A240828 -id:A240828 - cp's OEIS frontend

A201629 a(n) = n if n is even and otherwise its nearest multiple of 4.

0, 0, 2, 4, 4, 4, 6, 8, 8, 8, 10, 12, 12, 12, 14, 16, 16, 16, 18, 20, 20, 20, 22, 24, 24, 24, 26, 28, 28, 28, 30, 32, 32, 32, 34, 36, 36, 36, 38, 40, 40, 40, 42, 44, 44, 44, 46, 48, 48, 48, 50, 52, 52, 52, 54, 56, 56, 56, 58, 60, 60, 60, 62, 64, 64, 64, 66, 68, 68

Offset: 0

Vaclav Kotesovec, Dec 03 2011

For n > 1, the maximal number of nonattacking knights on a 2 x (n-1) chessboard.
Compare this with the binary triangle construction of A240828.
Minimal number of straight segments in a rook circuit of an (n-1) X n board (see example). - Ruediger Jehn, Feb 26 2021

			G.f. = 2*x^2 + 4*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + 8*x^9 + ...
From _Ruediger Jehn_, Feb 26 2021: (Start)
a(5) = 4:
   +----+----+----+----+----+
   |  __|____|_   |   _|__  |
   | /  |    | \  |  / |  \ |
   +----+----+----+----+----+
   | \__|__  | |  |  | |  | |
   |    |  \ | \__|__/ |  | |
   +----+----+----+----+----+
   |  __|__/ |  __|__  |  | |
   | /  |    | /  |  \ |  | |
   +----+----+----+----+----+
   | \  |    | |  |  | |  | |
   |  \_|____|_/  |  \_|__/ |
   +----+----+----+----+----+
There are at least 4 squares on the 4 X 5 board with straight lines (here in squares a_12, a_25, a_35 and a_42).  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Rüdiger Jehn, Properties of Hamiltonian Circuits in Rectangular Grids, arXiv:2103.15778 [math.GM], 2021.
Craig Knecht, Row sums of superimposed and added binary filled triangles.
Vaclav Kotesovec, Non-attacking chess pieces

Cf. A004524, A085801, A189889, A190394.

a201629 = (* 2) . a004524 . (+ 1) -- Reinhard Zumkeller, Aug 05 2014

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x^2/((1-x)^2*(1+x^2)))); // G. C. Greubel, Aug 13 2018

seq(n-sin(Pi*n/2), n=0..30); # Robert Israel, Jul 14 2015

Table[2*(Floor[(Floor[(n + 1)/2] + 1)/2] + Floor[(Floor[n/2] + 1)/2]), {n, 1, 100}]
Table[If[EvenQ[n], n, 4*Round[n/4]], {n, 0, 68}] (* Alonso del Arte, Jan 27 2012 *)
CoefficientList[Series[2 x^2/((-1 + x)^2 (1 + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 06 2014 *)
a[ n_] := n - KroneckerSymbol[ -4, n]; (* Michael Somos, Jul 18 2015 *)

a(n)=n\4*4+[0, 0, 2, 4][n%4+1] \\ Charles R Greathouse IV, Jan 27 2012

{a(n) = n - kronecker( -4, n)}; /* Michael Somos, Jul 18 2015 */

a(n) = n - sin(n*Pi/2).
G.f.: 2*x^2/((1-x)^2*(1+x^2)).
a(n) = 2*A004524(n+1). - R. J. Mathar, Feb 02 2012
a(n) = n+(1-(-1)^n)*(-1)^((n+1)/2)/2. - Bruno Berselli, Aug 06 2014
E.g.f.: x*exp(x) - sin(x). - G. C. Greubel, Aug 13 2018

Formula corrected by Robert Israel, Jul 14 2015

A186421 Even numbers interleaved with repeated odd numbers.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 6, 3, 8, 5, 10, 5, 12, 7, 14, 7, 16, 9, 18, 9, 20, 11, 22, 11, 24, 13, 26, 13, 28, 15, 30, 15, 32, 17, 34, 17, 36, 19, 38, 19, 40, 21, 42, 21, 44, 23, 46, 23, 48, 25, 50, 25, 52, 27, 54, 27, 56, 29, 58, 29, 60, 31, 62, 31, 64, 33, 66, 33, 68, 35, 70, 35, 72, 37, 74, 37, 76, 39, 78, 39, 80, 41, 82, 41, 84, 43, 86, 43

Offset: 0

Reinhard Zumkeller, Feb 21 2011

A005843 interleaved with A109613.

Row sum of superimposed binary filled triangle. - Craig Knecht, Aug 07 2015

			A005843: 0   2   4   6   8   10   12   14   16   18   20    22
A109613:   1   1   3   3   5    5    7    7    9    9    11    11
this   : 0 1 2 1 4 3 6 3 8 5 10 5 12 7 14 7 16 9 18 9 20 11 22 ... .

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Craig Knecht, Row sums of superimposed binary triangles.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A005843, A109043, A109613.

Cf. A186422 (first differences), A186423 (partial sums), A240828 (row sum of superimposed binary triangle).

a186421 n = a186421_list !! n
a186421_list = interleave [0,2..] $ rep [1,3..] where
   interleave (x:xs) ys = x : interleave ys xs
   rep (x:xs) = x : x : rep xs

[IsEven(n) select n else 2*Floor(n/4)+1: n in [0..100]]; // Vincenzo Librandi, Jul 13 2015

A186421:=n->n-(1-(-1)^n)*(n+(-1)^(n*(n+1)/2))/4: seq(A186421(n), n=0..100); # Wesley Ivan Hurt, Aug 07 2015

Table[n - (1 - (-1)^n)*(n + I^(n (n + 1)))/4, {n, 0, 87}] (* or *)
CoefficientList[Series[x (1 + 2 x + 2 x^3 + x^4)/((1 + x^2) (x - 1)^2 (1 + x)^2), {x, 0, 87}], x] (* or *)
n = 88; Riffle[Range[0, n, 2], Flatten@ Transpose[{Range[1, n, 2], Range[1, n, 2]}]] (* Michael De Vlieger, Jul 14 2015 *)

makelist(n-(1-(-1)^n)*(n+%i^(n*(n+1)))/4, n, 0, 90); /* Bruno Berselli, Mar 04 2013 */

def A186421(n): return (n>>1)|1 if n&1 else n # Chai Wah Wu, Jan 31 2023

a(2*k) = 2*k, a(4*k+1) = a(4*k+3) = 2*k+1.
a(n) = n if n is even, else 2*floor(n/4)+1.
a(2*n-(2*k+1)) + a(2*n+2*k+1) = 2*n, 0 <= k < n.
a(n+2) = A109043(n) - a(n).
G.f.: x*(1+2*x+2*x^3+x^4) / ( (1+x^2)*(x-1)^2*(1+x)^2 ). - R. J. Mathar, Feb 23 2011
a(n) = n-(1-(-1)^n)*(n+i^(n(n+1)))/4, where i=sqrt(-1). - Bruno Berselli, Feb 23 2011
a(n) = a(n-2)+a(n-4)-a(n-6) for n>5. - Wesley Ivan Hurt, Aug 07 2015
E.g.f.: (x*cosh(x) + sin(x) + 2*x*sinh(x))/2. - Stefano Spezia, May 09 2021

Edited by Bruno Berselli, Mar 04 2013

A240830 a(n)=1 for n <= s+k; thereafter a(n) = Sum(a(n-i-s-a(n-i-1)),i=0..k-1) where s=0, k=7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 7, 7, 7, 7, 7, 7, 7, 13, 13, 13, 13, 13, 13, 19, 13, 19, 19, 19, 19, 25, 19, 25, 19, 25, 25, 31, 25, 31, 25, 31, 25, 31, 31, 37, 31, 37, 31, 37, 37, 37, 37, 43, 37, 43, 43, 43, 43, 43, 43, 49, 49, 49, 49, 49, 49, 49, 55, 55, 55, 55, 55, 55, 61, 55, 61, 61, 61, 61, 67, 61, 67, 61, 67, 67, 73

Offset: 1

N. J. A. Sloane, Apr 16 2014

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Joseph Callaghan, John J. Chew III, and Stephen M. Tanny, On the behavior of a family of meta-Fibonacci sequences, SIAM Journal on Discrete Mathematics 18.4 (2005): 794-824. See Table 2.1.
Index entries for Hofstadter-type sequences

Same recurrence as A240828, A120503 and A046702.
See also A240831, A240832.
Callaghan et al. (2005)'s sequences T_{0,k}(n) for k=1 through 7 are A000012, A046699, A046702, A240835, A241154, A241155, A240830.

#T_s,k(n) from Callaghan et al. Eq. (1.7).
s:=0; k:=7;
a:=proc(n) option remember; global s,k;
if n <= s+k then 1
else
    add(a(n-i-s-a(n-i-1)),i=0..k-1);
fi; end;
t1:=[seq(a(n),n=1..100)];

A240830[n_]:=A240830[n]=If[n<=7,1,Sum[A240830[n-i-A240830[n-i-1]],{i,0,6}]];
Array[A240830,100] (* Paolo Xausa, Dec 06 2023 *)

A185048 Length of the continued fraction for floor(Fibonacci(n)*(1+sqrt(5))/2) / Fibonacci(n).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 8, 4, 10, 6, 12, 6, 14, 8, 16, 8, 18, 10, 20, 10, 22, 12, 24, 12, 26, 14, 28, 14, 30, 16, 32, 16, 34, 18, 36, 18, 38, 20, 40, 20, 42, 22, 44, 22, 46, 24, 48, 24, 50, 26, 52, 26, 54, 28, 56, 28, 58, 30, 60, 30, 62, 32, 64, 32, 66, 34, 68, 34

Offset: 1

Benoit Cloitre, Feb 15 2011

A240828 is an essentially identical sequence. - Bruno Berselli, Apr 18 2014

Bruno Berselli, Table of n, a(n) for n = 1..20000

Cf. A000045, A001622, A240828.

Table[Length[ContinuedFraction[Floor[Fibonacci[n]*GoldenRatio]/Fibonacci[n]]], {n, 70}]

PARI
```
a(n)=if(n<3,1,if(n%2,n-1,2*floor(n/4)))
```

a(1)=a(2)=1, for k>=2 we have a(2k)=2*floor(k/2) and a(2k-1)=2*k-2.

G.f.: x*(x^7+x^6-x^5+x^4+x^3+x^2+x+1) / ((x-1)^2*(x+1)^2*(x^2+1)). - Colin Barker, Jun 20 2013

A240829 a(1)=-1, a(2)=0, a(3)=1; thereafter a(n) = Sum(a(n-i-s-a(n-i-1)),i=0..k-1) where s=0, k=3.

Original entry on oeis.org

-1, 0, 1, 3, 2, 4, 4, 7, 4, 7, 7, 9, 8, 9, 11, 10, 10, 13, 15, 13, 13, 13, 18, 15, 18, 18, 18, 18, 18, 23, 23, 20, 19, 23, 28, 27, 23, 25, 27, 28, 25, 26, 28, 30, 31, 32, 33, 33, 32, 34, 33, 38, 36, 39, 34, 36, 36, 39, 39, 39, 39, 44, 46, 46, 43, 46, 46, 44, 44, 49, 49, 49, 46, 51, 48, 51, 51, 54, 54, 54, 54, 54

Offset: 1

N. J. A. Sloane, Apr 16 2014

Callaghan, Joseph, John J. Chew III, and Stephen M. Tanny. "On the behavior of a family of meta-Fibonacci sequences." SIAM Journal on Discrete Mathematics 18.4 (2005): 794-824. See Fig. 1.7.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Index entries for Hofstadter-type sequences

Same recurrence as A240828, A120503 and A046702.

#T_s,k(n) from Callaghan et al. Eq. (1.6).
s:=0; k:=3;
a:=proc(n) option remember; global s,k;
if n <= 3 then n-2
else
    add(a(n-i-s-a(n-i-1)),i=0..k-1);
fi; end;
t1:=[seq(a(n),n=1..100)];

A257857 Sequentially filled binary triangle rotated 180 degrees and then superimposed and added to the original triangle.

Original entry on oeis.org

2, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 0, 2, 0, 2, 1, 1, 1, 1, 1, 1, 0, 2, 0, 2, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 2, 0, 2, 0, 2, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 2, 0, 2

Offset: 1

Craig Knecht, Jul 12 2015

The integers in the LINKS illustration hang like ornaments on a tree.

			Triangle T(n,k) begins:       Row sums
2;                                2
1,  1;                            2
0,  2,  0;                        2
1,  1,  1,  1;                    4
2,  0,  2,  0,  2;                6
1,  1,  1,  1,  1,  1;            6
0,  2,  0,  2,  0,  2,  0;        6
1,  1,  1,  1,  1,  1,  1,  1;    8

Craig Knecht, binary triangle rotated 180 degrees and superimposed on itself
Craig Knecht, Color coded contributions of the individual triangles

For row sums for the three other variations of this build process, see A186421, A201629, A240828.

A257857 := proc(n,k)
    if type(n,'even') then
        1 ;
    elif type((n+1)/2+k,'even') then
        2 ;
    else
        0;
    end if;
end proc:

T(n,k)=1 if n even, 1<=k<=n.
T(n,k)=2 if n odd and (n+1)/2+k even, 1<=k<=n.
T(n,k)=0 if n odd and (n+1)/2+k odd, 1<=k<=n.

cp's OEIS Frontend

A201629 a(n) = n if n is even and otherwise its nearest multiple of 4.

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A186421 Even numbers interleaved with repeated odd numbers.

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A240830 a(n)=1 for n <= s+k; thereafter a(n) = Sum(a(n-i-s-a(n-i-1)),i=0..k-1) where s=0, k=7.

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A185048 Length of the continued fraction for floor(Fibonacci(n)*(1+sqrt(5))/2) / Fibonacci(n).

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A240829 a(1)=-1, a(2)=0, a(3)=1; thereafter a(n) = Sum(a(n-i-s-a(n-i-1)),i=0..k-1) where s=0, k=3.

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A257857 Sequentially filled binary triangle rotated 180 degrees and then superimposed and added to the original triangle.

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