A114752 -id:A114752 - cp's OEIS frontend

A133566 Triangle read by rows: (1,1,1,...) on the main diagonal and (0,1,0,1,...) on the subdiagonal.

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Sep 16 2007

Usually regarded as a square matrix T when combined with other matrices and column vectors.
Then T * V, where V = any sequence regarded as a column vector with offset 1 is a new sequence S [called an interpolation transform] given by S(2n) = V(2n), S(2n-1) = V(2n) + V(2n-1). Example: If T * [1,2,3,...], S = [1, 2, 5, 4, 9, 6, 13, 8, 17, ...) = A114752. A133080 is identical to A133566 except that the subdiagonal = (1,0,1,0,...). A133080 * [1,2,3,...] = A114753: (1, 3, 3, 7, 5, 11, 7, 15, 9, 19, ...).
Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,-1,0,0,0,0,0,0,...] DELTA [1,0,-2,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 15 2007

			First few rows of the triangle:
  1;
  0, 1;
  0, 1, 1;
  0, 0, 0, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 1;
  ...

Cf. A133080, A114752, A114753, A084938, A040001.

A133566 := proc(n,k)
    if n = k then
        1;
    elif  k=n-1 and type(n,odd) then
        1;
    else
        0 ;
    end if;
end proc: # R. J. Mathar, Jun 20 2015

T[n_, k_] := Which[n == k, 1, k == n - 1 && OddQ[n], 1, True, 0];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 24 2023 *)

Odd rows: (n-2) zeros followed by 1, 1. Even rows: (n-1) zeros followed by 1.
Sum_{k=0..n} T(n,k) = A040001(n). - Philippe Deléham, Dec 15 2007
G.f.: (-1-x*y-x^2*y)*x*y/((-1+x*y)*(1+x*y)). - R. J. Mathar, Aug 11 2015

Entry revised by N. J. A. Sloane, Jun 20 2015

A114753 First column of A114751.

Original entry on oeis.org

1, 3, 3, 7, 5, 11, 7, 15, 9, 19, 11, 23, 13, 27, 15, 31, 17, 35, 19, 39, 21, 43, 23, 47, 25, 51, 27, 55, 29, 59, 31, 63, 33, 67, 35, 71, 37, 75, 39, 79, 41, 83, 43, 87, 45, 91, 47, 95, 49, 99, 51, 103, 53, 107, 55, 111, 57, 115, 59, 119, 61, 123, 63, 127, 65, 131, 67, 135, 69

Offset: 1

Amarnath Murthy, Nov 15 2005

A114752, (1, 2, 5, 4, 9, 6, ...) + A114753 - (1,1,1,...) = 3n+1: (1, 4, 7, 10, 13, ...). - Gary W. Adamson, Sep 16 2007

First column of table A210530. - Boris Putievskiy, Jan 29 2013

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A114751, A114752.

Cf. A133080, A210530.

Table[If[OddQ[n],n,2n-1],{n,1,120}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011*)

a(2n+1) = 2n+1, a(2n) = 4n-1.
a(n) = 2*a(n-2) - a(n-4). - Joerg Arndt, Apr 02 2011
Equals A133080 * [1,2,3,...]. - Gary W. Adamson, Sep 08 2007
G.f. x*(1+3*x+x^2+x^3) / ( (x-1)^2*(1+x)^2 ). - R. J. Mathar, Apr 04 2012
a(n) = (3*n-1-(n-1)*(-1)^(n-1))/2. - Boris Putievskiy, Jan 29 2013

More terms from Joshua Zucker, May 05 2006

A178414 Least odd number in the Collatz (3x+1) preimage of odd numbers not a multiple of 3.

Original entry on oeis.org

1, 3, 9, 7, 17, 11, 25, 15, 33, 19, 41, 23, 49, 27, 57, 31, 65, 35, 73, 39, 81, 43, 89, 47, 97, 51, 105, 55, 113, 59, 121, 63, 129, 67, 137, 71, 145, 75, 153, 79, 161, 83, 169, 87, 177, 91, 185, 95, 193, 99, 201, 103, 209, 107, 217, 111, 225, 115, 233, 119, 241, 123, 249

Offset: 1

T. D. Noe, May 28 2010

The odd non-multiples of 3 are 1, 5, 7, 11,... (A007310). The odd multiples of 3 have no odd numbers their Collatz pre-image. The next odd number in the Collatz iteration of a(2n) is 6n-1. The next odd number in the Collatz iteration of a(2n+1) is 6n+1. For each non-multiple of 3, there are an infinite number of odd numbers in its Collatz pre-image. For example:
Odd pre-images of 1: 1, 5, 21, 85, 341,... (A002450)
Odd pre-images of 5: 3, 13, 53, 213, 853,... (A072197)
Odd pre-images of 7: 9, 37, 149, 597, 2389,...
Odd pre-images of 11: 7, 29, 117, 469, 1877,...(A072261)
In each case, the pre-image sequence is t(k+1) = 4*t(k) + 1 with t(0)=a(n). The array of pre-images is in A178415.
a(n) = A047529(P(n)), with the permutation P(n) = A006368(n-1) + 1, for n >= 1. This shows that this sequence gives the numbers {1, 3, 7} (mod 8) uniquely. - Wolfdieter Lang, Sep 21 2021

Matthew House, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A002450, A006368, A007310, A022998, A047529, A072197, A072261, A114752.

Riffle[1+8*Range[0,50], 3+4*Range[0,50]]

a(n) = (n - 1)*(3 - (-1)^n) + 1. [Bogart B. Strauss, Sep 20 2013, adapted to the offset by Matthew House, Feb 14 2017]
From Matthew House, Feb 14 2017: (Start)
G.f.: x*(1 + 3*x + 7*x^2 + x^3)/((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4). (End)
From Philippe Deléham, Nov 06 2023: (Start)
a(2*n) = 4*n-1, a(2*n+1) = 8*n+1.
a(n) = 2*A022998(n-1)+1.
a(n) = 2*A114752(n)-1. (End)

A114751 The following triangle contains n consecutive numbers beginning from n in ascending order if n is odd else in descending order. Sequence contains the triangle by rows.

Original entry on oeis.org

1, 3, 2, 3, 4, 5, 7, 6, 5, 4, 5, 6, 7, 8, 9, 11, 10, 9, 8, 7, 6, 7, 8, 9, 10, 11, 12, 13, 15, 14, 13, 12, 11, 10, 9, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12

Offset: 1

Amarnath Murthy, Nov 15 2005

			1
3 2
3 4 5
7 6 5 4
5 6 7 8 9
11 10 9 8 7 6
...

Cf. A114752, A114753, A210530.

for n from 1 to 14 do if n mod 2 = 1 then print(seq(k,k=n..2*n-1)) else print(seq(2*n-k,k=1..n)) fi od; # yields sequence in triangular form # Emeric Deutsch, Jan 26 2006

a(n) = (3*t+2-t*(-1)^(t-1))/2-(1+(-1)^t)*(j-1)/2+(1-(-1)^t)*(j-1)/2, where j = (t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Jan 30 2013

More terms from Emeric Deutsch, Jan 26 2006

A133087 A133080 * A007318.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 5, 4, 1, 1, 4, 6, 4, 1, 2, 9, 16, 14, 6, 1, 1, 6, 15, 20, 15, 6, 1, 2, 13, 36, 55, 50, 27, 8, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 2, 17, 64, 140, 196, 182, 112, 44, 10, 1

Offset: 0

Gary W. Adamson, Sep 08 2007

Row sums = A084221: (1, 3, 4, 12, 16, 48, 64, 192, ...).

Subtriangle of (0, 2, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012

			First few rows of the triangle:
  1;
  2,  1;
  1,  2,  1;
  2,  5,  4,  1;
  1,  4,  6,  4,  1;
  2,  9, 16, 14,  6,  1;
  1,  6, 15, 20, 15,  6,  1;
  2, 13, 36, 55, 50, 27,  8,  1;
  1,  8, 28, 56, 70, 56, 28,  8,  1;
  ...
Triangle (0, 2, -3/2, -1/2, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0,  2,  1;
  0,  1,  2,  1;
  0,  2,  5,  4,  1;
  0,  1,  4,  6,  4,  1;
  0,  2,  9, 16, 14,  6,  1;
  0,  1,  6, 15, 20, 15,  6,  1;
  0,  2, 13, 36, 55, 50, 27,  8,  1;
  0,  1,  8, 28, 56, 70, 56, 28,  8,  1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A133080, A084221.

Columns: A000007, A114752, A133092.

CoefficientList[CoefficientList[Series[(1 + 2*x + y*x)/(1 - (1 + y)^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 21 2017 *)

A133080 * A007318 as infinite lower triangular matrices.
G.f.: (1+2*x+y*x)/(1-(1+y)^2*x^2). - Philippe Deléham, Mar 03 2012
T(n,k) = T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 2, T(1,1) = 1. - Philippe Deléham, Mar 03 2012
Sum_{k=0..n} T(n,k)*x^k = A059841(n), A019590(n+1), A000034(n), A084221(n), A133125(n) for x = -2, -1, 0, 1, 2 respectively. - Philippe Deléham, Mar 03 2012

A321220 a(n) = n+2 if n is even, otherwise a(n) = 2*n+1 if n is odd.

Original entry on oeis.org

2, 3, 4, 7, 6, 11, 8, 15, 10, 19, 12, 23, 14, 27, 16, 31, 18, 35, 20, 39, 22, 43, 24, 47, 26, 51, 28, 55, 30, 59, 32, 63, 34, 67, 36, 71, 38, 75, 40, 79, 42, 83, 44, 87, 46, 91, 48, 95, 50, 99, 52, 103, 54, 107, 56, 111, 58, 115, 60, 119, 62, 123, 64, 127, 66

Offset: 0

Michel Marcus, Oct 31 2018

For n >= 3, a(n) is the Harborth Constant for the Dihedral groups D2n. See Balachandra link, Theorem 1 p. 2.

Colin Barker, Table of n, a(n) for n = 0..1000
Niranjan Balachandran, Eshita Mazumdar, Kevin Zhao, The Harborth Constant for Dihedral Groups, arXiv:1803.08286 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

A299174 and A004767 interleaved.

Cf. A043547, A114752.

[IsOdd(n) select (2*n+1) else n+2: n in [0..80]]; // Vincenzo Librandi, Nov 01 2018

a:=n->`if`(modp(n,2)=0,n+2,2*n+1): seq(a(n),n=0..70); # Muniru A Asiru, Oct 31 2018

CoefficientList[Series[(2 + 3 x + x^3)/(1 - x^2)^2, {x, 0, 64}], x] (* Michael De Vlieger, Oct 31 2018 *)
Table[If[OddQ[n], (2 n + 1), n + 2], {n, 0, 80}] (* Vincenzo Librandi, Nov 01 2018 *)

PARI
```
a(n) = if (n%2, 2*n+1, n+2);
```

Vec((2 + 3*x + x^3) / ((1 - x)^2*(1 + x)^2) + O(x^80)) \\ Colin Barker, Oct 31 2018

a(n) = A043547(n+1) + 1.
From Colin Barker, Oct 31 2018: (Start)
G.f.: (2 + 3*x + x^3) / (1-x^2)^2.
a(n) = 2*a(n-2) - a(n-4) for n > 3.
(End)

cp's OEIS Frontend

A133566 Triangle read by rows: (1,1,1,...) on the main diagonal and (0,1,0,1,...) on the subdiagonal.

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A114753 First column of A114751.

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A178414 Least odd number in the Collatz (3x+1) preimage of odd numbers not a multiple of 3.

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A114751 The following triangle contains n consecutive numbers beginning from n in ascending order if n is odd else in descending order. Sequence contains the triangle by rows.

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A133087 A133080 * A007318.

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A321220 a(n) = n+2 if n is even, otherwise a(n) = 2*n+1 if n is odd.

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