A020988 -id:A020988 - cp's OEIS frontend

A351122 Irregular triangle read by rows in which row n lists the number of divisions by 2 after tripling steps in the Collatz 3x+1 trajectory of 2n+1 until it reaches 1.

1, 4, 4, 1, 1, 2, 3, 4, 2, 1, 1, 2, 3, 4, 1, 2, 3, 4, 3, 4, 1, 1, 1, 5, 4, 2, 3, 4, 1, 3, 1, 2, 3, 4, 6, 1, 1, 5, 4, 2, 1, 3, 1, 2, 3, 4, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 2, 2, 4, 3, 1, 1, 5, 4, 3, 1, 2, 3, 4

Offset: 1

Flávio V. Fernandes, Feb 01 2022

			Triangle starts at T(1,0):
   n\k   0   1   2   3   4   5   6   7   8 ...
   1:    1   4
   2:    4
   3:    1   1   2   3   4
   4:    2   1   1   2   3   4
   5:    1   2   3   4
   6:    3   4
   7:    1   1   1   5   4
   8:    2   3   4
   9:    1   3   1   2   3   4
  10:    6
  11:    1   1   5   4
  12:    2   1   3   1   2   3   4
  13:    1   2   1   1   1   1   2   2   1   2   1   1   2  ... (see A372362)
  ...
For n=6, the trajectory of 2*n+1 = 13 is as follows. The tripling steps ("=>") are followed by runs of 3 and then 4 halvings ("->"), so row n=6 is 3, 4.
  13  =>  40 -> 20 -> 10 -> 5  =>  16 -> 8 -> 4 -> 2 -> 1
    triple   \------------/   triple  \---------------/
               3 halvings                4 halvings
Runs of halvings are divisions by 2^T(n,k). Row n=11 is 1, 1, 5, 4 and its steps starting from 2*n+1 = 23 reach 1 by a nested expression
  (((((((23*3+1)/2^1)*3+1)/2^1)*3+1)/2^5)*3+1)/2^4 = 1.

Cf. A075680 (row lengths), A166549 (row sums), A351123 (row partial sums).
Cf. A256598.
Cf. A020988 (where row is [2*n]).
Cf. A198584 (where row length is 2), A228871 (where row is [1, x]).
Cf. A372362 (row 13, the first 41 terms).

row(n) = my(m=2*n+1, list=List()); while (m != 1, if (m%2, m = 3*m+1, my(nb = valuation(m,2)); m/=2^nb; listput(list, nb));); Vec(list); \\ Michel Marcus, Jul 18 2022

T(n,k) = log_2( (3*A256598(n,k)+1) / A256598(n,k+1) ).

Corrected by Michel Marcus, Jul 18 2022

A353157 a(n) is the distance from n to the nearest integer whose binary expansion has no common 1-bits with that of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 5, 4, 3, 2, 1, 1, 3, 5, 7, 9, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25

Offset: 0

Rémy Sigrist, Apr 27 2022

Equivalently the distance to the nearest integer that can be added without carries in base 2.

			For n = 42 ("101010" in binary):
- 21 ("10101") is the greatest number <= 42 that has no common 1-bits with 42,
- 128 ("1000000") is the least number >= 42 that has no common 1-bits with 42,
- so a(42) = min(42-21, 128-42) = min(21, 86) = 21.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A006257, A020988, A097110, A080079, A108019, A353158.

a(n) = { my (high=2^#binary(n), low=high-1-n); min(n-low, high-n) }

a(n) = min(A006257(n), A080079(n)) for any n > 0.
a(n) = 1 iff n belongs to A097110.
a(n) = n/2 iff n belongs to A020988.
a(n) = n/4 iff n belongs to A108019.
2*a(n) - a(2*n) = 0 or 1.

A358550 Depth of the ordered rooted tree with binary encoding A014486(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 4, 4, 5, 2, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 4, 4, 5, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 2, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 4, 4, 5, 3, 3, 3, 3, 4, 3, 3, 3

Offset: 1

Gus Wiseman, Nov 22 2022

nonn

The binary encoding of an ordered tree (A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

			The first few rooted trees in binary encoding are:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   44: (o(o))
   50: ((o)o)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  172: (oo(o))
  178: (o(o)o)
  180: (o(oo))
  184: (o((o)))

Positions of first appearances are A014137.
Leaves of the ordered tree are counted by A057514, standard A358371.
Branches of the ordered tree are counted by A057515.
Edges of the ordered tree are counted by A072643.
The Matula-Goebel number of the ordered tree is A127301.
Positions of 2's are A155587, indices of A020988.
The standard ranking of the ordered tree is A358523.
Nodes of the ordered tree are counted by A358551, standard A358372.
For standard instead of binary encoding we have A358379.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists all binary encodings.
Cf. A001263, A057122, A358373, A358505, A358524.

binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[Depth[bint[k]]-1,{k,Select[Range[0,1000],binbalQ]}]

A364214 Numbers whose canonical representation as a sum of distinct Jacobsthal numbers (A280049) is palindromic.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 12, 15, 18, 21, 22, 30, 34, 42, 44, 49, 58, 63, 66, 71, 80, 85, 86, 102, 110, 126, 130, 146, 154, 170, 172, 183, 198, 209, 218, 229, 244, 255, 258, 269, 284, 295, 304, 315, 330, 341, 342, 374, 390, 422, 430, 462, 478, 510, 514, 546, 562, 594

Offset: 1

Amiram Eldar, Jul 14 2023

The even-indexed Jacobsthal numbers A001045(2*n) = A002450(n) = (4^n-1)/3, for n >= 1, are terms since their representation is 2*n-1 1's.
A001045(2*n+1) - 1 = A020988(n) = (2/3)*(4^n-1) is a term for n >= 1, since its representation is 2*n 1's.
A001045(n) + 1 = A128209(n) is a term for n >= 0, since its representation for n = 0 is 1 and its representation for n >= 1 is n-1 0's between 2 1's.
A160156(n) is a term for n >= 0 since its representation is n 0's interleaved with n+1 1's.

			The first 10 terms are:
   n  a(n)  A280049(a(n))
  --  ----  -------------
   1     1              1
   2     2             11
   3     4            101
   4     5            111
   5     6           1001
   6    10           1111
   7    12          10001
   8    15          10101
   9    18          11011
  10    21          11111

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001045, A002450, A003159, A020988, A280049.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341.

Position[Select[Range[1000], EvenQ[IntegerExponent[#, 2]] &], _?(PalindromeQ[IntegerDigits[#, 2]] &)] // Flatten

s(n) = if(n < 2, n > 0, n = s(n-1); until(valuation(n, 2)%2 == 0, n++); n); \\ A003159
is(n) = {my(d = binary(s(n))); d == Vecrev(d);}

A014131 a(n) = 2a(n-1) if n odd else 2a(n-1) + 6.

Original entry on oeis.org

0, 6, 12, 30, 60, 126, 252, 510, 1020, 2046, 4092, 8190, 16380, 32766, 65532, 131070, 262140, 524286, 1048572, 2097150, 4194300, 8388606, 16777212, 33554430, 67108860, 134217726, 268435452, 536870910

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 1, -2).

Cf. A000975.

[2^(n+2)-3-(-1)^n: n in [0..30]]; // Vincenzo Librandi, Apr 03 2012

Table[2^(n+2)-3-(-1)^n,{n,0,40}] (* or *) CoefficientList[Series[6x/((1-2x)(1-x)(1+x)),{x,0,30}],x] (* Vincenzo Librandi, Apr 03 2012 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],2a,2a+6]}; NestList[nxt,{1,0},30][[;;,2]] (* or *) LinearRecurrence[ {2,1,-2},{0,6,12},30] (* Harvey P. Dale, Aug 26 2024 *)

a(n) = 3*A026644(n), n > 0. [moved from A020988 by R. J. Mathar, Oct 21 2008]
From R. J. Mathar, Oct 21 2008: (Start)
G.f.: 6x/((1-2x)(1-x)(1+x)).
a(n) = 2^(n+2) - 3 - (-1)^n. (End)

A120462 Expansion of -2x(-3-2x+4x^2) / ((x-1)(2x+1)(2x-1)*(1+x)).

Original entry on oeis.org

0, 6, 4, 22, 20, 86, 84, 342, 340, 1366, 1364, 5462, 5460, 21846, 21844, 87382, 87380, 349526, 349524, 1398102, 1398100, 5592406, 5592404, 22369622, 22369620, 89478486, 89478484, 357913942, 357913940, 1431655766, 1431655764, 5726623062

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jun 28 2006

Top element of the vector obtained by multiplying the n-th power of the 6 X 6 matrix [[0, 1, 0, 0, 0, 1], [1, 0, 1, 0, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 0, 1], [1, 0, 0, 0, 1, 0]] by the column vector [0, 1, 1, 2, 3, 5].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

M = {{0, 1, 0, 0, 0, 1}, {1, 0, 1, 0, 0, 0}, {0, 1, 0, 1, 0, 0}, {0, 0, 1, 0, 1, 0}, {0, 0, 0, 1, 0, 1}, {1, 0, 0, 0, 1, 0}} v[1] = {0, 1, 1, 2, 3, 5} v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]
LinearRecurrence[{0,5,0,-4},{0,6,4,22},40] (* Harvey P. Dale, Jul 28 2024 *)

concat(0, Vec(2*x*(3+2*x-4*x^2)/((1-x)*(1+x)*(1-2*x)*(1+2*x)) + O(x^40))) \\ Colin Barker, Sep 09 2016

a(2*n+1) = A047849(n+2). a(2*n)= 2*A020988(n). - R. J. Mathar, Nov 07 2011
From Colin Barker, Sep 09 2016: (Start)
a(n) = -2*(1/6 + (-2)^n/3 + (-1)^n/2 - 2^n).
a(n) = 5*a(n-2)-4*a(n-4) for n>3.
(End)

A140950 a(n) = A140944(n+1) - 3*A140944(n).

Original entry on oeis.org

1, -3, -1, 5, -6, 3, -11, 10, -12, -5, 21, -22, 20, -24, 11, -43, 42, -44, 40, -48, -21, 85, -86, 84, -88, 80, -96, 43, -171, 170, -172, 168, -176, 160, -192, -85, 341, -342, 340, -344, 336, -352, 320, -384, 171, -683, 682, -684, 680, -688

Offset: 0

Paul Curtz, Jul 25 2008

Jacobsthal numbers appear twice: 1) A001045(n+2) signed, terms 0, 1, 3, 6, 10 (A000217); 2) A001045(n+1) signed, terms 0, 2, 5, 9 (n*(n+3)/2=A000096); between them are -3; 5, -6; -11, 10, -12; which appear (opposite sign) by rows in A140503 (1, -1, 2, 3, -2, 4) square.
Consider the permutation of the nonnegative numbers
0, 2, 5,  9, 14, 20, 27,
1, 3, 6, 10, 15, 21, 28,
   4, 7, 11, 16, 22, 29,
      8, 12, 17, 23, 30,
         13, 18, 24, 31,
             19, 25, 32,
                 26, 33,
                     34, etc.
The corresponding distribution of a(n) is
1,  -1,   3,  -5,  11, -21,   43,
-3,  5, -11,  21, -43,  85, -171,
    -6,  10, -22,  42, -86,  170,
        -12,  20, -44,  84, -172,
             -24,  40, -88,  168,
                  -48,  80, -176,
                       -96,  160,
                            -192, etc.
Column sums: -2, -2, -10, -10, -42, -42, -170, ... duplicate of a bisection of -A078008(n+2).
b(n)= 1, -1, 3, -5, 11, 21, ... = (-1)^n*A001045(n+1) = A077925(n). Every row is b(n) or b(n+2) multiplied by 1, -1, -2, -4, -8, -16, ..., essentially -A011782(n).

Cf. A000096, A000217, A005015, A001045, A007283, A020988, A077925, A078008, A140503, A146523, A151575, A175805, A084247.

T[0, 0] = 0; T[1, 0] = T[0, 1] = 1; T[0, n_] := T[0, n] = T[0, n - 1] + 2*T[0, n - 2]; T[d_, d_] = 0; T[d_, n_] := T[d, n] = T[d - 1, n + 1] - T[d - 1, n]; A140944 = Table[T[d, n], {d, 0, 10}, {n, 0, d}] // Flatten; a[n_] := A140944[[n + 2]] - 3*A140944[[n + 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 18 2014 *)

More terms and a(19)=-48 instead of 42 corrected by Jean-François Alcover, Dec 22 2014

A165275 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-power summands in its base 2 representation.

Original entry on oeis.org

1, 4, 2, 5, 3, 10, 16, 6, 11, 42, 17, 7, 14, 43, 170, 20, 8, 15, 46, 171, 682, 21, 9, 26, 47, 174, 683, 2730, 64, 12, 27, 58, 175, 686, 2731, 10922, 65, 13, 30, 59, 186, 687, 2734, 10923, 43690, 68, 18, 31, 62, 187, 698, 2735, 10926, 43691, 174762, 69, 19, 34

Offset: 1

Clark Kimberling, Sep 12 2009

For n>=0, row n is the ordered sequence of positive integers m such that the number of odd powers of 2 in the base 2 representation of m is n.
Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
For even powers, see A165274. For the number of even powers of 2 in the base 2 representation of n, see A139351; for odd, see A139352.
Essentially, (Row 0)=A000695, (Column 1)=A020988, also possibly (Column 2)=A007583.
It appears that, for n>=3, a(t(n)) = 4*a(t(n-1))+2, where t(n) is the n-th triangular number t(n)=n(n+1)/2 (A000217). [John W. Layman, Sep 15 2009]

			Northwest corner:
  1....4....5...16...17...20...21...64
  2....3....6....7....8....9...12...13
  10..11...14...26...27...30...31...34
  42..43...46...47...58...59...62...63
Examples:
20 = 16 + 4 = 2^4 + 2^2, so that 20 is in row 0.
13 = 8 + 4 + 1 = 2^3 + 2^2 + 2^0, so that 13 is in row 1.

Cf. A139351, A139352, A165274, A165276, A165277, A165278, A165279.

Cf. A000217.

f[n_] := Total[(Reverse@IntegerDigits[n, 2])[[2 ;; -1 ;; 2]]]; T = GatherBy[ SortBy[Range[10^5], f], f]; Table[Table[T[[n - k + 1, k]], {k, n, 1, -1}], {n, 1, Length[T]}] // Flatten (* Amiram Eldar, Feb 04 2020*)

a(27) corrected and a(28)-a(54) added by John W. Layman, Sep 15 2009

More terms from Amiram Eldar, Feb 04 2020

A167193 a(n) = (1/3)(1 - (-2)^n + 3(-1)^n ) = (-1)^(n+1)*A167030(n).

Original entry on oeis.org

1, 0, 0, 2, -4, 10, -20, 42, -84, 170, -340, 682, -1364, 2730, -5460, 10922, -21844, 43690, -87380, 174762, -349524, 699050, -1398100, 2796202, -5592404, 11184810, -22369620, 44739242, -89478484, 178956970, -357913940, 715827882, -1431655764, 2863311530, -5726623060, 11453246122, -22906492244, 45812984490

Offset: 0

Paul Curtz, Oct 30 2009

This is the inverse binomial transform of 1, 1, 1, 3, 5, 11,.. (continued as in A001045 and conjectured to be equal to A152046).

Any sequence (like this one) which obeys a(n)= -2a(n-1)+a(n-2)+2a(n-3) also obeys a(n)=5a(n-2)-4a(n-4), proved by telescoping; see A101622.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,1,2).

[( 1-(-1)^n*2^n)/3+(-1)^n: n in [0..40] ]; // Vincenzo Librandi, Aug 06 2011

LinearRecurrence[{-2,1,2},{1,0,0}, 25] (* or *) Table[(1/3)*(1 + 3*(-1)^n - (-2)^n), {n,0,25}] (* G. C. Greubel, Jun 04 2016 *)

a(2n) = (-1)^n* A084240(n). a(2n+1) = A020988(n).
G.f.: ( -1 - 2*x + x^2 ) / ( (x-1)*(1+2*x)*(1+x) ).
a(n) = -a(n-1) + 2*a(n-2) - 2*(-1)^n.
a(n) = -2*a(n-1) + a(n-2) + 2*a(n-3).
E.g.f.: (1/3)*(exp(x) + 3*exp(-x) - exp(-2*x)). - G. C. Greubel, Jun 04 2016

A330720 a(n) is the number of ways of writing the binary expansion of n as a product (or concatenation) of nonpalindromes.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 0, 2, 1, 1, 2, 2, 0, 1, 1, 1, 0, 1, 3, 3, 3, 3, 3, 3, 2, 2, 3, 4, 2, 2, 2, 3, 1, 1, 3, 2, 1, 2, 2, 3, 1, 1, 2, 2, 1, 1, 1, 1, 0, 1, 4, 4, 4, 3, 5, 5, 3, 3, 4, 4, 4, 4, 5, 5, 2, 2, 5, 4, 4, 4, 0, 4

Offset: 0

Rémy Sigrist, Dec 28 2019

This sequence is a variant of A215244.

			For n = 41:
- the binary expansion of 41 is "101001",
- the possible products of nonpalindromes are "101001", "1010"."01", and "10"."10"."01",
- hence a(41) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A020988, A154809, A215244, A301453.

ispali:= proc(L) L = ListTools:-Reverse(L) end proc:
g:= proc(L) option remember; local m;
    add(procname(L[m+1..-1]), m= remove(t -> ispali(L[1..t]),[$1..nops(L)]))
end proc:
g([]):= 1:
seq(g(convert(n,base,2)),n=0..100); # Robert Israel, Dec 29 2019

a(n) = my (b=binary(n), v=b!=Vecrev(b)); for (s=1, #b, my (z=b[s..#b]); if (z!=Vecrev(z), v+=a(fromdigits(b[1..s-1],2)))); v

a(2^k-1) = 0 for any k >= 0.

a(A020988(k+1)) = 2^k for any k >= 0.

cp's OEIS Frontend

A351122 Irregular triangle read by rows in which row n lists the number of divisions by 2 after tripling steps in the Collatz 3x+1 trajectory of 2n+1 until it reaches 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A353157 a(n) is the distance from n to the nearest integer whose binary expansion has no common 1-bits with that of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A358550 Depth of the ordered rooted tree with binary encoding A014486(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A364214 Numbers whose canonical representation as a sum of distinct Jacobsthal numbers (A280049) is palindromic.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A014131 a(n) = 2*a(n-1) if n odd else 2*a(n-1) + 6.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A120462 Expansion of -2*x*(-3-2*x+4*x^2) / ((x-1)*(2*x+1)*(2*x-1)*(1+x)).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula

A140950 a(n) = A140944(n+1) - 3*A140944(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A165275 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-power summands in its base 2 representation.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A014131 a(n) = 2a(n-1) if n odd else 2a(n-1) + 6.

A120462 Expansion of -2x(-3-2x+4x^2) / ((x-1)(2x+1)(2x-1)*(1+x)).

A167193 a(n) = (1/3)(1 - (-2)^n + 3(-1)^n ) = (-1)^(n+1)*A167030(n).