A001590 -id:A001590 - cp's OEIS frontend

A027053 a(n) = T(n,n+2), T given by A027052.

1, 4, 9, 18, 35, 66, 123, 228, 421, 776, 1429, 2630, 4839, 8902, 16375, 30120, 55401, 101900, 187425, 344730, 634059, 1166218, 2145011, 3945292, 7256525, 13346832, 24548653, 45152014, 83047503, 152748174, 280947695, 516743376

Offset: 2

Clark Kimberling

Second differences of (A027026(n)-1)/2.
Pairwise sums of A089068.
a(n) is also the number of fixed polyominoes with n cells of height (or width) 2. - David Bevan, Sep 09 2009

G. C. Greubel, Table of n, a(n) for n = 2..1001
Doron Zeilberger, The generating functions and series expansions for 2D lattice-animals of globally bounded width. [Local copies of generating functions and series expansions].
Index entries for sequences related to polyominoes
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).

2nd column of A308359.

a:=[1,4,9,18];; for n in [5..30] do a[n]:=2*a[n-1]-a[n-4]; od; a; # G. C. Greubel, Nov 05 2019

R:=PowerSeriesRing(Integers(), 32); Coefficients(R!( x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3)) )); // G. C. Greubel, Nov 05 2019

seq(coeff(series(x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3)), x, n+1), x, n), n = 2 ..30); # G. C. Greubel, Nov 05 2019

LinearRecurrence[{2,0,0,-1}, {1,4,9,18}, 30] (* G. C. Greubel, Nov 05 2019 *)

my(x='x+O('x^32)); Vec(x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3))) \\ G. C. Greubel, Nov 05 2019

def A027053_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3))).list()
a=A027053_list(32); a[2:] # G. C. Greubel, Nov 05 2019

G.f.: x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3)).
a(n) = A089068(n-1) + A089068(n).
a(n) = a(n-1) + a(n-2) + a(n-3) + 4. - David Bevan, Sep 09 2009
a(n) = A001590(n+3) - 2. - David Bevan, Sep 09 2009
a(n+1) - a(n) = A000213(n+1). - R. J. Mathar, Aug 04 2013

A196700 Number of n X 1 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 3,1,0,4,2 for x=0,1,2,3,4.

Original entry on oeis.org

1, 2, 4, 6, 12, 22, 40, 74, 136, 250, 460, 846, 1556, 2862, 5264, 9682, 17808, 32754, 60244, 110806, 203804, 374854, 689464, 1268122, 2332440, 4290026, 7890588, 14513054, 26693668, 49097310, 90304032, 166095010, 305496352, 561895394

Offset: 1

R. H. Hardin, Oct 05 2011

nonn

Every 0 is next to zero 3's, every 1 is next to one 1, every 2 is next to two 0's, every 3 is next to three 4's, every 4 is next to four 2's.
Column 1 of A196707.
The perimeter of cuboids with the dimensions of consecutive tribonacci numbers, signature (0,1,0). - Peter M. Chema, Feb 03 2017

			All solutions for n=4:
  0    0    1    0    0    0
  0    0    1    1    0    2
  1    0    0    1    2    0
  1    0    0    0    0    0

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A000073, A001590, A196707.

Empirical: a(n) = a(n-1) + a(n-2) + a(n-3) for n > 4.
G.f.: 1 - 1/x - 1/x^2 + 1/x^2/G(0), where G(k)= 1 - (2*k+1)*x/(1 - x/(x - (2*k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 09 2013
Empirical: a(n) = 2*(A001590(n) + A001590(n-1) + A001590(n-2)) for n > 1. - Peter M. Chema, Feb 03 2017
From Gregory L. Simay, Jun 23 2017: (Start)
a(n) = A000073(n+2) - A000073(n-2), the difference of two tribonacci numbers. The corresponding g.f. is (1 - x^4)/(1 - x - x^2 - x^3). E.g.: a(10) = A000073(12) - A000073(8) = 274 - 24 = 250.
The tribonacci formula arises from considering the number of compositions of n where only the order of parts 1,2,3 matters (part of an upcoming paper), which may be denoted by C(n [4). We are convolving the number of partitions of n with parts >3 with the tribonacci numbers. The number of partitions of n with parts greater than 3 is P(n) - P(n-1) - P(n-2) + P(n-4) + P(n-5) - P(n-6). (Derived from the corresponding gf which is (1-x)(1-x^2)(1-x^3)gfP(x).) The rest is algebra. It looks like C(n, [4) = P(n) + Sum_{j=0..n-3} P(n-3-j)*A196700(j+1). (End)

A242324 Indices of primes in the tribonacci-like sequence A214827.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 14, 15, 18, 39, 42, 46, 128, 319, 501, 645, 749, 785, 924, 1786, 1810, 3032, 3053, 3913, 4444, 5611, 6290, 20526, 20850, 23431, 44281, 45981, 103816, 133938

Offset: 1

Robert Price, May 10 2014

a(37) > 2*10^5.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A001590, A100683, A231574, A231575, A232542, A214899, A230607, A020992, A232498, A214727, A081172, A214752, A141523,A214825, A235862, A214827.

a={1,5,5}; Print[1];Print[2]; For[n=3, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[n]]; a=RotateLeft[a]; a[[3]]=sum]

A247561 Indices of primes in the tetranacci sequence A000288.

Original entry on oeis.org

5, 6, 10, 11, 12, 13, 18, 30, 31, 36, 38, 97, 108, 150, 196, 221, 277, 532, 596, 2468, 2691, 3773, 4303, 5755, 8925, 10083, 11708, 14080, 19990, 24102, 34767, 35973, 39238, 49760, 97706

Offset: 1

Robert Price, Sep 27 2014

a(36) > 2*10^5.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A001590, A001631, A100683, A231574, A231575, A232542, A214899, A230607, A020992, A232498, A214727, A081172, A214752, A141523, A214825, A235862, A000288.

a={1,1,1,1}; For[n=4, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[n]]; a=RotateLeft[a]; a[[4]]=sum]
Flatten[Position[LinearRecurrence[{1,1,1,1},{1,1,1,1},10^5], ?PrimeQ]]- 1 (* _Harvey P. Dale, Dec 20 2016 *)

A290821 Side length of largest equilateral triangle that can be made from n or fewer equilateral triangles with integer sides s_k, subject to gcd(s_1,s_2,...,s_n) = 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 39, 49

Offset: 1

Hugo Pfoertner, Aug 11 2017

No construction from 2, 3 or 5 equilateral triangles exists. The first difference from the Padovan numbers occurs for a(15)=39, where the corresponding term A000931(19)=37. a(16)=A000931(20)=49. a(n) >= A000931(n+3). From the growth behavior of A290697 it is conjectured that a(k) > A000931(k+3) for all k > 20.
a(19) is at least 130. This compares with A000931(23) = 114. It hints of growth behavior similar to sqrt(A014529) or sqrt(A001590). Ceiling(sqrt(A001590(n))) matches a(n) to n=14, then runs 38, 52, 70, 95, 128, ... . - Peter Munn, Mar 10 2018
From Peter Munn, Mar 14 2018 re monotonicity: (Start)
For n >= 6, a(n+1) > a(n).
Sketch of proof (inductive step) expressed in terms of tiling:
Given a triangle of side a(n) tiled with n equilateral triangular tiles. Let X, Y and Z be the tiles incident on its vertices, with X being not smaller than Y or Z.
Case 1: Y and Z have no vertices coincident. Remove Y and Z, thereby reducing the tiled area to a pentagon that has edges A and C that were previously internal to the area, and an edge B between A and C. Fit a new tile T against edge B, thereby extending edges A and C. Make the tiled area triangular by fitting a new tile against each of the extended edges.
Case 2: X, Y and Z have pairwise coincident vertices. It follows that these tiles are the same size. Remove Y and Z, thereby reducing the tiled area to a rhombus. Remove the tile at the rhombus vertex opposite X. The remaining area is a pentagon, since n >= 6. Extend the area by resiting Y against X, and Z against Y so that X and Z have external edges aligned. Make the area trapezoidal by fitting a new tile against the area's edge that includes an edge of Y. Fit another tile T against the smaller of the trapezoid's parallel edges.
In each case, we now have n+1 tiles, tiling an equilateral triangle with side length a(n) plus the side of T. As the sides of new and removed tiles can be calculated by adding sides of tiles that stayed in place, the GCD of the sides is unchanged.
(End)

			a(12) = 16:
                                  *
                                 / \
                                +   +
                               /     \
                              +       +
                             /         \
                            +           +
                           /             \
                          +               +
                         /                 \
                        +                   +
                       /                     \
                      +                       +
                     /                         \
                    +                           +
                   /                             \
                  +                               +
                 /                                 \
                *---+---*---+---+---+---+---+---+---*
               / \     / \                         / \
              +   +   +   +                       +   +
             /     \ /     \                     /     \
            *---*---*       +                   +       +
           / \ / \ /         \                 /         \
          +   *---*---+---+---*               +           +
         /     \             / \             /             \
        +       +           +   +           +               +
       /         \         /     \         /                 \
      +           +       +       +       +                   +
     /             \     /         \     /                     \
    +               +   +           +   +                       +
   /                 \ /             \ /                         \
  *---+---+---+---+---*---+---+---+---*---+---+---+---+---+---+---*

Stuart Anderson, An Introduction to Triangled Equilateral Triangles
Ales Drapal and Carlo Hamalainen, An enumeration of equilateral triangle dissections, arXiv:0910.5199 [math.CO], 2009-2010.
Hugo Pfoertner, Illustration of a(15)=39.

Cf. A000931, A001590, A167123, A290653, A290697, A290820.
A014529 gives greatest area of any convex polygon constructable from such triangles.
A089047 is this sequence's equivalent for squares.

Definition modified and 5 terms prepended by Peter Munn, Mar 14 2018

A113238 Differences of nonzero tribonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 9, 11, 12, 17, 20, 22, 23, 31, 37, 40, 42, 43, 57, 68, 74, 77, 79, 80, 105, 125, 136, 142, 145, 147, 148, 193, 230, 250, 261, 267, 270, 272, 273, 355, 423, 460, 480, 491, 497, 500, 502, 503, 653, 778, 846, 883, 903, 914, 920, 923, 925, 926, 1201

Offset: 1

Jonathan Vos Post, Oct 19 2005

Cf. A000073, A001590 (subset), A113239 (primes in this sequence).

Union[Flatten[Differences/@Subsets[Drop[LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 16],2],{2}]]] (* James C. McMahon, Jun 22 2024 *)

{a(n)} = { A000073(i) - A000073(j) such that i>=j>=2 }.

A180668 a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 5, 14, 32, 67, 133, 256, 484, 905, 1681, 3110, 5740, 10579, 19481, 35856, 65976, 121377, 223277, 410702, 755432, 1389491, 2555709, 4700720, 8646012, 15902537, 29249369, 53798022, 98950036, 181997539, 334745713

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+2) represent the Kn13 and Kn23 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (3,-2,0,-1,1).

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).

nmax:=31: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+4*n-8 od: seq(a(n),n=0..nmax);

LinearRecurrence[{3,-2,0,-1,1},{0,0,1,5,14},40] (* Harvey P. Dale, Dec 15 2023 *)

a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.
a(n) = a(n-1)+A001590(n+3)-2 with a(0)=0.
a(n) = sum(A008574(m)*A000073(n-m),m=0..n).
a(n+2) = add(A008288(n-k+2,k+2),k=0..floor(n/2)).
GF(x) = (x^2*(1+x)^2)/((1-x)^2*(1-x-x^2-x^3)).
Contribution from Bruno Berselli, Sep 23 2010: (Start)
a(n) = 2*a(n-1)-a(n-4)+4 for n>4.
a(n)-3*a(n-1)+2a(n-2)+a(n-4)-a(n-5) = 0 for n>4. (End)

A180669 a(n) = a(n-1)+a(n-2)+a(n-3)+4n^2-16n+18 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 7, 26, 72, 171, 371, 760, 1500, 2889, 5475, 10266, 19116, 35435, 65495, 120832, 222664, 410017, 754671, 1388650, 2554784, 4699707, 8644907, 15901336, 29248068, 53796617, 98948523, 181995914, 334743972, 615691547

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+2) represent the Kn14 and Kn24 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,-1,2,-1).

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).

Cf. A000073, A005899, A008288.

nmax:=30: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+4*(n-2)^2+2 od: seq(a(n),n=0..nmax);

nxt[{n_,a_,b_,c_}]:={n+1,b,c,a+b+c+4n(n-2)+6}; NestList[nxt,{2,0,0,1},30][[;;,2]] (* or *) LinearRecurrence[{4,-5,2,-1,2,-1},{0,0,1,7,26,72},40] (* Harvey P. Dale, Jul 13 2024 *)

a(n) = a(n-1)+a(n-2)+a(n-3)+4*(n-2)^2+2 with a(0)=0, a(1)=0 and a(2)=1.
a(n) = a(n-1)+A001590(n+5)-2-4*n with a(0)=0.
a(n) = Sum_{m=0..n} A005899(m)*A000073(n-m).
a(n+2) = Sum_{k=0..floor(n/2)} A008288(n-k+3,k+3).
GF(x) = (x^2*(1+x)^3)/((1-x)^3*(1-x-x^2-x^3)).
From Bruno Berselli, Sep 23 2010: (Start)
a(n) = 3*a(n-1)-2a(n-2)-a(n-4)+a(n-5)+8 for n>4.
a(n)-4*a(n-1)+5a(n-2)-2*a(n-3)+a(n-4)-2*a(n-5)+a(n-6) = 0 for n>5. (End)

A180670 a(n) = a(n-1)+a(n-2)+a(n-3)+(8n^3-48n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 9, 42, 140, 383, 925, 2056, 4316, 8705, 17069, 32810, 62192, 116743, 217673, 404000, 747496, 1380177, 2544865, 4688186, 8631620, 15886111, 29230725, 53776968, 98926372, 181971057, 334716197, 615660634, 1132400520

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+2) represent the Kn15 and Kn25 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-3,3,-3,1).

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).

nmax:=29: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 od: seq(a(n),n=0..nmax);

RecurrenceTable[{a[0]==a[1]==0,a[2]==1,a[n]==a[n-1]+a[n-2]+a[n-3]+(8n^3-48n^2+112n-96)/3},a,{n,30}] (* or *) LinearRecurrence[{5,-9,7,-3,3,-3,1},{0,0,1,9,42,140,383},30] (* Harvey P. Dale, Dec 04 2019 *)

a(n) = a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.
a(n) = a(n-1)+A001590(n+7)-(12+4*n+4*n^2) with a(0)=0.
a(n) = sum(A008412(m)*A000073(n-m),m=0..n).
a(n+2) = add(A008288(n-k+4,k+4),k=0..floor(n/2)).
GF(x) = (x^2*(1+x)^4)/((1-x)^4*(1-x-x^2-x^3)).

A232561 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 3*x are in S, and duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 3, 6, 4, 9, 7, 18, 5, 12, 10, 27, 8, 21, 19, 54, 15, 13, 36, 11, 30, 28, 81, 24, 22, 63, 20, 57, 55, 162, 16, 45, 14, 39, 37, 108, 33, 31, 90, 29, 84, 82, 243, 25, 72, 23, 66, 64, 189, 60, 58, 171, 56, 165, 163, 486, 17, 48, 46, 135, 42, 40, 117, 38

Offset: 1

Clark Kimberling, Nov 26 2013

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x + 1 and 3*x are in S. Then S is the set of all positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2,3), g(3) = (6,4,9), g(4) = (7,18,5,8,10,27), etc. Concatenating these gives A232561, a permutation of the positive integers. The number of numbers in g(n) is A001590(n), the n-th tribonacci number. It is helpful to show the results as a tree with the terms of S as nodes and edges from x to x + 1 if x + 1 has not already occurred, and an edge from x to 3*x if 3*x has not already occurred.

			Each x begets x + 1 and 3*x, but if either has already occurred it is deleted.  Thus, 1 begets 2 and 3; in the next generation, 2 begets only 6, and 3 begets 4 and 9.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A232560, A232562, A001590.

z = 8; g[1] = {1}; g[2] = {2, 3};
g[n_] := Riffle[g[n - 1] + 1, 3 g[n - 1]]
j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]];
g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]];
g1[1] = g[1]; g1[2] = g[2];
t = Flatten[Table[g1[n], {n, 1, z}]]  (* A232561 *)
Table[Length[g1[n]], {n, 1, z}]  (* A001590 *)
t1 = Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232562 *)

cp's OEIS Frontend

A027053 a(n) = T(n,n+2), T given by A027052.

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A196700 Number of n X 1 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 3,1,0,4,2 for x=0,1,2,3,4.

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A242324 Indices of primes in the tribonacci-like sequence A214827.

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A247561 Indices of primes in the tetranacci sequence A000288.

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A290821 Side length of largest equilateral triangle that can be made from n or fewer equilateral triangles with integer sides s_k, subject to gcd(s_1,s_2,...,s_n) = 1.

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A113238 Differences of nonzero tribonacci numbers.

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A180668 a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.

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A180669 a(n) = a(n-1)+a(n-2)+a(n-3)+4*n^2-16*n+18 with a(0)=0, a(1)=0 and a(2)=1.

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A180669 a(n) = a(n-1)+a(n-2)+a(n-3)+4n^2-16n+18 with a(0)=0, a(1)=0 and a(2)=1.

A180670 a(n) = a(n-1)+a(n-2)+a(n-3)+(8n^3-48n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.