A093148 -id:A093148 - cp's OEIS frontend

A047246 Numbers that are congruent to {0, 1, 2, 3} mod 6.

0, 1, 2, 3, 6, 7, 8, 9, 12, 13, 14, 15, 18, 19, 20, 21, 24, 25, 26, 27, 30, 31, 32, 33, 36, 37, 38, 39, 42, 43, 44, 45, 48, 49, 50, 51, 54, 55, 56, 57, 60, 61, 62, 63, 66, 67, 68, 69, 72, 73, 74, 75, 78, 79, 80, 81, 84, 85, 86, 87, 90, 91, 92, 93, 96, 97, 98

Offset: 1

N. J. A. Sloane

The sequence is the interleaving of A047238 with A047241. - Guenther Schrack, Feb 12 2019

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

Cf. A045331 (primes congruent to {1,2,3} mod 6), A047238, A047241.

Complement: A047257.

Filtered([0..100],n->n mod 6 = 0 or n mod 6 = 1 or n mod 6 = 2 or n mod 6 = 3); # Muniru A Asiru, Feb 20 2019

a047246 n = a047246_list !! (n-1)
a047246_list = [0..3] ++ map (+ 6) a047246_list
-- Reinhard Zumkeller, Jan 15 2013

[Floor((6/5)*Floor(5*(n-1)/4)) : n in [1..100]]; // Wesley Ivan Hurt, May 21 2016

A047246:=n->(6*n-9-I^(2*n)-(1-I)*I^(-n)-(1+I)*I^n)/4: seq(A047246(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[(6n-9-I^(2n)-(1-I)*I^(-n)-(1+I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)

my(x='x+O('x^70)); concat([0], Vec(x^2*(1+x+x^2+3*x^3)/((1-x)*(1-x^4)))) \\ G. C. Greubel, Feb 16 2019

a=(x^2*(1+x+x^2+3*x^3)/((1-x)*(1-x^4))).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

G.f.: x^2*(1+x+x^2+3*x^3) / ((1+x)*(1-x)^2*(1+x^2)). - R. J. Mathar, Oct 08 2011
a(n) = floor((6/5)*floor(5*(n-1)/4)). - Bruno Berselli, May 03 2016
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6*n - 9 - i^(2*n) - (1-i)*i^(-n) - (1+i)*i^n)/4 where i=sqrt(-1).
a(2*n) = A047241(n), a(2*n-1) = A047238(n). (End)
E.g.f.: (6 + sin(x) - cos(x) + (3*x - 4)*sinh(x) + (3*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016
From Guenther Schrack, Feb 12 2019: (Start)
a(n) = (6*n - 9 - (-1)^n - 2*(-1)^(n*(n+1)/2))/4.
a(n) = a(n-4) + 6, a(1)=0, a(2)=1, a(3)=2, a(4)=3, for n > 4. (End)
Sum_{n>=2} (-1)^n/a(n) = Pi/(6*sqrt(3)) + 2*log(2)/3. - Amiram Eldar, Dec 16 2021
a(n)-a(n-1) = A093148(n-2). - R. J. Mathar, May 01 2024

More terms from Wesley Ivan Hurt, May 21 2016

A167816 Numerator of x(n) = x(n-1) + x(n-2), x(0)=0, x(1)=1/3; denominator=A167817.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 8, 13, 7, 34, 55, 89, 48, 233, 377, 610, 329, 1597, 2584, 4181, 2255, 10946, 17711, 28657, 15456, 75025, 121393, 196418, 105937, 514229, 832040, 1346269, 726103, 3524578, 5702887, 9227465, 4976784, 24157817, 39088169, 63245986, 34111385

Offset: 0

Reinhard Zumkeller, Nov 13 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Fibonacci number
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 7, 0, 0, 0, -1).

Cf. A000045, A167808.

[0,1,1] cat [Numerator(Fibonacci(n)/Fibonacci(2*n-4)): n in [3..40]]; // Vincenzo Librandi, Jun 28 2016

Numerator[LinearRecurrence[{1,1},{0,1/3},40]] (* Harvey P. Dale, Dec 07 2014 *)
LinearRecurrence[{0, 0, 0, 7, 0, 0, 0, -1},{0, 1, 1, 2, 1, 5, 8, 13},39] (* Ray Chandler, Aug 03 2015 *)

a(n) = (a(n-1)*A093148(n+2) + a(n-2)*A093148(n+1))/A093148(n-1) for n>1.
a(4*n) = A004187(n) = (a(4*n-1) + a(4*n-2))/3;
a(4*n+1) = A033889(n) = 3*a(4*n-1) + a(4*n-2);
a(4*n+2) = A033890(n) = a(4*n-1) + 3*a(4*n-2);
a(4*n+3) = A033891(n) = a(4*n-1) + a(4*n-2).
Numerator of Fibonacci(n) / Fibonacci(2n-4) for n>=3. - Gary Detlefs, Dec 20 2010

Definition corrected by D. S. McNeil, May 09 2010

A047227 Numbers that are congruent to {1, 2, 3, 4} mod 6.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 10, 13, 14, 15, 16, 19, 20, 21, 22, 25, 26, 27, 28, 31, 32, 33, 34, 37, 38, 39, 40, 43, 44, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 61, 62, 63, 64, 67, 68, 69, 70, 73, 74, 75, 76, 79, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 97, 98

Offset: 1

N. J. A. Sloane

a(k)^m is a term for k and m in N. - Jerzy R Borysowicz, Apr 18 2023

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

Cf. A008621, A047235, A047241, A047245, A047246, A047247, A093148.

Complement of A047264. Equals A203016 divided by 3.

[n: n in [0..100] | n mod 6 in [1..4]]; // Vincenzo Librandi, Jan 06 2013

A047227:=n->(6*n-5-I^(2*n)+(1+I)*I^(1-n)+(1-I)*I^(n-1))/4: seq(A047227(n), n=1..100); # Wesley Ivan Hurt, May 20 2016

Complement[Range[100], Flatten[Table[{6n - 1, 6n}, {n, 0, 15}]]] (* Alonso del Arte, Jul 07 2011 *)
Select[Range[100], MemberQ[{1, 2, 3, 4}, Mod[#, 6]]&] (* Vincenzo Librandi, Jan 06 2013 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,1,0,0,1]^(n-1)*[1;2;3;4;7])[1,1] \\ Charles R Greathouse IV, May 03 2023

From Johannes W. Meijer, Jul 07 2011: (Start)
a(n) = floor((n+2)/4) + floor((n+1)/4) + floor(n/4) + 2*floor((n-1)/4) + floor((n+3)/4).
G.f.: x*(1 + x + x^2 + x^3 + 2*x^4)/(x^5 - x^4 - x + 1). (End)
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6n - 5 - i^(2n) + (1+i)*i^(1-n) + (1-i)*i^(n-1))/4 where i=sqrt(-1).
a(2n) = A047235(n), a(2n-1) = A047241(n). (End)
E.g.f.: (4 + sin(x) - cos(x) + (3*x - 2)*sinh(x) + 3*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = A047246(n) + 1.
a(n+2) - a(n+1) = A093148(n) for n>0.
a(1-n) = - A047247(n). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/12 + 2*log(2)/3 - log(3)/4. - Amiram Eldar, Dec 17 2021

A178591 Decimal expansion of (9 + sqrt(165))/14.

Original entry on oeis.org

1, 5, 6, 0, 3, 7, 3, 7, 5, 5, 6, 1, 8, 9, 3, 7, 7, 8, 7, 1, 5, 1, 2, 0, 7, 0, 3, 4, 1, 1, 7, 1, 3, 8, 6, 0, 1, 3, 9, 0, 5, 2, 5, 1, 7, 9, 3, 9, 7, 1, 6, 1, 7, 0, 5, 0, 2, 0, 4, 6, 2, 2, 3, 1, 0, 1, 8, 4, 6, 5, 6, 9, 3, 1, 9, 8, 4, 1, 5, 9, 5, 7, 9, 3, 0, 1, 3, 1, 1, 6, 7, 4, 5, 5, 5, 7, 9, 8, 0, 4, 0, 5, 2, 4, 6

Offset: 1

Klaus Brockhaus, May 30 2010

Continued fraction expansion of (9 + sqrt(165))/14 is A093148.

			(9 + sqrt(165))/14 = 1.56037375561893778715...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A178592 (decimal expansion of sqrt(165)), A093148 (repeat 1, 1, 1, 3).

SetDefaultRealField(RealField(100)); (9+Sqrt(165))/14; // G. C. Greubel, Jan 30 2019

RealDigits[(9+Sqrt[165])/14, 10, 100][[1]] (* G. C. Greubel, Jan 30 2019 *)

default(realprecision, 100); (9+sqrt(165))/14 \\ G. C. Greubel, Jan 30 2019

numerical_approx((9+sqrt(165))/14, digits=100) # G. C. Greubel, Jan 30 2019

A106740 Triangle read by rows: greatest common divisors of pairs of Fibonacci numbers greater than 1: T(n, k) = gcd(Fibonacci(n), Fibonacci(k)).

Original entry on oeis.org

2, 1, 3, 1, 1, 5, 2, 1, 1, 8, 1, 1, 1, 1, 13, 1, 3, 1, 1, 1, 21, 2, 1, 1, 2, 1, 1, 34, 1, 1, 5, 1, 1, 1, 1, 55, 1, 1, 1, 1, 1, 1, 1, 1, 89, 2, 3, 1, 8, 1, 3, 2, 1, 1, 144, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 233, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 377, 2, 1, 5, 2, 1, 1, 2, 5, 1, 2, 1, 1, 610

Offset: 3

Reinhard Zumkeller, May 15 2005

			Triangle begins as:
  2;
  1, 3;
  1, 1, 5;
  2, 1, 1, 8;
  1, 1, 1, 1, 13;
  1, 3, 1, 1,  1, 21;
  2, 1, 1, 2,  1,  1, 34;
  1, 1, 5, 1,  1,  1,  1, 55;
  1, 1, 1, 1,  1,  1,  1,  1, 89;
  2, 3, 1, 8,  1,  3,  2,  1,  1, 144;
  1, 1, 1, 1,  1,  1,  1,  1,  1,   1, 233;
  1, 1, 1, 1, 13,  1,  1,  1,  1,   1,   1, 377;
  2, 1, 5, 2,  1,  1,  2,  5,  1,   2,   1,   1, 610;

G. C. Greubel, Rows n = 3..52 of the triangle, flattened

Cf. A000012, A000045, A010125, A060904, A061347, A093148, A131534.

T[n_, k_]:= GCD[Fibonacci[n], Fibonacci[k]];
Table[T[n, k], {n,3,18}, {k,3,n}]//Flatten (* G. C. Greubel, Sep 11 2021 *)

def T(n,k): return gcd(fibonacci(n), fibonacci(k))
flatten([[T(n,k) for k in (3..n)] for n in (3..18)]) # G. C. Greubel, Sep 11 2021

T(n, k) = gcd(A000045(n), A000045(k)) for n >= 3 and 3 <= k <= n.
T(n, 3) = abs(A061347(n)).
T(n, 4) = A093148(n-1).
T(n, n) = A000045(n).
From G. C. Greubel, Sep 11 2021: (Start)
T(n, 3) = A131534(n-2).
T(n, 5) = A060904(n).
T(n, 6) = A010125(n).
T(n, n-1) = T(n, n-2) = A000012(n).
T(n, n-3) = A093148(n-5).
T(n, n-4) = A093148(n-5).
T(n, n-5) = A060904(n-5).
T(n, n-6) = A010125(n-6). (End)

A179041 Partial sums of ceiling(Fibonacci(n)/3).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 14, 21, 33, 52, 82, 130, 208, 334, 538, 867, 1400, 2262, 3656, 5911, 9560, 15464, 25017, 40473, 65482, 105947, 171420, 277357, 448767, 726114, 1174871, 1900974, 3075834, 4976797, 8052619

Offset: 0

Mircea Merca, Jan 04 2011

nonn

			a(9) = 0 + 1 + 1 + 1 + 1 + 2 + 3 + 5 + 7 + 12 = 33.

Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.

A179041 := proc(n) add( ceil(combinat[fibonacci](i)/3),i=0..n) ; end proc:

a(n) = round(Fibonacci(n+2)/3 + 3*n/8 - 1/24).
a(n) = floor(Fibonacci(n+2)/3 + 3*n/8 + 1/4).
a(n) = ceiling(Fibonacci(n+2)/3 + 3*n/8 - 1/3).
a(n) = a(n-8) + Fibonacci(n-1) + Fibonacci(n-3) + 3, n > 7.
a(n) = 2*a(n-1) - a(n-3) + a(n-8) - 2*a(n-9) + a(n-11), n > 10.
G.f.: (x^9 + x^8 + x^4 + x^3 - x)/((x+1)*(x^2+1)*(x^2+x-1)*(x-1)^2*(x^4+1)).
a(n) = -1/16 + 3*n/8 - (-1)^n/16 + Fibonacci(n+2)/3 - A057077(n)/8 + (-1)^floor((n-1)/4)*A093148(n+1)/12. - R. J. Mathar, Jan 08 2011

A293990 a(n) = (3*n + ((n-2) mod 4))/2.

Original entry on oeis.org

1, 3, 3, 5, 7, 9, 9, 11, 13, 15, 15, 17, 19, 21, 21, 23, 25, 27, 27, 29, 31, 33, 33, 35, 37, 39, 39, 41, 43, 45, 45, 47, 49, 51, 51, 53, 55, 57, 57, 59, 61, 63, 63, 65, 67, 69, 69, 71, 73, 75, 75, 77, 79, 81, 81, 83, 85, 87, 87, 89, 91, 93, 93

Offset: 0

Dimitris Valianatos, Oct 21 2017

The product (2/3) * (4/3) * (6/5) * (6/7) * (8/9) * (10/9) * (12/11) * (12/13) * ... = Pi/(2*sqrt(3)). The denominators are a(n) for n >= 1 and numerators are a(n-1) + A093148(n) for n >= 1 -> [2, 4, 6, 6, 8, 10, 12, 12, ...].
Let r(n) = (a(n)-1)/(a(n)+1) if a(n) mod 4 = 1, (a(n)+1)/(a(n)-1) otherwise; then Product_{n>=1} r(n) = (2/1) * (2/1) * (2/3) * (4/3) * (4/5) * (4/5) * (6/5) * (6/7) * ... = Pi*sqrt(3)/2 = 2.72069904635132677...
The odd numbers of partial sums this sequence, are identified with the A003215 sequence. Also the prime numbers that appear in partial sums in this sequence, are identified with the A002407 sequence.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A007310, A047298, A063305, A093148, A134967, A157932, A162330, A168329, A214549, A285869, A003215, A002407.

[(3*n+((n-2) mod 4))/2 : n in [0..100]]; // Wesley Ivan Hurt, Oct 29 2017

A293990:=n->(3*n+((n-2) mod 4))/2: seq(A293990(n), n=0..100); # Wesley Ivan Hurt, Oct 29 2017

Table[(3*n + Mod[(n - 2), 4])/2, {n, 0, 100}] (* Wesley Ivan Hurt, Oct 29 2017 *)
f[n_] := (3n + Mod[n - 2, 4])/2; Array[f, 65, 0] (* or *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 3, 5, 7}, 65] (* or *)
CoefficientList[ Series[(x^4 + 2x^3 + 2x + 1)/((x - 1)^2 (x^3 + x^2 + x + 1)), {x, 0, 64}], x] (* Robert G. Wilson v, Nov 28 2017 *)

PARI
```
a(n) = (3*n + (n-2)%4) / 2
```

Vec(x*(1 + 2*x + 2*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^30)) \\ Colin Barker, Oct 21 2017

first(n) = my(start=[1,3,3,5,7,9,9,11]); if(n<=8, return(start)); my(res=vector(n)); for (i=1, 8, res[i] = start[i]); for(i = 1, n-8 ,res[i+8] = res[i] + 12); res \\ David A. Corneth, Oct 21 2017

Sum_{n>=0} 1/a(n)^2 = 5*Pi^2/36 = 1.3707783890401886970... = 10*A086729.
(a(n) - n) * (-1)^(n+1) = A134967(n) for n >= 0.
a(n) - n = A162330(n) for n >= 0.
a(n) - n = A285869(n+1) for n >= 0.
a(n) + a(n+1) = A157932(n+2) for n >= 0.
a(n) + (2*n+1) = A047298(n+1) for n >= 0.
From Colin Barker, Oct 21 2017: (Start)
G.f.: x*(1 + 2*x + 2*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
(End)
a(n + 8) = a(n) + 12. - David A. Corneth, Oct 21 2017
a(4*k+4) * a(4*k+3) - a(4*k+2) * a(4*k+1) = 2*A063305(k+3) for k >= 0.
Sum_{n>=0} 1/(a(n) + a(n+2))^2 = (4*Pi^2 - 27) / 108 = (A214549 - 1) / 4.

cp's OEIS Frontend

A047246 Numbers that are congruent to {0, 1, 2, 3} mod 6.

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A167816 Numerator of x(n) = x(n-1) + x(n-2), x(0)=0, x(1)=1/3; denominator=A167817.

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A047227 Numbers that are congruent to {1, 2, 3, 4} mod 6.

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A178591 Decimal expansion of (9 + sqrt(165))/14.

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A106740 Triangle read by rows: greatest common divisors of pairs of Fibonacci numbers greater than 1: T(n, k) = gcd(Fibonacci(n), Fibonacci(k)).

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A179041 Partial sums of ceiling(Fibonacci(n)/3).

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A293990 a(n) = (3*n + ((n-2) mod 4))/2.

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