A083706 -id:A083706 - cp's OEIS frontend

A130295 Erroneous duplicate of A125026.

1, 3, 2, 5, 7, 3, 7, 15, 13, 4, 9, 26, 34, 21, 5, 11, 40, 70, 65, 31, 6, 13, 57, 125, 155, 111, 43, 7, 15, 77, 203, 315, 301, 175, 57, 8, 17, 100, 308, 574, 686, 532, 260, 73, 9, 19, 126, 444, 966, 1386, 1344, 876369, 91, 10

Offset: 1

Gary W. Adamson, May 20 2007

dead

This sequence was initially defined as "A051340 * A007318". However, the matrix product A051340 * A007318 is not well defined, because all elements of A051340 are strictly positive integers, as are all elements of the lower left of A007318. Therefore the matrix A051340 must be truncated to its lower left (setting A[i,j]=0 if j>i), which actually equals A130296. Then the product yields this sequence, which is identical to A125026.

Row sums = A099035 (not A083706 as stated initially): (1, 5, 15, 39, 95, 223, 511, ...).

			First few rows of the triangle A125026:
   1;
   3,  2;
   5,  7,   3;
   7, 15,  13,   4;
   9, 26,  34,  21,   5;
  11, 40,  70,  65,  31,  6;
  13, 57, 125, 155, 111, 43, 7;
  ...

Cf. A051340, A099035.

(A051340) * A007318 as infinite lower triangular matrices. [Here (A051340) is that matrix with the upper right triangle set to zero, which is actually A130296. - M. F. Hasler, Aug 15 2015]

Restored and edited by M. F. Hasler, Aug 15 2015

A131817 a(n) = A051340(n) + A130321(n) - A000012(n).

Original entry on oeis.org

1, 2, 2, 4, 2, 3, 8, 4, 2, 4, 16, 8, 4, 2, 5, 32, 16, 8, 4, 2, 6, 64, 32, 16, 8, 4, 2, 7, 128, 64, 32, 16, 8, 4, 2, 8, 256, 128, 64, 32, 16, 8, 4, 2, 9, 512, 256, 128, 64, 32, 16, 8, 4, 2, 10, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 11, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 12

Offset: 0

Gary W. Adamson, Jul 18 2007

			First few rows of the triangle:
   1;
   2,  2;
   4,  2,  3;
   8,  4,  2,  4;
  16,  8,  4,  2,  5;
  32, 16,  8,  4,  2,  6;
  64, 32, 16,  8,  4,  2,  7;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)

Cf. A051340, A083706 (row sums), A130321.

Array[Append[ConstantArray[1, # - 1], #] + 2^Range[# - 1, 0, -1] - 1 &, 12] // Flatten (* Michael De Vlieger, Aug 19 2021 *)

A051340 + A130321 - A000012 as infinite lower triangular matrices.

More terms from Jinyuan Wang, Aug 19 2021

A179571 Number of permutations of 1..n+4 with the number moved left exceeding the number moved right by n.

Original entry on oeis.org

31, 66, 134, 267, 529, 1048, 2080, 4137, 8243, 16446, 32842, 65623, 131173, 262260, 524420, 1048725, 2097319, 4194490, 8388814, 16777443, 33554681, 67109136, 134218024, 268435777, 536871259, 1073742198, 2147484050, 4294967727

Offset: 1

R. H. Hardin, g.f. from R. J. Mathar in the Sequence Fans Mailing List, Jul 19 2010

nonn

Recurrence would also extend to an a(0) if the definition were made to exclude the identity permutation.

R. H. Hardin, Table of n, a(n) for n = 1..99

Cf. A083706.

Drop[Accumulate[Table[BitSet[n, (n + 2)], {n, 0, 100}]], 2] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)

Empirical: a(n)=5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4) ; G.f.: -x*(-31+89*x-83*x^2+26*x^3) / ( (2*x-1)*(x-1)^3 ).

Empirical: a(n) = (n^2+3*n-6)/2 +2^(n+4) = 2^(n+4)+A046691(n-1). - R. J. Mathar, May 26 2016

A343949 Shortest distance from curve start to end along the segments of dragon curve expansion level n, and which is the diameter of the curve as a graph.

Original entry on oeis.org

1, 2, 4, 8, 12, 18, 26, 36, 52, 70, 102, 136, 200, 266, 394, 524, 780, 1038, 1550, 2064, 3088, 4114, 6162, 8212, 12308, 16406, 24598, 32792, 49176, 65562, 98330, 131100, 196636, 262174, 393246, 524320, 786464, 1048610, 1572898, 2097188, 3145764, 4194342, 6291494

Offset: 0

Kevin Ryde, May 05 2021

Expansion level n is the first 2^n segments of the curve, and can be taken as a graph with visited points as vertices and segments as edges.

			Curve n=4:
     *--*  *--*
     |  |  |  |        Start S to end E along segments.
     *--*--*  *--*     Distance a(4) = 12,
        |        |     which is also graph diameter.
  E  *--*     S--*
  |  |
  *--*

Kevin Ryde, Table of n, a(n) for n = 0..1000
Kevin Ryde, Iterations of the Dragon Curve, see index "Diameter".
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-2,2).

Cf. A275970, A083706, A228693.

Cf. A332383, A332384 (curve coordinates).

a(n) = if(n==0,1, my(t=n%2); (3+t)<<(n>>1) + n-4 + t);

a(0) = 1.
a(2*n)   = 3*2^n + 2*n - 4 = 2*A275970(n-1), for n>=1.
a(2*n+1) = 4*2^n + 2*n - 2 = 2*A083706(n).
a(n+1) - a(n) = 2*A228693(n), for n>=1.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 2*a(n-4) + 2*a(n-5) for n >= 6.
G.f.: (1 + x - x^2 + x^3 - 4*x^5) / ((1+x) * (1-x)^2 * (1-2*x^2)).
G.f.: 2 - (1/2)/(1+x) - (9/2)/(1-x) + 1/(1-x)^2  + (3 + 4*x)/(1 - 2*x^2).

A091264 Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 3, 2, 2, 7, 4, 3, 3, 15, 8, 5, 4, 4, 31, 16, 9, 6, 5, 5, 63, 32, 17, 10, 7, 6, 6, 127, 64, 33, 18, 11, 8, 7, 7, 255, 128, 65, 34, 19, 12, 9, 8, 8, 511, 256, 129, 66, 35, 20, 13, 10, 9, 9, 1023, 512, 257, 130, 67, 36, 21, 14, 11, 10, 10, 2047, 1024, 513, 258, 131, 68, 37, 22

Offset: 0

Ross La Haye, Feb 23 2004

			{0};
{1,1};
{3,2,2};
{7,4,3,3};
{15,8,5,4,4};
{31,16,9,6,5,5};
{63,32,17,10,7,6,6};
a(5,3) = 34 because 2^5 + (3-1) = 34.

Rows: a(0, k) = A001477(k), a(1, k) = A000027(k+1) etc. etc. Columns: a(n, 0) = A000225(n). a(n, 1) = A000079(n). a(n, 2) = A000051(n). a(n, 3) = A052548(n). a(n, 4) = A062709(n). Diagonals: a(n, n+3) = A052968(n+1). a(n, n+2) = A005126(n). a(n, n+1) = A006127(n). a(n, n) = A052944(n). a(n, n-1) = A083706(n-1). Also note that the sums of the antidiagonals = the partial sums of the main diagonal, i.e., a(n, n).

Flatten[ Table[ Table[ a[i, n - i], {i, n, 0, -1}], {n, 0, 11}]] (* both from Robert G. Wilson v, Feb 26 2004 *)
Table[a[n, k], {n, 0, 10}, {k, 0, 10}] // TableForm (* to view the table *)

For k > 0, a(n, k)= a(n, k-1) + 1.

a(n, k) = 2^n + (k-1).

More terms from Robert G. Wilson v, Feb 23 2004

cp's OEIS Frontend

A130295 Erroneous duplicate of A125026.

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A131817 a(n) = A051340(n) + A130321(n) - A000012(n).

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A179571 Number of permutations of 1..n+4 with the number moved left exceeding the number moved right by n.

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Mathematica

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A343949 Shortest distance from curve start to end along the segments of dragon curve expansion level n, and which is the diameter of the curve as a graph.

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A091264 Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.

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Mathematica

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Extensions