A035344 -id:A035344 - cp's OEIS frontend

A007052 Number of order-consecutive partitions of n.

1, 3, 10, 34, 116, 396, 1352, 4616, 15760, 53808, 183712, 627232, 2141504, 7311552, 24963200, 85229696, 290992384, 993510144, 3392055808, 11581202944, 39540700160, 135000394752, 460920178688, 1573679925248, 5372879343616, 18344157523968, 62630871408640, 213835170586624

Offset: 0

Colin Mallows, N. J. A. Sloane, and Simon Plouffe

After initial terms, first differs from A291292 at a(6) = 1352, A291292(8) = 1353.
Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 3) is "size of raises in pot-limit poker, one blind, maximum raising".
It appears that this sequence is the BinomialMean transform of A001653 (see A075271). - John W. Layman, Oct 03 2002
Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 3, s(2n+1) = 4. - Herbert Kociemba, Jun 12 2004
Equals the INVERT transform of (1, 2, 5, 13, 34, 89, ...). - Gary W. Adamson, May 01 2009
a(n) is the number of compositions of n when there are 3 types of ones. - Milan Janjic, Aug 13 2010
a(n)/a(n-1) tends to (4 + sqrt(8))/2 = 3.414213.... Gary W. Adamson, Jul 30 2013
a(n) is the first subdiagonal of array A228405. - Richard R. Forberg, Sep 02 2013
Number of words of length n over {0,1,2,3,4} in which binary subwords appear in the form 10...0. - Milan Janjic, Jan 25 2017
From Gus Wiseman, Mar 05 2020: (Start)
Also the number of unimodal sequences of length n + 1 covering an initial interval of positive integers, where a sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. For example, the a(0) = 1 through a(2) = 10 sequences are:
  (1)  (1,1)  (1,1,1)
       (1,2)  (1,1,2)
       (2,1)  (1,2,1)
              (1,2,2)
              (1,2,3)
              (1,3,2)
              (2,1,1)
              (2,2,1)
              (2,3,1)
              (3,2,1)
Missing are: (2,1,2), (2,1,3), (3,1,2).
Conjecture: Also the number of ordered set partitions of {1..n + 1} where no element of any block is greater than any element of a non-adjacent consecutive block. For example, the a(0) = 1 through a(2) = 10 ordered set partitions are:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{2}}  {{1},{2,3}}
         {{2},{1}}  {{1,2},{3}}
                    {{1,3},{2}}
                    {{2},{1,3}}
                    {{2,3},{1}}
                    {{3},{1,2}}
                    {{1},{2},{3}}
                    {{1},{3},{2}}
                    {{2},{1},{3}}
Cf. A000670, A056242, A332673, A332872. (End)
a(n-1) is the number of hexagonal directed-column convex polyominoes having area n (see Baril et al. at page 4). - Stefano Spezia, Oct 14 2023

			G.f. = 1 + 3*x + 10*x^2 + 34*x^3 + 116*x^4 + 396*x^5 + 1352*x^6 + 4616*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Barbero, U. Cerruti, and N. Murru, A Generalization of the Binomial Interpolated Operator and its Action on Linear Recurrent Sequences, J. Int. Seq. 13 (2010) # 10.9.7, proposition 16.
Jean-Luc Baril, José L. Ramírez, and Fabio A. Velandia, Bijections between Directed-Column Convex Polyominoes and Restricted Compositions, September 29, 2023.
Tyler Clark and Tom Richmond, The Number of Convex Topologies on a Finite Totally Ordered Set, 2013, Involve, Vol. 8 (2015), No. 1, 25-32.
Pamela Fleischmann, Jonas Höfer, Annika Huch, and Dirk Nowotka, alpha-beta-Factorization and the Binary Case of Simon's Congruence, arXiv:2306.14192 [math.CO], 2023.
Juan B. Gil and Jessica A. Tomasko, Fibonacci colored compositions and applications, arXiv:2108.06462 [math.CO], 2021.
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, Preprint. (Annotated scanned copy)
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 164
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
N. J. A. Sloane, Transforms
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
Index entries for sequences related to poker
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A006012, A003480, A056236.
First differences of A007070.
Cf. A000129, A000670, A001523, A001653, A007068, A035344, A060223, A075271, A227038, A291292, A328509, A332577, A332743, A332873.

[Floor((2+Sqrt(2))^n*(1/2+Sqrt(2)/4)+(2-Sqrt(2))^n*(1/2-Sqrt(2)/4)): n in [0..30] ] ; // Vincenzo Librandi, Aug 20 2011

a[n_]:=(MatrixPower[{{3,1},{1,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
a[ n_] := ((2 + Sqrt[2])^(n + 1) + (2 - Sqrt[2])^(n + 1)) / 4 // Simplify; (* Michael Somos, Jan 25 2017 *)
LinearRecurrence[{4, -2}, {1, 3}, 24] (* Jean-François Alcover, Jan 07 2019 *)
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Union@@Permutations/@allnorm[n],unimodQ]],{n,6}] (* Gus Wiseman, Mar 06 2020 *)

{a(n) = real((2 + quadgen(8))^(n+1)) / 2}; /* Michael Somos, Mar 06 2003 */

a(n+1) = 4*a(n) - 2*a(n-1).
G.f.: (1-x)/(1-4*x+2*x^2).
Binomial transform of Pell numbers 1, 2, 5, 12, ... (A000129).
a(n) = A006012(n+1)/2 = A056236(n+1)/4. - Michael Somos, Mar 06 2003
a(n) = (A035344(n)+1)/2; a(n) = (2+sqrt(2))^n(1/2+sqrt(2)/4)+(2-sqrt(2))^n(1/2-sqrt(2)/4). - Paul Barry, Jul 16 2003
Second binomial transform of (1, 1, 2, 2, 4, 4, ...). a(n) = Sum_{k=1..floor(n/2)}, C(n, 2k)*2^(n-k-1). - Paul Barry, Nov 22 2003
a(n) = ( (2-sqrt(2))^(n+1) + (2+sqrt(2))^(n+1) )/4. - Herbert Kociemba, Jun 12 2004
a(n) = both left and right terms in M^n * [1 1 1], where M = the 3 X 3 matrix [1 1 1 / 1 2 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A007070(n) a(n)]. E.g., a(3) = 34. M^3 * [1 1 1] = [34 48 34] (center term is A007070(3)). - Gary W. Adamson, Dec 18 2004
The i-th term of the sequence is the entry (2, 2) in the i-th power of the 2 X 2 matrix M = ((1, 1), (1, 3)). - Simone Severini, Oct 15 2005
E.g.f.: exp(2*x)*(cosh(sqrt(2)*x)+sinh(sqrt(2)*x)/sqrt(2)). - Paul Barry, Nov 20 2003
a(n) = A007068(2*n), n>0. - R. J. Mathar, Aug 17 2009
If p[i]=Fibonacci(2i-1) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
a(n-1) = Sum_{k=-floor(n/4)..floor(n/4)} (-1)^k*binomial(2*n,n+4*k)/2. - Mircea Merca, Jan 28 2012
G.f.: G(0)*(1-x)/(2*x) + 1 - 1/x, where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - (1-x)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = 3*a(n-1) + a(n-2) + a(n-3) + a(n-4) + ... + a(0). - Gary W. Adamson, Aug 12 2013
a(n) = a(-2-n) * 2^(n+1) for all n in Z. - Michael Somos, Jan 25 2017

A264237 Sum of values of vertices at level n of the hyperbolic Pascal pyramid.

Original entry on oeis.org

1, 3, 9, 33, 165, 1137, 9837, 95193, 962541, 9884889, 102049197, 1055383929, 10921055661, 113032307769, 1169952636525, 12109971475065, 125349031354029, 1297477519769145, 13430093334225645, 139013932289379321, 1438923355509080877, 14894194022848480185

Offset: 0

Michel Marcus, Nov 09 2015

Colin Barker, Table of n, a(n) for n = 0..987
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.02067 [math.CO], 2015 (6th line of Table 2).
Index entries for linear recurrences with constant coefficients, signature (18,-99,226,-224,92,-12).

Cf. A035344, A264236.

CoefficientList[Series[-(20*x^5 - 8*x^4 + 58*x^3 - 54*x^2 + 15*x - 1)/((x - 1)*(2*x^2 - 4*x + 1)*(6*x^3 - 28*x^2 + 13*x - 1)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)

Vec(-(20*x^5-8*x^4+58*x^3-54*x^2+15*x-1)/((x-1)*(2*x^2-4*x+1)*(6*x^3-28*x^2+13*x-1)) + O(x^30)) \\ Colin Barker, Nov 09 2015

a(n) = 18*a(n-1) - 99*a(n-2) + 226*a(n-3) - 224*a(n-4) + 92*a(n-5) - 12*a(n-6), for n >= 7.

G.f.: -(20*x^5-8*x^4+58*x^3-54*x^2+15*x-1) / ((x-1)*(2*x^2-4*x+1)*(6*x^3-28*x^2+13*x-1)). - Colin Barker, Nov 09 2015

Definition edited by Eric M. Schmidt, Sep 17 2017

A210754 Triangle of coefficients of polynomials v(n,x) jointly generated with A210753; see the Formula section.

Original entry on oeis.org

1, 3, 2, 6, 9, 4, 10, 25, 24, 8, 15, 55, 85, 60, 16, 21, 105, 231, 258, 144, 32, 28, 182, 532, 833, 728, 336, 64, 36, 294, 1092, 2241, 2720, 1952, 768, 128, 45, 450, 2058, 5301, 8361, 8280, 5040, 1728, 256, 55, 660, 3630, 11385, 22363, 28610, 23920

Offset: 1

Clark Kimberling, Mar 25 2012

Column 1: triangular numbers, A000217
Coefficient of v(n,x):  2^(n-1)
Row sums:  A035344
Alternating row sums: 1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.
Appears to be the reversed row polynomials of A165241 with the unit diagonal removed. If so, the o.g.f. is [1-(1+y)x]/[1-2(1+y)x+(1+y)x^2] - 1/(1-x) and the triangular matrix here may be formed by adding each column of the matrix of A056242, presented in the example section with the additional zeros, to its subsequent column with the first row ignored. - Tom Copeland, Jan 09 2017

			First five rows:
1
3....2
6....9....4
10...25...24...8
15...55...85...60...16
First three polynomials v(n,x): 1, 3 + 2x, 6 + 9x +4x^2

Cf. A210753, A208510.

Cf. A056242, A165241.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210754 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A007070 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A035344 *)

u(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A158525 Number of connected spanning subgraphs and number of forests of the wheel graph W_n.

Original entry on oeis.org

38, 134, 462, 1582, 5406, 18462, 63038, 215230, 734846, 2508926, 8566014, 29246206, 99852798, 340918782, 1163969534, 3974040574, 13568223230, 46324811774, 158162800638, 540001579006, 1843680714750, 6294719700990, 21491517374462, 73376630095870, 250523485634558

Offset: 4

Alois P. Heinz, Mar 20 2009

The wheel graph W_n has n vertices and 2n-2 edges. A single vertex is connected to all vertices of an (n-1)-cycle.

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Eric Weisstein's World of Mathematics, Wheel Graph
Wikipedia, Wheel graph
Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.
Index entries for linear recurrences with constant coefficients, signature (5,-6,2).

Cf. A035344.

a:= n-> `if`(n<4, 0, (Matrix([[5, 1, 0], [ -6, 0, 1], [2, 0, 0]])^n)[3, 2]): seq(a(n), n=4..30);

CoefficientList[Series[((1 / x^4) (38 - 56 x + 20 x^2) x^4 / (6 x^2 + 1 - 5 x - 2 x^3)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 06 2013 *)

G.f.: (38-56*x+20*x^2)*x^4 / (6*x^2+1-5*x-2*x^3).

a(n) = 2 * A035344(n-2).

A210742 Triangle of coefficients of polynomials v(n,x) jointly generated with A210741; see the Formula section.

Original entry on oeis.org

1, 3, 2, 5, 9, 5, 7, 20, 27, 13, 9, 35, 73, 80, 34, 11, 54, 151, 252, 234, 89, 13, 77, 269, 597, 837, 677, 233, 15, 104, 435, 1199, 2225, 2702, 1941, 610, 17, 135, 657, 2158, 4956, 7943, 8533, 5523, 1597, 19, 170, 943, 3590, 9796, 19387, 27435, 26479

Offset: 1

Clark Kimberling, Mar 24 2012

Row n starts with 2n-1 and ends with an odd-indexed
Fibonacci number.
Row sums: A035344
Alternate row sums: 1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
3...2
5...9....5
7...20...27...13
9...35...73...80...34
First three polynomials v(n,x): 1, 3 + 2x, 5 + 9x + 5x^2

Cf. A210741, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]     (* A210741 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A210742 *)

u(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A210753 Triangle of coefficients of polynomials u(n,x) jointly generated with A210754; see the Formula section.

Original entry on oeis.org

1, 2, 2, 3, 7, 4, 4, 16, 20, 8, 5, 30, 61, 52, 16, 6, 50, 146, 198, 128, 32, 7, 77, 301, 575, 584, 304, 64, 8, 112, 560, 1408, 1992, 1616, 704, 128, 9, 156, 966, 3060, 5641, 6328, 4272, 1600, 256, 10, 210, 1572, 6084, 14002, 20330, 18880, 10912, 3584

Offset: 1

Clark Kimberling, Mar 25 2012

Row n starts with n and ends with 2^(n-1).
Row sums: A007070
Alternating row sums: 1,0,0,0,0,0,0,0,0,....
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2...2
3...7....4
4...16...20...8
5...30...61...52...16
First three polynomials u(n,x): 1, 2 + 2x, 3 + 7x + 4x^2.

Cf. A210754, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210754 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A007070 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A035344 *)

u(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

cp's OEIS Frontend

A007052 Number of order-consecutive partitions of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A264237 Sum of values of vertices at level n of the hyperbolic Pascal pyramid.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A210754 Triangle of coefficients of polynomials v(n,x) jointly generated with A210753; see the Formula section.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A158525 Number of connected spanning subgraphs and number of forests of the wheel graph W_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A210742 Triangle of coefficients of polynomials v(n,x) jointly generated with A210741; see the Formula section.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A210753 Triangle of coefficients of polynomials u(n,x) jointly generated with A210754; see the Formula section.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula