(n) = 2*k + 3.

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