A008937 -id:A008937 - cp's OEIS frontend

A209972 Number of binary words of length n avoiding the subword given by the binary expansion of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 4, 1, 1, 1, 2, 4, 5, 5, 1, 1, 1, 2, 4, 7, 8, 6, 1, 1, 1, 2, 4, 7, 12, 13, 7, 1, 1, 1, 2, 4, 7, 12, 20, 21, 8, 1, 1, 1, 2, 4, 7, 12, 21, 33, 34, 9, 1, 1, 1, 2, 4, 8, 13, 20, 37, 54, 55, 10, 1, 1, 1, 2, 4, 8, 15, 24, 33, 65, 88, 89, 11, 1, 1

Offset: 0

Alois P. Heinz, Mar 16 2012

			Square array begins:
  1,  1,  1,   1,   1,   1,   1,   1,   1, ...
  1,  1,  2,   2,   2,   2,   2,   2,   2, ...
  1,  1,  3,   3,   4,   4,   4,   4,   4, ...
  1,  1,  4,   5,   7,   7,   7,   7,   8, ...
  1,  1,  5,   8,  12,  12,  12,  13,  15, ...
  1,  1,  6,  13,  20,  21,  20,  24,  28, ...
  1,  1,  7,  21,  33,  37,  33,  44,  52, ...
  1,  1,  8,  34,  54,  65,  54,  81,  96, ...
  1,  1,  9,  55,  88, 114,  88, 149, 177, ...

Alois P. Heinz, Antidiagonals n = 0..150, flattened

Columns give: 0, 1: A000012, 2: A001477(n+1), 3: A000045(n+2), 4, 6: A000071(n+3), 5: A005251(n+3), 7: A000073(n+3), 8, 12, 14: A008937(n+1), 9, 11, 13: A049864(n+2), 10: A118870, 15: A000078(n+4), 16, 20, 24, 26, 28, 30: A107066, 17, 19, 23, 25, 29: A210003, 18, 22: A209888, 21: A152718(n+3), 27: A210021, 31: A001591(n+5), 32: A001949(n+5), 33, 35, 37, 39, 41, 43, 47, 49, 53, 57, 61: A210031.

Main diagonal equals A234005 or column k=0 of A233940.

A[n_, k_] := Module[{bb, cnt = 0}, Do[bb = PadLeft[IntegerDigits[j, 2], n]; If[SequencePosition[bb, IntegerDigits[k, 2], 1]=={}, cnt++], {j, 0, 2^n-1 }]; cnt];
Table[A[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 01 2021 *)

A126198 Triangle read by rows: T(n,k) (1 <= k <= n) = number of compositions of n into parts of size <= k.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 5, 7, 8, 1, 8, 13, 15, 16, 1, 13, 24, 29, 31, 32, 1, 21, 44, 56, 61, 63, 64, 1, 34, 81, 108, 120, 125, 127, 128, 1, 55, 149, 208, 236, 248, 253, 255, 256, 1, 89, 274, 401, 464, 492, 504, 509, 511, 512, 1, 144, 504, 773, 912, 976, 1004, 1016, 1021, 1023, 1024

Offset: 1

N. J. A. Sloane, Mar 09 2007

Also has an interpretation as number of binary vectors of length n-1 in which the length of the longest run of 1's is <= k (see A048004). - N. J. A. Sloane, Apr 03 2011

Higher Order Fibonacci numbers: A126198(n,k) = Sum_{h=0..k} A048004(n,h); for example, A126198(7,3) = Sum_{h=0..3} A048004(7,h) or A126198(7,3) = 1 + 33 + 47 + 27 = 108, the 7th tetranacci number. A048004 row(7) produces A126198 row(7) list of 1,34,81,108,120,125,127,128 which are 1, the 7th Fibonacci, the 7th tribonacci, ... 7th octanacci numbers. - Richard Southern, Aug 04 2017

			Triangle begins:
  1;
  1,  2;
  1,  3,  4;
  1,  5,  7,  8;
  1,  8, 13, 15, 16;
  1, 13, 24, 29, 31, 32;
  1, 21, 44, 56, 61, 63, 64;
Could also be extended to a square array:
  1  1  1  1  1  1  1 ...
  1  2  2  2  2  2  2 ...
  1  3  4  4  4  4  4 ...
  1  5  7  8  8  8  8 ...
  1  8 13 15 16 16 16 ...
  1 13 24 29 31 32 32 ...
  1 21 44 56 61 63 64 ...
which when read by antidiagonals (downwards) gives A048887.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 154-155.

Alois P. Heinz, Rows n = 1..141

Rows are partial sums of rows of A048004. Cf. A048887, A092921 for other versions.
2nd column = Fibonacci numbers, next two columns are A000073, A000078; last three diagonals are 2^n, 2^n-1, 2^n-3.
Cf. A082267.

A126198 := proc(n,k) coeftayl( x*(1-x^k)/(1-2*x+x^(k+1)),x=0,n); end: for n from 1 to 11 do for k from 1 to n do printf("%d, ",A126198(n,k)); od; od; # R. J. Mathar, Mar 09 2007
# second Maple program:
T:= proc(n, k) option remember;
      if n=0 or k=1 then 1
    else add(T(n-j, k), j=1..min(n, k))
      fi
    end:
seq(seq(T(n, k), k=1..n), n=1..15);  # Alois P. Heinz, Oct 23 2011

rows = 11; t[n_, k_] := Sum[ (-1)^i*2^(n-i*(k+1))*Binomial[ n-i*k, i], {i, 0, Floor[n/(k+1)]}] - Sum[ (-1)^i*2^((-i)*(k+1)+n-1)*Binomial[ n-i*k-1, i], {i, 0, Floor[(n-1)/(k+1)]}]; Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 1, n}]](* Jean-François Alcover, Nov 17 2011, after Max Alekseyev *)

G.f. for column k: (x-x^(k+1))/(1-2*x+x^(k+1)). [Riordan]
T(n,3) = A008937(n) - A008937(n-3) for n>=3. T(n,4) = A107066(n-1) - A107066(n-5) for n>=5. T(n,5) = A001949(n+4) - A001949(n-1) for n>=5. - R. J. Mathar, Mar 09 2007
T(n,k) = A181695(n,k) - A181695(n-1,k). - Max Alekseyev, Nov 18 2010
Conjecture: Sum_{k=1..n} T(n,k) = A039671(n), n>0. - L. Edson Jeffery, Nov 29 2013

More terms from R. J. Mathar, Mar 09 2007

A189161 T(n,k)=Number of nXk binary arrays without the pattern 0 0 1 1 diagonally, vertically or horizontally.

Original entry on oeis.org

2, 4, 4, 8, 16, 8, 15, 64, 64, 15, 28, 225, 512, 225, 28, 52, 784, 3375, 3375, 784, 52, 96, 2704, 21952, 37976, 21952, 2704, 96, 177, 9216, 140608, 424401, 424401, 140608, 9216, 177, 326, 31329, 884736, 4597967, 8210464, 4597967, 884736, 31329, 326, 600

Offset: 1

R. H. Hardin Apr 17 2011

Table starts
...2......4.........8..........15.............28...............52
...4.....16........64.........225............784.............2704
...8.....64.......512........3375..........21952...........140608
..15....225......3375.......37976.........424401..........4597967
..28....784.....21952......424401........8210464........151889331
..52...2704....140608.....4597967......151889331.......4708773306
..96...9216....884736....48460966.....2710884810.....139543984265
.177..31329...5545233...507947896....48083827685....4101554902927
.326.106276..34645976..5289780940...844714479714..118889716429242
.600.360000.216000000.54890361712.14772128520081.3425594682105277

			Some solutions for 6X4
..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0
..1..0..1..0....1..0..1..0....1..0..0..0....0..1..1..1....1..0..1..0
..1..0..1..0....1..0..1..0....0..1..0..1....0..0..1..0....0..0..0..0
..1..0..0..0....0..1..1..0....1..0..1..0....0..1..0..0....1..0..0..1
..0..0..1..0....0..0..1..0....0..0..0..0....0..0..1..0....0..1..0..0
..1..1..0..1....0..1..0..1....0..0..0..0....0..0..0..0....1..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..219

Column 1 is A008937(n+1)
Column 2 is column 1 squared
Column 3 is column 1 cubed

A352105 Numbers whose maximal tribonacci representation (A352103) is palindromic.

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 14, 18, 23, 27, 36, 40, 51, 52, 62, 69, 78, 88, 95, 102, 110, 130, 148, 156, 176, 181, 194, 211, 229, 242, 246, 264, 277, 294, 312, 325, 326, 363, 397, 411, 448, 463, 477, 514, 548, 562, 599, 617, 650, 674, 682, 715, 739, 770, 803, 827, 838, 862

Offset: 1

Amiram Eldar, Mar 05 2022

A027084(n) is a term since its maximal tribonacci representation is n-1 1's and no 0's.

The pairs {A008937(3*k+1)-1, A008937(3*k+1)} = {0, 1}, {7, 8}, {51, 52}, ... are consecutive terms in this sequence: the maximal tribonacci representation of A008937(3*k+1)-1 is 3*k 1's and no 0's (except for k=0 where the representation is 0), and the maximal tribonacci representation of A008937(3*k+1) is of the form 100100...1001 with k blocks of 100 followed by a 1 at the end.

			The first 10 terms are:
   n  a(n)  A352103(a(n))
  --  ----  -------------
   1    0               0
   2    1               1
   3    3              11
   4    5             101
   5    7             111
   6    8            1001
   7   14            1111
   8   18           10101
   9   23           11011
  10   27           11111

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000073, A008937, A352103.
A027084 is a subsequence.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; trib[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; IntegerDigits[Total[2^(s - 1)], 2]]; q[n_] := Module[{v = trib[n]}, nv = Length[v]; i = 1; While[i <= nv - 3, If[v[[i ;; i + 3]] == {1, 0, 0, 0}, v[[i ;; i + 3]] = {0, 1, 1, 1}; If[i > 3, i -= 4]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, True, PalindromeQ[FromDigits[v[[i[[1, 1]] ;; -1]]]]]]; Select[Range[0, 1000], q]

A027084 G.f.: x^2(x^2 + x + 1)/(x^4 - 2x + 1).

Original entry on oeis.org

1, 3, 7, 14, 27, 51, 95, 176, 325, 599, 1103, 2030, 3735, 6871, 12639, 23248, 42761, 78651, 144663, 266078, 489395, 900139, 1655615, 3045152, 5600909, 10301679, 18947743, 34850334, 64099759, 117897839, 216847935, 398845536

Offset: 2

Clark Kimberling

Lengths of palindromic prefixes of the ternary tribonacci word A080843 [A. Glen]. - N. J. A. Sloane, Jun 09 2019

Original definition was: a(n) = (1/2)*T(n,n+2), T given by A027082.

Hamoon Mousavi, Jeffrey Shallit, Mechanical Proofs of Properties of the Tribonacci Word, arXiv:1407.5841 [cs.FL], 2014.
Bo Tan and Zhi-Ying Wen, Some properties of the Tribonacci sequence, European Journal of Combinatorics, 28 (2007) 1703-1719. See Prop. 2.9, |D_n|.
Index entries for linear recurrences with constant coefficients, signature (2, 0, 0, -1).

Cf. A000073, A008937, A018921, A027082, A080843, A317197 (D_n).

Vec(x^2*(x^2 + x + 1)/(x^4 - 2*x + 1) + O(x^50)) \\ Michel Marcus, Dec 29 2014

Positive numbers of the form (t_n + t_{n+2} - 3)/2, n>1, where {t_n} are the tribonacci numbers A000073 [A. Glen]. See Mousavi-Shallit, 2014. - N. J. A. Sloane, Jun 09 2019
a(n) = A008937(n-1) - 1 = A018921(n-3) - 1.
2*a(n) = A000213(n+2)-3. - R. J. Mathar, Jun 24 2020

Entry revised by N. J. A. Sloane, Aug 05 2018

A055216 Triangle T(n,k) by rows, n >= 0, 0<=k<=n: T(n,k) = Sum_{i=0..n-k} binomial(n-k,i) *Sum_{j=0..k-i} binomial(i,j).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 3, 1, 1, 5, 10, 8, 3, 1, 1, 6, 15, 17, 9, 3, 1, 1, 7, 21, 31, 23, 9, 3, 1, 1, 8, 28, 51, 50, 26, 9, 3, 1, 1, 9, 36, 78, 96, 66, 27, 9, 3, 1, 1, 10, 45, 113, 168, 147, 76, 27, 9, 3, 1, 1, 11, 55, 157, 274, 294, 192, 80, 27, 9, 3, 1

Offset: 0

Clark Kimberling, May 07 2000

T(n,k) is the maximal number of different sequences that can be obtained from a ternary sequence of length n by deleting k symbols.
T(i,j) is the number of paths from (0,0) to (i-j,j) using steps (1 unit right) or (1 unit right and 1 unit up) or (1 unit right and 2 units up).
If m >= 1 and n >= 2, then T(m+n-1,m) is the number of strings (s(1),s(2),...,s(n)) of nonnegative integers satisfying s(n)=m and 0<=s(k)-s(k-1)<=2 for k=2,3,...,n.
T(n,k) is the number of 1100-avoiding 0-1 sequences of length n containing k good 1's. A 1 is bad if it is immediately followed by two or more 1's and then a 0; otherwise it is good. In particular, a 1 with no 0's to its right is good. For example, 110101110111 is 1100-avoiding and only the 1 in position 6 is bad and T(4,3) counts 0111, 1011, 1101. - David Callan, Jul 25 2005
The matrix inverse starts:
1;
-1,1;
1,-2,1;
-1,3,-3,1;
1,-4,6,-4,1;
-2,8,-13,11,-5,1;
8,-30,45,-36,18,-6,1;
-36,137,-207,163,-78,27,-7,1;
192,-732,1112,-884,425,-144,38,-8,1;
- R. J. Mathar, Mar 12 2013

			8=T(5,2) counts these strings: 013, 023, 113, 123, 133, 223, 233, 333.
Rows:
1;
1,1;
1,2,1;
1,3,3,1;
1,4,6,3,1;
...

D. S. Hirschberg, Algorithms for the longest subsequence problem, J. ACM, 24 (1977), 664-675.
C. Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 1E.
V. I. Levenshtein, Efficient reconstruction of sequences from their subsequences or supersequences, J. Combin. Theory Ser. A 93 (2001), no. 2, 310-332.

Row sums: A008937. Central numbers: T(2n, n)=A027914(n) for n >= 0.

A055216 := proc(n,k)
    a := 0 ;
    for i from 0 to n-k do
        a := a+binomial(n-k,i)*add(binomial(i,j),j=0..k-i) ;
    end do:
    a ;
end proc: # R. J. Mathar, Mar 13 2013

T[n_, k_] := Sum[Binomial[n - k, i]*Sum[Binomial[i, j], {j, 0, k - i}], {i, 0, n - k}];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 28 2019 *)

T(i, 0)=T(i, i)=1 for i >= 0; T(i, 1)=T(i, i-1)=i for i >= 2; T(i, j)=T(i-1, j)+T(i-2, j-1)+T(i-3, j-2) for 2<=j<=i-2, i >= 3.

Better description and references from N. J. A. Sloane, Aug 05 2000

A113300 Sum of even-indexed terms of tribonacci numbers.

Original entry on oeis.org

0, 1, 3, 10, 34, 115, 389, 1316, 4452, 15061, 50951, 172366, 583110, 1972647, 6673417, 22576008, 76374088, 258371689, 874065163, 2956941266, 10003260650, 33840788379, 114482567053, 387291750188, 1310198605996, 4432370135229, 14994600761871, 50726371026838

Offset: 0

Jonathan Vos Post, Oct 24 2005

Partial sums of A099463. a(n+1) gives row sums of Riordan array (1/(1-x)^2,(1+x)^2/(1-x)^2). Congruent to 0,1,1,0,0,1,1,0,0,... modulo 2. - Paul Barry, Feb 07 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,1).

Cf. A000073, A008937, A099463, A113301.

I:=[0,1,3]; [n le 3 select I[n] else 3*Self(n-1) +Self(n-2) +Self(n-3): n in [1..61]]; // G. C. Greubel, Nov 19 2021

Accumulate[Take[LinearRecurrence[{1,1,1},{0,0,1},60],{1,-1,2}]] (* Harvey P. Dale, Nov 06 2011 *)
LinearRecurrence[{3,1,1},{0,1,3},40] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)
a[ n_] := Sum[ SeriesCoefficient[ SeriesCoefficient[ x / (1 - x - y - x y) , {x, 0, n - k}]^2 , {y, 0, k}], {k, 0, n}]; (* Michael Somos, Jun 27 2017 *)

@CachedFunction
def T(n): # T(n) = A000073(n)
    if (n<2): return 0
    elif (n==2): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def a(n): return sum( T(2*j) for j in (0..n) )
[a(n) for n in (0..60)] # G. C. Greubel, Nov 19 2021

a(n) = Sum_{i=0..n} A000073(2*n).
a(n) = Sum_{i=0..n} A099463(n).
a(n) + A113301(n) = A008937(n).
From Paul Barry, Feb 07 2006: (Start)
G.f.: x/(1 - 3*x - x^2 - x^3).
a(n) = 3*a(n-1) + a(n-2) + a(n-3). (End)

A172316 7th column of the array A172119.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 127, 252, 500, 992, 1968, 3904, 7744, 15361, 30470, 60440, 119888, 237808, 471712, 935680, 1855999, 3681528, 7302616, 14485344, 28732880, 56994048, 113052416, 224248833, 444816138, 882329660

Offset: 0

Richard Choulet, Jan 31 2010

			a(3) = binomial(3,3)*2^3 = 8.
a(7) = binomial(7,7)*2^7 - binomial(1,0)*2^0 = 127.

O. Dunkel, Solutions of a probability difference equation, Amer. Math. Monthly, 32 (1925), 354-370; see p. 356 with r = 6.
Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,0,0,-1)

Cf. A000071, A001949, A008937, A107066, A172119.

Partial sums of A001592.

for k from 0 to 20 do for n from 0 to 30 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od:k: seq(b(n),n=0..30):od; k:=6:taylor(1/(1-2*z+z^(k+1)),z=0,30);

G.f.: 1/(1 - 2*z + z^7).
Recurrence formula: a(n+7) = 2*a(n+6) - a(n).
a(n) = Sum_{j=0..floor(n/(k+1))} ((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j)) with k=6.

A172317 8th column of A172119.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 255, 508, 1012, 2016, 4016, 8000, 15936, 31744, 63233, 125958, 250904, 499792, 995568, 1983136, 3950336, 7868928, 15674623, 31223288, 62195672, 123891552, 246787536, 491591936, 979233536

Offset: 0

Richard Choulet, Jan 31 2010

			a(4) = binomial(4,4)*2^4 = 16.
a(9) = binomial(9,9)*2^9 - binomial(2,1)*2^1 = 512 - 4 = 508.

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

Cf. A172316, A001949, A107066, A008937, A172119.

Partial sums of A066178.

k:=7:taylor(1/(1-2*z+z^(k+1)),z=0,30); for k from 0 to 20 do for n from 0 to 30 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od:k: seq(b(n),n=0..30):od;

The generating function is f such that: f(z)=1/(1-2*z+z^8). Recurrence relation: a(n+8)=2*a(n+7)-a(n). General term: a(n) = Sum_{j=0..floor(n/(k+1))} ((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j)) with k=7.

A018921 Define the generalized Pisot sequence T(a(0),a(1)) by: a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n). This is T(4,8).

Original entry on oeis.org

4, 8, 15, 28, 52, 96, 177, 326, 600, 1104, 2031, 3736, 6872, 12640, 23249, 42762, 78652, 144664, 266079, 489396, 900140, 1655616, 3045153, 5600910, 10301680, 18947744, 34850335, 64099760, 117897840, 216847936, 398845537, 733591314, 1349284788, 2481721640

Offset: 0

R. K. Guy

Not to be confused with the Pisot T(4,8) sequence, which is A020707. - R. J. Mathar, Feb 13 2016

Colin Barker, Table of n, a(n) for n = 0..1000
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).
Index entries for Pisot sequences

Cf. A008937.

Tiv:=[4,8]; [n le 2 select Tiv[n] else Ceiling(Self(n-1)^2/Self(n-2))-1: n in [1..40]]; // Bruno Berselli, Feb 17 2016

RecurrenceTable[{a[1] == 4, a[2] == 8, a[n] == Ceiling[a[n-1]^2/a[n-2]] - 1}, a, {n, 40}] (* Bruno Berselli, Feb 17 2016 *)
LinearRecurrence[{2,0,0,-1},{4,8,15,28},40] (* Harvey P. Dale, Mar 05 2019 *)

Vec((4-x^2-2*x^3)/((1-x)*(1-x-x^2-x^3)) + O(x^40)) \\ Colin Barker, Feb 13 2016

T(a0, a1, maxn) = a=vector(maxn); a[1]=a0; a[2]=a1; for(n=3, maxn, a[n]=ceil(a[n-1]^2/a[n-2])-1); a
T(4, 8, 30) \\ Colin Barker, Feb 14 2016

a(n) = 2*a(n-1) - a(n-4).
G.f.: (4-x^2-2*x^3) / ((1-x)*(1-x-x^2-x^3)). - Colin Barker, Feb 08 2012
a(n) = A008937(n+3) = A027084(n+3)+1. [first index correct by R. J. Mathar, Jun 24 2020]
a(n) = 2*a(n-1) - A008937(n). - Vincenzo Librandi, Feb 12 2016

Comments moved to formula, and typo in data fixed by Colin Barker, Feb 13 2016

cp's OEIS Frontend

A209972 Number of binary words of length n avoiding the subword given by the binary expansion of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A126198 Triangle read by rows: T(n,k) (1 <= k <= n) = number of compositions of n into parts of size <= k.

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A189161 T(n,k)=Number of nXk binary arrays without the pattern 0 0 1 1 diagonally, vertically or horizontally.

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A352105 Numbers whose maximal tribonacci representation (A352103) is palindromic.

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A027084 G.f.: x^2*(x^2 + x + 1)/(x^4 - 2*x + 1).

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A055216 Triangle T(n,k) by rows, n >= 0, 0<=k<=n: T(n,k) = Sum_{i=0..n-k} binomial(n-k,i) *Sum_{j=0..k-i} binomial(i,j).

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A113300 Sum of even-indexed terms of tribonacci numbers.

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A172316 7th column of the array A172119.

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A027084 G.f.: x^2(x^2 + x + 1)/(x^4 - 2x + 1).