A004695 -id:A004695 - cp's OEIS frontend

A178982 Partial sums of floor(Fibonacci(n)/2).

0, 0, 0, 1, 2, 4, 8, 14, 24, 41, 68, 112, 184, 300, 488, 793, 1286, 2084, 3376, 5466, 8848, 14321, 23176, 37504, 60688, 98200, 158896, 257105, 416010, 673124, 1089144, 1762278, 2851432, 4613721, 7465164, 12078896, 19544072, 31622980

Offset: 0

Mircea Merca, Jan 02 2011

nonn

Partial sums of A004695.

			a(4) = 0 + 0 + 0 + 1 + 1 = 2.

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

Cf. A004695, A164397.

seq(round(fibonacci(n+2)/2-(n+2)/3),n=0..40).

f[n_] := Floor[Fibonacci@n/2]; Accumulate@ Array[f, 38, 0]

a(n) = round(Fibonacci(n+2)/2 - (n+2)/3).
a(n) = round(Fibonacci(n+2)/2 - n/3 - 1/2).
a(n) = floor(Fibonacci(n+2)/2 - n/3 - 1/2).
a(n) = ceiling(Fibonacci(n+2)/2 - (n+1)/3 - 1/2).
a(n) = a(n-3) + Fibonacci(n) - 1, n > 3.
a(n) = 2*a(n-1) - 2*a(n-4) + a(n-6), n > 5.
G.f.: -x^3 / ( (x^2+x+1)*(x^2+x-1)*(x-1)^2 ).
a(n) = (1/2) * (Fibonacci(n+2) + floor(n/3) - n - 1). - Ralf Stephan, Jan 19 2014

A104221 a(n) = Fibonacci(n) - (Fibonacci(n) mod 2).

Original entry on oeis.org

0, 0, 0, 2, 2, 4, 8, 12, 20, 34, 54, 88, 144, 232, 376, 610, 986, 1596, 2584, 4180, 6764, 10946, 17710, 28656, 46368, 75024, 121392, 196418, 317810, 514228, 832040, 1346268, 2178308, 3524578, 5702886, 9227464, 14930352, 24157816, 39088168

Offset: 0

Roger L. Bagula, Mar 14 2005

Also the circumference of the (n-2)-Fibonacci cube graph for n > 4. - Eric W. Weisstein, Sep 03 2017

Robert Israel, Table of n, a(n) for n = 0..4781
Eric Weisstein's World of Mathematics, Fibonacci Cube Graph.
Eric Weisstein's World of Mathematics, Graph Circumference.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).

Cf. A000045, A004695, A011655, A049347.

[2*Floor(Fibonacci(n)/2): n in [0..40]]; // G. C. Greubel, Jul 08 2022

f:= gfun:-rectoproc({a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5),
seq(a(i)=[0, 0, 0, 2, 2][i+1],i=0..4)},a(n),remember):
map(f, [$0..50]); # Robert Israel, Mar 25 2018

2*Floor[Fibonacci[Range[0, 50]]/2] (* Eric W. Weisstein, Sep 03 2017 *)
Table[2/3 (Cos[2n*Pi/3] -1) +Fibonacci[n], {n,0,50}] (* Eric W. Weisstein, Sep 03 2017 *)
Table[(I^n*LucasL[n,I] -2)/3 +Fibonacci[n], {n,0,50}] (* Eric W. Weisstein, Mar 25 2018 *)
LinearRecurrence[{1,1,1,-1,-1}, {0,0,0,2,2}, 51] (* Eric W. Weisstein, Sep 03 2017 *)
CoefficientList[Series[(2x^3)/(1-x-x^2-x^3+x^4+x^5), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 03 2017 *)

a(n) = 2*(fibonacci(n)\2); \\ Altug Alkan, Mar 25 2018

[2*(fibonacci(n)//2) for n in (0..40)] # G. C. Greubel, Jul 08 2022

a(n) = 2*A004695(n). - R. J. Mathar, Jul 23 2010
G.f.: 2*x^3/((1-x)*(1+x+x^2)*(1-x-x^2)). - R. J. Mathar, Jul 23 2010
a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5). - Eric W. Weisstein, Sep 03 2017
a(n) = Fibonacci(n) - A011655(n). - David A. Corneth, Mar 25 2018
a(n) = (1/3)*(-2 +3*Fibonacci(n) + 2*ChebyshevU(n, -1/2) + ChebyshevU(n-1, -1/2)). - G. C. Greubel, Jul 08 2022

A124234 Riordan array (1/(1-x), x(1+x)^2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 4, 11, 7, 1, 1, 4, 15, 22, 9, 1, 1, 4, 16, 42, 37, 11, 1, 1, 4, 16, 57, 93, 56, 13, 1, 1, 4, 16, 63, 163, 176, 79, 15, 1, 1, 4, 16, 64, 219, 386, 299, 106, 17, 1, 1, 4, 16, 64, 247, 638, 794, 470, 137, 19, 1

Offset: 0

Paul Barry, Oct 22 2006

Row sums are A077864. Diagonal sums are A004695(n+3). T(2n,n) is A032443.

			Triangle begins
1,
1, 1,
1, 3, 1,
1, 4, 5, 1,
1, 4, 11, 7, 1,
1, 4, 15, 22, 9, 1,
1, 4, 16, 42, 37, 11, 1

Cf. A004695, A032443, A077864.

tabl(nn) = for (n=0, nn, for (k=0, n, print1(sum(j=0, n-k, binomial(2*k, j)), ", ")); print();); \\ Michel Marcus, Nov 05 2016

T(n,k) = Sum_{j=0..n-k} C(2k,j).

A173714 Floor(Lucas(n+1)/2), Lucas(n) = A000032(n).

Original entry on oeis.org

0, 1, 2, 3, 5, 9, 14, 23, 38, 61, 99, 161, 260, 421, 682, 1103, 1785, 2889, 4674, 7563, 12238, 19801, 32039, 51841, 83880, 135721, 219602, 355323, 574925, 930249, 1505174

Offset: 0

Gary Detlefs, Nov 25 2010

Sequences of the form a(0)=1, a(1)=b,
a(n) = a(n-1) + a(n-2) + 1 if n mod 3 =2, else
a(n) = a(n-1) + a(n-2) have a closed form of
a(n) = F(n-1)*a + F(n)*b + floor(F(n+1)/2),
where F(n)= Fibonacci(n) = A000045(n), floor(F(n+1)/2) = A004695(n+1).
We can generalize the definition of this sequence by changing the added 1 to any value of k and changing the last term of the formula to floor(F(n+1)/2)*k.
Two variants: if we add the constant at n mod 3 = 0, then a(n)=F(n-1)*a + F(n)*b + floor(F(n)/2), and if for n mod 3 =1, then a(n)=F(n-1)*a + F(n)*b + floor(F(n-1)/2).

			a(5) = a(4) + a(3) + 1 = 5 +3 +1 =9 because 5 mod 3 = 2.
a(6) = a(5) + a(4) = 9 +5 =14 because 6 mod 3 <>2.

Vincenzo Librandi, Table of n, a(n) for n = 0..280
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).

[Floor(Lucas(n+1)/2): n in [0..50]]; // Vincenzo Librandi, Apr 24 2011

with(combinat):
g:=(a,b,n)->fibonacci(n-1)*a+fibonacci(n)*b + floor(fibonacci(n+1)/2):
seq(g(0,1,n),n=0..30)

Table[Floor[LucasL[n + 1]/2], {n,0,50}] (* G. C. Greubel, Nov 24 2016 *)

a(0)= 0, a(1)=1, a(n)=a(n-1)+a(n-2)+1 if n mod 3 =2, else a(n)=a(n-1)+a(n-2).
G.f.: x*(1+x-x^3)/[(1-x-x^2)*(1-x^3)].
a(n) = a(n-1) +a(n-2) +(1+(-1)^Fib(n+1))/2.
a(n) = A000204(n+1)/2 + A099837(n+1)/6 - 1/3. - R. J. Mathar, Nov 26 2010
a(n) = Fibonacci(n) + floor(Fibonacci(n+1)/2). - Gary Detlefs, Dec 10 2010

A214286 a(n) = floor(Fibonacci(n)/7).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 7, 12, 20, 33, 53, 87, 141, 228, 369, 597, 966, 1563, 2530, 4093, 6624, 10717, 17341, 28059, 45401, 73461, 118862, 192324, 311187, 503511, 814698, 1318209, 2132907, 3451116, 5584024, 9035140, 14619165

Offset: 0

Vincenzo Librandi, Jul 10 2012

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

Cf. A004695-A004699.

Magma
```
[Floor(Fibonacci(n)/7): n in [0..40]];
```

Floor[Fibonacci[Range[0, 40]]/7] (* modified by G. C. Greubel, May 22 2019 *)
LinearRecurrence[{1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1},{0,0,0,0,0,0,1,1,3,4,7,12,20,33,53,87,141,228},50] (* Harvey P. Dale, Dec 01 2020 *)

vector(40, n, n--; fibonacci(n)\7 ) \\ G. C. Greubel, May 22 2019

[floor(fibonacci(n)/7) for n in (0..40)] # G. C. Greubel, May 22 2019

G.f.: x^6*(1+x^2+x^5+x^6+x^7+x^9+x^10) / ( (1-x-x^2)*(1-x^16) ). - R. J. Mathar, Jul 14 2012

a(n) = (A000045(n) - A105870(n))/7. - R. J. Mathar, Jul 14 2012

A260710 Expansion of 1/(1 - x - x^2 - x^4 + x^5 + x^7).

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 25, 43, 69, 116, 188, 313, 511, 846, 1386, 2288, 3756, 6191, 10174, 16756, 27552, 45357, 74604, 122787, 201996, 332414, 546901, 899946, 1480699, 2436459, 4008858, 6596366, 10853563, 17858788, 29384804, 48350401, 79555943, 130902711

Offset: 0

David Neil McGrath, Jul 30 2015

This sequence counts partially ordered partitions of (n) into parts 1,2,3,4 where the order (position) of adjacent pairs of numbers (1,2);(2,3);(3,4) is unimportant. Alternatively the order of the complementary pairs (1,4);(1,3);(2,4) is important.

			There are 25 partially ordered partitions of 7, i.e., a(7) = 25. These are (43=34),(421=412),(124=214),(241),(142),(4111),(1411),(1141),(1114),(331),(313),(133),(1132=1123),(2131=1231),(1312=1321),(2311=3211),(31111),(13111),(11311),(11131),(11113),(2221=four),(22111=ten),(211111=six),(1111111).

Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,0,1,-1,0,-1).

Cf. A023435, A080239, A023434, A254685, A116732, A004695.

I:=[1,1,2,3,6,9,16]; [n le 7 select I[n] else Self(n-1)+Self(n-2)+Self(n-4)-Self(n-5)-Self(n-7): n in [1..40]]; // Vincenzo Librandi, Aug 04 2015

LinearRecurrence[{1, 1, 0, 1, -1, 0, -1}, {1, 1, 2, 3, 6, 9, 16}, 50] (* Vincenzo Librandi, Aug 04 2015 *)

Vec(1/(1 - x - x^2 - x^4 + x^5 + x^7) + O(x^50)) \\ Michel Marcus, Aug 06 2015

G.f: 1/(1 - x - x^2 - x^4 + x^5 + x^7).
a(n) = a(n-1) + a(n-2) + a(n-4) - a(n-5) - a(n-7).
Construct the matrix array T(n,j) = [A^*j]*[S^*(j-1)] where A=(1,1,0,1,-1,0,-1) and S=(0,1,0,...) (A063524). [* is convolution operation] Define S^*0=I with I=(1,0,...). a(n) = Sum_{j=1..n} T(n,j).

A293505 a(n) is the integer k that minimizes |k/Fibonacci(n) - 1/2|.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 4, 6, 10, 17, 28, 44, 72, 116, 188, 305, 494, 798, 1292, 2090, 3382, 5473, 8856, 14328, 23184, 37512, 60696, 98209, 158906, 257114, 416020, 673134, 1089154, 1762289, 2851444, 4613732, 7465176, 12078908, 19544084, 31622993, 51167078

Offset: 0

Clark Kimberling, Oct 12 2017

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

Cf. A000045, A004695, A293505.

z = 120; r = 1/2; f[n_] := Fibonacci[n];
Table[Floor[r*f[n]], {n, 0, z}];   (* A004695 *)
Table[Ceiling[r*f[n]], {n, 0, z}]; (* A173173 *)
Table[Round[r*f[n]], {n, 0, z}];   (* A293505 *)

G.f.: -((x^3 (-1 - x + x^2))/((-1 + x) (1 + x) (1 - x + x^2) (-1 + x + x^2) (1 + x + x^2))).
a(n) = a(n-1) + a(n-2) + a(n-6) - a(n-7) - a(n-8) for n >= 9.
a(n) = floor(1/2 + Fibonacci(n)/2).
a(n) = A004695(n) if (fractional part of Fibonacci(n)/2) < 1/2, otherwise a(n) = A293419(n).

A131255 A004070 * A000012(signed).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 0, 1, 1, 3, 1, 0, 1, 1, 3, 4, 1, 1, 0, 2, 2, 6, 5, 1, 0, 1, 1, 3, 5, 10, 6, 1, 0, 1, 1, 3, 5, 11, 15, 7, 1, 1, 0, 2, 2, 6, 10, 21, 21, 8, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A004695 starting (1, 1, 2, 4, 6, 10, 17, 27, 44, 72, ...). A058393 = A000012(signed) * A004070.

			First few rows of the triangle:
  1;
  0,  1;
  0,  1,  1;
  1,  0,  2,  1;
  0,  1,  1,  3,  1;
  0,  1,  1,  3,  4,  1;
  1,  0,  2,  2,  6,  5,  1;
  0,  1,  1,  3,  5, 10,  6,  1;
  0,  1,  1,  3,  5, 11, 15,  7,  1;
  1,  0,  2,  2,  6, 10,  2, 21,  8,  1;
  ...

Cf. A004070, A000012, A004695, A058393.

A004070 * A000012, where A000012 = (1; -1,1; 1,-1,1; ...).

A147997 Number of nonnegative even integers <= Fibonacci(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 7, 11, 18, 28, 45, 73, 117, 189, 306, 494, 799, 1293, 2091, 3383, 5474, 8856, 14329, 23185, 37513, 60697, 98210, 158906, 257115, 416021, 673135, 1089155, 1762290, 2851444, 4613733, 7465177, 12078909

Offset: 0

Giovanni Teofilatto, Nov 19 2008

Cf. A000045

Table[f=Fibonacci[n]; If[EvenQ[f], f = f/2, f = (f-1)/2];  f+1, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Nov 22 2010 *)

a(n) = 1+floor( A000045(n)/2) = 1+A004695(n). - R. J. Mathar, Jan 30 2010
From Chai Wah Wu, Sep 23 2016: (Start)
a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5) for n > 4.
G.f.: (1 + x)*(1 - x - x^3)/((1 - x)*(1 - x - x^2)*(1 + x + x^2)). (End)

Definition and offset corrected by R. J. Mathar, Jan 30 2010

Definition corrected by Joel B. Lewis, Nov 14 2012

A279890 Expansion of x(1 - x + 2x^3 - x^4)/((1 - x)(1 + x)(1 - x + x^2)*(1 - x - x^2)).

Original entry on oeis.org

0, 1, 1, 2, 4, 7, 12, 19, 31, 50, 82, 133, 216, 349, 565, 914, 1480, 2395, 3876, 6271, 10147, 16418, 26566, 42985, 69552, 112537, 182089, 294626, 476716, 771343, 1248060, 2019403, 3267463, 5286866, 8554330, 13841197, 22395528, 36236725, 58632253, 94868978, 153501232, 248370211, 401871444, 650241655, 1052113099, 1702354754

Offset: 0

Ilya Gutkovskiy, Dec 22 2016

The integer part of the harmonic mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2).
The o.g.f. for the numerators of the fractional part of the harmonic mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2) is 6*x/((1 + x - x^2)*(1 - 4*x - x^2)).
The o.g.f. for the denominators of the fractional part of the harmonic mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2) is (1 + 3*x - x^2)/((1 + x)*(1 - 3*x + x^2)).
Convolution of Fibonacci numbers and periodic sequence [1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, ...].

			a(1) = floor(3/(1/F(1)+1/F(2)+1/F(3))) = floor(3/(1/1+1/1+1/2)) = 1;
a(2) = floor(3/(1/F(2)+1/F(3)+1/F(4))) = floor(3/(1/1+1/2+1/3)) = 1;
a(3) = floor(3/(1/F(3)+1/F(4)+1/F(5))) = floor(3/(1/2+1/3+1/5)) = 2, etc.

Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Harmonic Mean
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,0,-1).

Cf. A000045, A001622, A004695, A065563, A236428.
Cf. A062114 (the integer part of the harmonic mean of Fibonacci(n+1) and Fibonacci(n+2) for n>0).
Cf. A074331 (the integer part of the geometric mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2)).

LinearRecurrence[{2, 0, -2, 2, 0, -1}, {0, 1, 1, 2, 4, 7}, 46]
Table[Floor[3 Fibonacci[n] Fibonacci[n + 1] Fibonacci[n + 2]/(2 Fibonacci[n + 1] Fibonacci[n + 2] - (-1)^n)], {n, 0, 45}]

concat(0, Vec((x*(1-x+2*x^3-x^4)/((1-x)*(1+x)*(1-x+x^2))) + O(x^40))) \\ Felix Fröhlich, Dec 22 2016

G.f.: x*(1 - x + 2*x^3 - x^4)/((1 - x)*(1 + x)*(1 - x + x^2)*(1 - x - x^2)).
a(n) = 2*a(n-1) - 2*a(n-3) + 2*a(n-4) - a(n-6).
a(n) = (9*sqrt(5)*(((1 + sqrt(5))/2)^n - ((1 - sqrt(5))/2)^n) + 5*((-1)^n + 2*cos(Pi*n/3) - 3))/30.
a(n) = floor(3*F(n)*F(n+1)*F(n+2)/(2*F(n+1)*F(n+2)-(-1)^n)), where F(n) is the n-th Fibonacci number (A000045).
a(n) = floor(3*A065563(n)/A236428(n+1)).
a(n) = 3*A000045(n)/2 + ((-1)^n + 2*cos(Pi*n/3) - 3)/6.
a(n) ~ 3*phi^n/(2*sqrt(5)), where phi is the golden ratio (A001622).
Lim_{n->infinity} a(n+1)/a(n) = phi.

cp's OEIS Frontend

A178982 Partial sums of floor(Fibonacci(n)/2).

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A104221 a(n) = Fibonacci(n) - (Fibonacci(n) mod 2).

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A124234 Riordan array (1/(1-x), x(1+x)^2).

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A173714 Floor(Lucas(n+1)/2), Lucas(n) = A000032(n).

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A214286 a(n) = floor(Fibonacci(n)/7).

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A260710 Expansion of 1/(1 - x - x^2 - x^4 + x^5 + x^7).

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Magma

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A293505 a(n) is the integer k that minimizes |k/Fibonacci(n) - 1/2|.

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A279890 Expansion of x(1 - x + 2x^3 - x^4)/((1 - x)(1 + x)(1 - x + x^2)*(1 - x - x^2)).