A005578 -id:A005578 - cp's OEIS frontend

A191666 Dispersion of A042964 (numbers congruent to 2 or 3 mod 4), by antidiagonals.

1, 2, 4, 3, 7, 5, 6, 14, 10, 8, 11, 27, 19, 15, 9, 22, 54, 38, 30, 18, 12, 43, 107, 75, 59, 35, 23, 13, 86, 214, 150, 118, 70, 46, 26, 16, 171, 427, 299, 235, 139, 91, 51, 31, 17, 342, 854, 598, 470, 278, 182, 102, 62, 34, 20, 683, 1707, 1195, 939, 555, 363

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1:  A005578
Row 2:  A160113
For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The six sequences and dispersions are listed here:
...
A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
A191666=dispersion of A042964 (2 or 3 mod 4)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191663 has 1st col A042964, all else A042948
A054582 has 1st col A005408, all else A005843
A191664 has 1st col A042963, all else A014601
A191665 has 1st col A014601, all else A042963
A191448 has 1st col A005843, all else A005408
A191666 has 1st col A042948, all else A042964
...
There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
   a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
   a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.

			Northwest corner:
1...2...3....6...11
4...7...14....27...54
5...10...19...38...75
8...15..30...59...118
8...18..35...70...139

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A042963, A014601, A191426, A191663.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 2; b = 3; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A042964: (2+4k,3+4k) *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191666 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191666  *)

A213526 a(n) = 3*n AND n, where AND is the bitwise AND operator.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 5, 8, 9, 10, 1, 4, 5, 10, 13, 16, 17, 18, 17, 20, 21, 2, 5, 8, 9, 10, 17, 20, 21, 26, 29, 32, 33, 34, 33, 36, 37, 34, 37, 40, 41, 42, 1, 4, 5, 10, 13, 16, 17, 18, 17, 20, 21, 34, 37, 40, 41, 42, 49, 52, 53, 58, 61, 64, 65, 66, 65, 68

Offset: 0

Alex Ratushnyak, Jun 13 2012

Indices of 1's: A007583(n),
indices of 2's: A047849(n+1),
indices of 4's: A039301(n+2),
indices of 5's: A153643(n+3),
indices of 8's: A155701(n+2),
indices of 9's: A155701(n+2)+1 = A163868(n+2),
indices of 10's: A153643(n+4)+3^((n+1) mod 2),
indices of 13's: A039301(n+3)+3,
indices of 16's: A039301(n+3)+4,
indices of 17's: 17, 19, 27, 49, 51, 91, 177, 179, 347, 689, 691, 1371, 2737, 2739, 5467, 10929, 10931, 21851, 43697, 43699, 87387, 174769, 174771, 349531, 699057, 699059, 1398107, 2796209, 2796211, 5592411, 11184817, 11184819, 22369627, 44739249, 44739251, 89478491, ...
indices of 18's: A039301(n+3)+6,
n's such that a(n)<3: A005578, except the first term.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

a:= proc(n) local i, k, m, r;
      k, m, r:= n, 3*n, 0;
      for i from 0 while (m>0 or k>0) do
        r:= r +2^i* irem(m, 2, 'm') *irem(k, 2, 'k')
      od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 22 2012

Table[BitAnd[n, 3*n], {n, 0, 68}] (* Arkadiusz Wesolowski, Jun 23 2012 *)

a(n)=bitand(n,3*n) \\ Charles R Greathouse IV, Feb 05 2013

for n in range(99):
    print(3*n & n, end=',')

A241684 The total number of rectangles appearing in the Thue-Morse sequence logical matrices after n stages.

Original entry on oeis.org

0, 0, 4, 8, 32, 120, 464, 1848, 7312, 29240, 116624, 466488, 1864592, 7458360, 29827984, 119311928, 477225872, 1908903480, 7635526544, 30542106168, 122168075152, 488672300600, 1954687804304, 7818751217208, 31274999276432, 125099997105720, 500399966053264, 2001599864213048

Offset: 0

Kival Ngaokrajang, Apr 27 2014

a(n) is the total number of non-isolated "1s" (consecutive 1s on 2 rows, 1 column or 1 row, 2 columns) that appear as rectangles in the Thue-Morse logical matrices after n stages. See links for more details.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Wikipedia, Thue-Morse sequence
Index entries for linear recurrences with constant coefficients, signature (4,5,-20,-4,16)

Cf. A010060.

[(8+3*2^n+2*4^n+(-1)^n*(24-2^n))/18: n in [0..30]]; // Vincenzo Librandi, Sep 29 2017

CoefficientList[Series[-4*x^2*(8*x^3 - 5*x^2 - 2*x + 1)/((x - 1)*(x + 1)*(2*x - 1)*(2*x + 1)*(4*x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Sep 28 2017 *)

x='x+O('x^50); concat([0,0], Vec(-4*x^2*(8*x^3-5*x^2-2*x+1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)))) \\ G. C. Greubel, Sep 28 2017

a(n) = A007590(A005578(n+1)) - (A139598(A000975(n-2)) + A007590(A000975(n-1))).
G.f.: -4*x^2*(8*x^3-5*x^2-2*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)). - Colin Barker, Apr 27 2014
a(n) = (8 + 3*2^n + 2*4^n + (-1)^n*(24 - 2^n))/18, n>0. - R. J. Mathar, May 04 2014

Terms a(21) onward added by G. C. Greubel, Sep 28 2017

A085281 Expansion of (1 - 3x + x^2)/((1-2x)(1-3x)).

Original entry on oeis.org

1, 2, 5, 13, 35, 97, 275, 793, 2315, 6817, 20195, 60073, 179195, 535537, 1602515, 4799353, 14381675, 43112257, 129271235, 387682633, 1162785755, 3487832977, 10462450355, 31385253913, 94151567435, 282446313697, 847322163875, 2541932937193, 7625731702715, 22877060890417, 68630914235795, 205892205836473

Offset: 0

Paul Barry, Jun 25 2003

Binomial transform of A005578.

Binomial transform is A085282.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Cf. A005578, A007689, A085282.

[3^n/3+2^n/2+0^n/6: n in [0..40]]; // Vincenzo Librandi, May 29 2011

a[n_]:=3^n/3 + 2^n/2; Flatten[Join[{1, Array[a, 50]}]] (* or *)
CoefficientList[Series[(1 - 3*x + x^2)/((1-2*x)*(1-3*x)), {x, 0, 50}], x] (* Stefano Spezia, Sep 09 2018 *)
LinearRecurrence[{5,-6},{1,2,5},40] (* Harvey P. Dale, Jun 14 2022 *)

def A085281(n): return 2^(n-1) +3^(n-1) +int(n==0)/6
[A085281(n) for n in range(41)] # G. C. Greubel, Nov 11 2024

a(n) = 3^(n-1) + 2^(n-1) + 0^n/6.
a(n) = A007689(n-1), n > 0. - R. J. Mathar, Sep 12 2008
E.g.f.: (1/6)*(1 + 3*exp(2*x) + 2*exp(3*x)). - G. C. Greubel, Nov 11 2024

A107910 A107909(n+1) - A107909(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 6, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 11, 1, 2, 1, 1, 6, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 2, 1, 11, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 6

Offset: 0

Reinhard Zumkeller, May 28 2005

nonn

For n>0 A005578 gives record-values: a(A023548(n+2))=A005578(n), a(m) < A005578(n) for m < A023548(n).

A166692 Triangle T(n,k) read by rows: T(n,k) = 2^(k-1), k>0, T(n,0) = (n+1) mod 2.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 0, 1, 2, 4, 1, 1, 2, 4, 8, 0, 1, 2, 4, 8, 16, 1, 1, 2, 4, 8, 16, 32, 0, 1, 2, 4, 8, 16, 32, 64, 1, 1, 2, 4, 8, 16, 32, 64, 128, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024

Offset: 0

Paul Curtz, Oct 18 2009

Variant of A166918.

			Triangle begins as:
  1;
  0, 1;
  1, 1, 2;
  0, 1, 2, 4;
  1, 1, 2, 4, 8;
  0, 1, 2, 4, 8, 16;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A005578, A011782, A051049, A131577, A106624, A166494, A166918.

A166692:= func< n,k | k eq 0 select ((n+1) mod 2) else 2^(k-1) >;
[A166692(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 24 2023

Join[{1,0},Flatten[Riffle[Table[2^Range[0,n],{n,0,10}],{1,0}]]] (* Harvey P. Dale, Jan 18 2015 *)

def A166692(n,k): return ((n+1)%2) if (k==0) else 2^(k-1)
flatten([[A166692(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Apr 24 2023

T(2n, k) = A011782(k).
T(2n+1, k) = A131577(k).
Sum_{k=0..n} T(n,k) = A051049(n).
From G. C. Greubel, Apr 24 2023: (Start)
T(2*n, n) = A011782(n).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A005578(n).
Sum_{k=0..n} T(n-k, k) = A106624(n). (End)

More terms from Harvey P. Dale, Jan 18 2015

A178746 Binary counter with intermittent bits. Starting at zero the counter attempts to increment by 1 at each step but each bit in the counter alternately accepts and rejects requests to toggle.

Original entry on oeis.org

0, 1, 3, 6, 6, 7, 13, 12, 12, 13, 15, 26, 26, 27, 25, 24, 24, 25, 27, 30, 30, 31, 53, 52, 52, 53, 55, 50, 50, 51, 49, 48, 48, 49, 51, 54, 54, 55, 61, 60, 60, 61, 63, 106, 106, 107, 105, 104

Offset: 0

David Scambler, Jun 08 2010

A simple scatter plot reveals a self-similar structure that resembles flying geese.
Ignoring the initial zero term, split the sequence into rows of increasing binary magnitude such that the terms in row m satisfy 2^m <= a(n) < 2^(m+1).
0: 1,
1: 3,
2: 6,6,7,
3: 13,12,12,13,15,
4: 26,26,27,25,24,24,25,27,30,30,31,
5: 53,52,52,53,55,50,50,51,49,48,48,49,51,54,54,55,61,60,60,61,63,
Then,
Row m starts at n = A005578(m+1) in the original sequence
The first term in row m is A081254(m)
The last term in row m is 2^(m+1)-1
The number of terms in row m is A001045(m+1)
The number of distinct terms in row m is A005578(m)
The number of ascending runs in row m is A005578(m)
The number of non-ascending runs in row m is A005578(m)
The number of descending runs in row m is A052950(m)
The number of non-descending runs in row m is A005578(m-1)
The sum of terms in row m is A178747(m)
The total number of '1' bits in the terms of row n is A178748(m)

			0 -> low bit toggles -> 1 -> should be 2 but low bit does not toggle -> 3 -> should be 4 but 2nd-lowest bit does not toggle -> 6 -> should be 7 but low bit does not toggle -> 6 -> low bit toggles -> 7

D. Scambler, Table of n, a(n) for n = 0..1024

Cf. A178747 sum of terms in rows of a(n), A178748 total number of '1' bits in the terms of rows of a(n).

seq(n)={my(a=vector(n+1), f=0, p=0); for(i=2, #a, my(b=bitxor(p+1,p)); f=bitxor(f,b); p=bitxor(p, bitand(b,f)); a[i]=p); a} \\ Andrew Howroyd, Mar 03 2020

If n is a power of 2, a(n) = n*3/2. Lim(a(n)/n) = 3/2.

A214613 Abelian complexity function of ordinary paperfolding word (A014707).

Original entry on oeis.org

2, 3, 4, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 5, 4, 3, 4, 5, 6, 5, 6, 7, 6, 5, 6, 5, 6, 5, 6, 5, 4, 3, 4, 5, 6, 5, 6, 7, 6, 5, 6, 7, 8, 7, 6, 7, 6, 5, 6, 7, 6, 5, 6, 7, 6, 5, 6, 7, 6, 5, 6, 5, 4, 3, 4, 5, 6, 5, 6, 7, 6, 5, 6, 7, 8, 7, 6, 7, 6, 5, 6, 7

Offset: 1

N. J. A. Sloane, Mar 08 2013

nonn

k first appears at position A005578(k-1). - Charlie Neder, Mar 03 2019

Charlie Neder, Table of n, a(n) for n = 1..1024
Blake Madill, Narad Rampersad, The abelian complexity of the paperfolding word, Discrete Math. 313 (2013), no. 7, 831--838. MR3017968.

Cf. A014707, A014577, A005578, A337120.

From Charlie Neder, Mar 03 2019 [Corrected by Kevin Ryde, Sep 05 2020]: (Start)
Madill and Rampersad provide the following recurrence:
a(1) = 2,
a(4n) = a(2n),
a(4n+2) = a(2n+1) + 1,
a(16n+1) = a(8n+1),
a(16n+{3,7,9,13}) = a(2n+1) + 2,
a(16n+5) = a(4n+1) + 2,
a(16n+11) = a(4n+3) + 2,
a(16n+15) = a(2n+2) + 1. (End)

a(21)-a(82) from Charlie Neder, Mar 03 2019

A241685 The total number of squares and rectangles appearing in the Thue-Morse sequence logical matrices after n stages.

Original entry on oeis.org

0, 2, 4, 18, 60, 242, 924, 3698, 14620, 58482, 233244, 932978, 3729180, 14916722, 59655964, 238623858, 954451740, 3817806962, 15271053084, 61084212338, 244336150300, 977344601202, 3909375608604

Offset: 0

Kival Ngaokrajang, Apr 27 2014

nonn

a(n) is the total number of unit squares (A241682), 2 X 2 squares (A241683), 2 X 1 and 1 X 2 rectangles (A241684) that appear in the Thue-Morse logical matrices after n stages. See links for more details.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Kival Ngaokrajang, Illustration for n = 6
Wikipedia, Thue-Morse sequence
Index entries for linear recurrences with constant coefficients, signature (4, 5, -20, -4, 16).

Cf. A010060.

Table[Floor[(2^(n + 2) + 3 - (-1)^n)^2/72], {n, 0, 50}] (* G. C. Greubel, Sep 29 2017 *)

{for (n=1,50, b=(2^(n+1)+3+(-1)^n)/6; a=floor(b^2/2); print1(a,","))}

a(n)  = A007590(A005578(n+1)).
Empirical g.f.: -2*x*(4*x^3-4*x^2-2*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)). - Colin Barker, Apr 27 2014
a(n) = floor((2^(n + 2) + 3 - (-1)^n)^2/72). - G. C. Greubel, Sep 29 2017

A135229 Triangle A000012(signed) * A103451 * A007318, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 3, 6, 7, 4, 1, 1, 3, 9, 13, 11, 5, 1, 1, 4, 12, 22, 24, 16, 6, 1, 1, 4, 16, 34, 46, 40, 22, 7, 1, 1, 5, 20, 50, 80, 86, 62, 29, 8, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

row sums = A005578 starting (1, 2, 3, 6, 11, 22, 43, 86, ...).

			First few rows of the triangle are:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,  1;
  1, 2,  4,  3,  1;
  1, 3,  6,  7,  4,  1;
  1, 3,  9, 13, 11,  5,  1;
  1, 4, 12, 22, 24, 16,  6, 1;
  1, 4, 16, 34, 46, 40, 22, 7, 1;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A005578, A007318, A103451.

function T(n,k)
  if k eq 0 then return 1;
  else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=0 then 1
    else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==0, 1, Sum[Binomial[n-1-2*j, k-1], {j, 0, Floor[(n-1)/2]}]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==0, 1, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) ); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n, k):
    if (k==0): 1
    else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

T(n,k) = A000012(signed) * A103451 * A007318 as infinite lower triangular matrices, where A000012(signed) = (1; -1,1; 1,-1,1; ...).

T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) = 1. - G. C. Greubel, Nov 20 2019

Offset changed by G. C. Greubel, Nov 20 2019

cp's OEIS Frontend

A191666 Dispersion of A042964 (numbers congruent to 2 or 3 mod 4), by antidiagonals.

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A213526 a(n) = 3*n AND n, where AND is the bitwise AND operator.

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A241684 The total number of rectangles appearing in the Thue-Morse sequence logical matrices after n stages.

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A085281 Expansion of (1 - 3*x + x^2)/((1-2*x)*(1-3*x)).

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A107910 A107909(n+1) - A107909(n).

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A166692 Triangle T(n,k) read by rows: T(n,k) = 2^(k-1), k>0, T(n,0) = (n+1) mod 2.

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A178746 Binary counter with intermittent bits. Starting at zero the counter attempts to increment by 1 at each step but each bit in the counter alternately accepts and rejects requests to toggle.

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A214613 Abelian complexity function of ordinary paperfolding word (A014707).

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A085281 Expansion of (1 - 3x + x^2)/((1-2x)(1-3x)).