A113127 -id:A113127 - cp's OEIS frontend

A181527 Binomial transform of A113127; (1, 1, 3, 7, 15, 31, ...) convolved with (1, 3, 7, 15, 31, 63, ...).

1, 4, 13, 38, 103, 264, 649, 1546, 3595, 8204, 18445, 40974, 90127, 196624, 426001, 917522, 1966099, 4194324, 8912917, 18874390, 39845911, 83886104, 176160793, 369098778, 771751963, 1610612764, 3355443229, 6979321886, 14495514655, 30064771104, 62277025825

Offset: 0

Gary W. Adamson, Oct 26 2010

A181527 = Partial sums of (A002064 Cullen numbers: n*2^n+1). - Vladimir Joseph Stephan Orlovsky, Jul 09 2011

Form a triangle with T(1,1) = n, T(2,1) = T(2,2) = n-1, T(3,1) = T(3,3) = n-2, ..., T(n,1) = T(n,n) = 1. The interior members are T(i,j) = T(i-1,j-1) + T(i-1,j). The sum of all members for a triangle of size n is a(n-1). Example for n = 5: row(1) = 5; row(2) = 4, 4; row(3) = 3, 8, 3; row(4) = 2, 11, 11, 2; row(5) = 1, 13, 22, 13, 1. The sum of all members is 103 = a(4). - J. M. Bergot, Oct 16 2012

			a(4) = 103 = (1, 1, 3, 7, 15) dot (31, 15, 7, 3, 1) = (31 + 15 + 21, + 21 + 15)
a(3) = 38 = (1, 3, 3, 1) dot (1, 3, 6, 10) = (1 + 9 + 18 + 10).

Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A002064, A113127.

Accumulate[Table[n*2^n + 1, {n, 0, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jul 09 2011 *)
LinearRecurrence[{6,-13,12,-4},{1,4,13,38},40] (* Harvey P. Dale, Apr 14 2016 *)

Binomial transform of A113127; (1, 3, 7, 15, 31, ...) convolved with (1, 1, 3, 7, 15, 31, ...).
From R. J. Mathar, Oct 30 2010: (Start)
a(n) = 3+  n + 2^(n+1)*(n-1) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4).
G.f.: ( 1-2*x+2*x^2 ) / ( (2*x-1)^2*(x-1)^2 ). (End)

A167381 The numbers read down the left-center column of an arrangement of the natural numbers in square blocks.

Original entry on oeis.org

1, 3, 6, 10, 14, 18, 23, 29, 35, 41, 47, 53, 60, 68, 76, 84, 92, 100, 108, 116, 125, 135, 145, 155, 165, 175, 185, 195, 205, 215, 226, 238, 250, 262, 274, 286, 298, 310, 322, 334, 346, 358, 371, 385, 399, 413, 427, 441, 455, 469, 483, 497, 511, 525, 539, 553

Offset: 1

Paul Curtz, Nov 02 2009

The natural numbers are filled into square blocks of edge length 2, 4, 6, 8, ...
by taking A016742(n+1) = 4, 16, 36, ... at a time:
.......1..2......
.......3..4......
....5..6..7..8...
....9.10.11.12...
...13.14.15.16...
...17.18.19.20...
21.22.23.24.25.26
27.28.29.30.31.32
33.34.35.36.37.38
39.40.41.42.43.44
Reading down the column just left from the center yields a(n).
The length of the rows is given by A001670.
The number of elements in each square block, 4, 16, 36, etc., are the first differences of A166464:
A016742(n) = A166464(n)-A166464(n-1).
Reading the blocks from right to left, row by row, we obtain a permutation of the integers, which starts similar to A166133.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A113127, A167991 (first differences).

r[1] = Range[4]; r[n_] := r[n] = Range[r[n-1][[-1]]+1, r[n-1][[-1]]+(2n)^2 ];
s[n_] := Partition[r[n], Sqrt[Length[r[n]]]][[All, n]];
A167381 = Table[s[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Mar 26 2017 *)
Module[{nn=7,c},c=TakeList[Range[(2/3)*nn(nn+1)(2*nn+1)],(2*Range[ nn])^2]; Table[Take[c[[n]],{n,-1,2*n}],{n,nn}]]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 18 2018 *)

Edited by R. J. Mathar, Aug 29 2010

More terms from Jean-François Alcover, Mar 26 2017

A113126 A simple 4-diagonal matrix.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 0, 2, 3, 4, 5, 0, 0, 3, 4, 5, 6, 0, 0, 0, 4, 5, 6, 7, 0, 0, 0, 0, 5, 6, 7, 8, 0, 0, 0, 0, 0, 6, 7, 8, 9, 0, 0, 0, 0, 0, 0, 7, 8, 9, 10, 0, 0, 0, 0, 0, 0, 0, 8, 9, 10, 11, 0, 0, 0, 0, 0, 0, 0, 0, 9, 10, 11, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 11, 12, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul Barry, Oct 14 2005

Row sums are A113127. Diagonal sums are A109613. Diagonals are A000027.

			Triangle begins
1;
1,2;
1,2,3;
1,2,3,4;
0,2,3,4,5;
0,0,3,4,5,6;
0,0,0,4,5,6,7;
0,0,0,0,5,6,7,8;
0,0,0,0,0,6,7,8,9;

Cf. A113125.

Number triangle where column k has g.f. (1+x+x^2+x^3)(k+1)x^k.

A133653 A007318^(-1) * A003261.

Original entry on oeis.org

1, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154

Offset: 1

Gary W. Adamson, Sep 19 2007

It appears this sequence gives the positive integers m such that the sum of the first m Fibonacci numbers divides their product. For example, if n=2 and m=a(2)=6, we have the sum 1+1+2+3+5+8=20 which clearly divides the corresponding product 480. See A175553 for the analogous sequence when using the triangular numbers. Sum_{k=1..n} Fibonacci(k) divides Product_{k=1..n} Fibonacci(k). - John W. Layman, Jul 10 2010

			a(4) = 14 = (1, 3, 3, 1) dot (1, 5, -1, 1) = (1, 15, -3, 1).

Cf. A003261, A007318, A175553.

Essentially the same as A130824, A113127, A111284, A073760, A016825.

Inverse binomial transform of A003261: (1, 7, 23, 63, 159, 383, ...).
Binomial transform of [1, 5, -1, 1, -1, 1, ...].
"1" followed by 2 * [3, 5, 7, 9, 11, ...].
O.g.f.: x*(1+4x-x^2)/(1-x)^2. a(n) = 4n-2, n > 1. - R. J. Mathar, Jun 08 2008
1/(1+1/(6+1/(10+1/(14+1/(...(continued fraction)))))) = (e-1)/2 with e = 2.718281...- Philippe Deléham, Mar 09 2013

More terms from R. J. Mathar, Jun 08 2008

A110190 Number of (1,0)-steps on the lines y=0 and y=1 in all Schroeder paths of length 2n (a Schroeder path of length 2n is a path from (0,0) to (2n,0), consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis).

Original entry on oeis.org

0, 1, 5, 24, 116, 568, 2820, 14184, 72180, 371112, 1925380, 10068728, 53023860, 280969560, 1497072132, 8016213960, 43114424308, 232817773640, 1261793848836, 6861179441880, 37421756333172, 204671007577464, 1122275850740996, 6168352091629864, 33977333521770996, 187539324760522728

Offset: 0

Emeric Deutsch, Jul 15 2005

nonn

			a(2)=5 because in the 6 (=A006318(2)) Schroeder paths of length 4, namely, HH, HUD, UDH, UDUD, UHD, UUDD, all 5 H-steps are at levels 0 or 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Cf. A006318, A110189.

R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*(1-z-2*z*R+z^2+2*z^2*R+z^2*R^2)/(1-3*z-z*R+z^2+z^2*R)^2: Gser:=series(G,z=0,30): 0,seq(coeff(Gser,z^n),n=1..26);

CoefficientList[Series[x*(1-x-2*x*((1-x-Sqrt[1-6*x+x^2])/(2*x))+x^2+2*x^2*((1-x-Sqrt[1-6*x+x^2])/(2*x))+x^2*((1-x-Sqrt[1-6*x+x^2])/(2*x))^2)/(1-3*x-x*((1-x-Sqrt[1-6*x+x^2])/(2*x))+x^2+x^2*((1-x-Sqrt[1-6*x+x^2])/(2*x)))^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)

a113127(n):=if n=0 then 1 else if n=1 then 3 else 4*n-2;
a(n):=sum((k+1)*sum(binomial(n+1,n-k-i)*binomial(n+i,n),i,0,n-k)/(n+1)*a113127(k),k,0,n); /* Vladimir Kruchinin, Mar 13 2016 */

x = 'x+O('x^66);
R = (1-x-sqrt(1-6*x+x^2))/(2*x);
gf = x*(1-x-2*x*R+x^2+2*x^2*R+x^2*R^2)/(1-3*x-x*R+x^2+x^2*R)^2;
concat([0],Vec(gf))
\\ Joerg Arndt, May 16 2013

a(n) = Sum_{k=0..n} k*A110189(n,k).
G.f.: x*(1-x-2*x*R+x^2+2*x^2*R+x^2*R^2)/(1-3*x-x*R+x^2+x^2*R)^2, where R = 1+x*R+x*R^2 = (1-x-sqrt(1-6*x+x^2))/(2*x) is the g.f. for the large Schroeder numbers (A006318).
Recurrence: (n+2)*(n+3)*a(n) = (5*n^2+29*n+10)*a(n-1) + (5*n^2-59*n+142)*a(n-2) - (n-6)*(n-5)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 3*2^(1/4)*(3+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012
G.f. A(x) satisfies x^2*A(x)^2 = (x^4 - 7*x^3 + 12*x^2 - 7*x + 1)*A(x) + (-x^3 + 2*x^2 - x). - Joerg Arndt, May 16 2013
a(n) = Sum_{k=0..n} ((k+1)*Sum_{i=0..n-k} (binomial(n+1,n-k-i)*binomial(n+i,n))/ (n+1)*a113127(k)). - Vladimir Kruchinin, Mar 13 2016

A131034 A129686 * A051340.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 6, 2, 1, 1, 8, 2, 2, 1, 1, 10, 2, 2, 2, 1, 12, 2, 2, 2, 2, 1, 1, 14, 2, 2, 2, 2, 2, 1, 1, 16, 2, 2, 2, 2, 2, 2, 1, 1

Offset: 0

Gary W. Adamson, Jun 10 2007

Row sums = A113127: (1, 3, 6, 10, 14, 18, 22, ...).

			First few rows of the triangle:
   1;
   2, 1;
   4, 1, 1;
   6, 2, 1, 1;
   8, 2, 2, 1, 1;
  10, 2, 2, 2, 1, 1;
  ...

Cf. A129686, A051340, A113127.

A129686 * A051340 as infinite lower triangular matrices.

A357778 Maximum number of edges in a 5-degenerate graph with n vertices.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235

Offset: 1

Allan Bickle, Oct 13 2022

nonn

A maximal 5-degenerate graph can be constructed from a 5-clique by iteratively adding a new 5-leaf (vertex of degree 5) adjacent to five existing vertices.

This is also the number of edges in a 5-tree with n>5 vertices. (In a 5-tree, the neighbors of a newly added vertex must form a clique.)

			For n < 7, the only maximal 5-degenerate graph is complete.

Allan Bickle, Fundamentals of Graph Theory, AMS (2020).
J. Mitchem, Maximal k-degenerate graphs, Util. Math. 11 (1977), 101-106.

Allan Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 4 (2012), 659-676.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
D. R. Lick and A. T. White, k-degenerate graphs, Canad. J. Math. 22 (1970), 1082-1096.

Number of edges in a maximal k-degenerate graph for k=2..6: A004273, A296515, A113127, A357778, A357779.

a(n) = C(n,2) for n < 7.

a(n) = 5*n-15 for n > 4.

A357779 Maximum number of edges in a 6-degenerate graph with n vertices.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279

Offset: 1

Allan Bickle, Oct 13 2022

nonn

A maximal 6-degenerate graph can be constructed from a 6-clique by iteratively adding a new 6-leaf (vertex of degree 6) adjacent to six existing vertices.

This is also the number of edges in a 6-tree with n>6 vertices. (In a 6-tree, the neighbors of a newly added vertex must form a clique.)

			For n < 8, the only maximal 6-degenerate graph is complete.

Allan Bickle, Fundamentals of Graph Theory, AMS (2020).
J. Mitchem, Maximal k-degenerate graphs, Util. Math. 11 (1977), 101-106.

Allan Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 4 (2012), 659-676.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
D. R. Lick and A. T. White, k-degenerate graphs, Canad. J. Math. 22 (1970), 1082-1096.

Number of edges in a maximal k-degenerate graph for k=2..6: A004273, A296515, A113127, A357778, A357779.

a(n) = C(n,2) for n < 8.

a(n) = 6*n-21 for n > 5.

cp's OEIS Frontend

A181527 Binomial transform of A113127; (1, 1, 3, 7, 15, 31, ...) convolved with (1, 3, 7, 15, 31, 63, ...).

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A167381 The numbers read down the left-center column of an arrangement of the natural numbers in square blocks.

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A113126 A simple 4-diagonal matrix.

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A133653 A007318^(-1) * A003261.

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A110190 Number of (1,0)-steps on the lines y=0 and y=1 in all Schroeder paths of length 2n (a Schroeder path of length 2n is a path from (0,0) to (2n,0), consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis).

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A131034 A129686 * A051340.

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A357778 Maximum number of edges in a 5-degenerate graph with n vertices.

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A357779 Maximum number of edges in a 6-degenerate graph with n vertices.

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