A052980 -id:A052980 - cp's OEIS frontend

A008998 a(n) = 2*a(n-1) + a(n-3), with a(0)=1 and a(1)=2.

1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064, 11169, 24634, 54332, 119833, 264300, 582932, 1285697, 2835694, 6254320, 13794337, 30424368, 67103056, 148000449, 326425266, 719953588, 1587907625, 3502240516, 7724434620, 17036776865, 37575794246

Offset: 0

N. J. A. Sloane

A transform of A000079 under the mapping g(x)->(1/(1-x^3))g(x/(1-x^3)). - Paul Barry, Oct 20 2004
The binomial transform yields 1,3,9,..., i.e., A049220 without the leading zeros. - R. J. Mathar, May 15 2008
a(n-3) is the top left entry of the n-th power of the 3 X 3 matrix [0, 0, 1; 1, 1, 1; 0, 1, 1] or of the 3 X 3 matrix [0, 1, 0; 0, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) equals the number of n-length words on {0,1,2} such that 0 appears only in a run which length is a multiple of 3. - Milan Janjic, Feb 17 2015
a(n) is the number of ways to fill a 1 X n strip of tiles, using only trominos, of length 3, and squares which can be chosen to have one of two possible colors. - Michael Tulskikh, Feb 12 2020
For x the real root of x^3 - 2*x^2 - 1 from A356035, then x^n = a(n-4)*x^2 + a(n-2)*x + a(n-3). - Greg Dresden and Qianhuai He, Jul 01 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 452
B. Rittaud, On the Average Growth of Random Fibonacci Sequences, Journal of Integer Sequences, 10 (2007), Article 07.2.4.
Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Cf. A077852, A077926. Partial sums of A052980.

a:=[1,2,4];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-3]; od; a; # G. C. Greubel, Feb 14 2020

[ n eq 1 select 1 else n eq 2 select 2 else n eq 3 select 4 else 2*Self(n-1)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Aug 21 2011

A008998 := proc(n) option remember; if n <= 2 then 2^n else 2*procname(n-1) +procname(n-3); fi; end proc;

LinearRecurrence[{2, 0, 1}, {1, 2, 4}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{a(n)=polcoeff(exp(sum(m=1,n+1,((1+sqrt(1+x+x*O(x^n)))^m + (1-sqrt(1+x+x*O(x^n)))^m)*x^m/m)),n)} /* Paul D. Hanna, Dec 21 2012 */

def A008998_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^3) ).list()
A008998_list(40) # G. C. Greubel, Feb 14 2020

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2k, k)*2^(n-3k). - Paul Barry, Oct 20 2004
O.g.f.: 1/(1-2*x-x^3). - R. J. Mathar, May 15 2008
O.g.f.: exp( Sum_{n>=1} ( (1 + sqrt(1+x))^n + (1 - sqrt(1+x))^n ) * x^n/n ). - Paul D. Hanna, Dec 21 2012
G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 + x^2)/( x*(4*k+4 + x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013
a(n) = Sum_{k=0..n} A052980(n). - Greg Dresden, May 28 2020

A219987 Number A(n,k) of tilings of a k X n rectangle using dominoes and right trominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 5, 5, 1, 1, 1, 0, 11, 8, 11, 0, 1, 1, 1, 24, 55, 55, 24, 1, 1, 1, 0, 53, 140, 380, 140, 53, 0, 1, 1, 1, 117, 633, 2319, 2319, 633, 117, 1, 1, 1, 0, 258, 1984, 15171, 21272, 15171, 1984, 258, 0, 1

Offset: 0

Alois P. Heinz, Dec 02 2012

			A(3,3) = 8, because there are 8 tilings of a 3 X 3 rectangle using dominoes and right trominoes:
  .___._.   .___._.   .___._.   .___._.
  |___| |   |___| |   |___| |   |_. | |
  | ._|_|   | | |_|   | |___|   | |_|_|
  |_|___|   |_|___|   |_|___|   |_|___|
  ._.___.   ._.___.   ._.___.   ._.___.
  | |___|   | | ._|   | |___|   | |___|
  |___| |   |_|_| |   |_|_. |   |_| | |
  |___|_|   |___|_|   |___|_|   |___|_|
Square array A(n,k) begins:
  1,  1,   1,    1,     1,       1,         1,          1, ...
  1,  0,   1,    0,     1,       0,         1,          0, ...
  1,  1,   2,    5,    11,      24,        53,        117, ...
  1,  0,   5,    8,    55,     140,       633,       1984, ...
  1,  1,  11,   55,   380,    2319,     15171,      96139, ...
  1,  0,  24,  140,  2319,   21272,    262191,    2746048, ...
  1,  1,  53,  633, 15171,  262191,   5350806,  100578811, ...
  1,  0, 117, 1984, 96139, 2746048, 100578811, 3238675344, ...

Liang Kai, Antidiagonals n = 0..35, flattened (Antidiagonals n = 0..31 from Alois P. Heinz)
Kai Liang, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025. See p. 25, Table 4.
Wikipedia, Domino
Wikipedia, Tromino

Columns (or rows) k=0-10 give: A000012, A059841, A052980, A165716, A165791, A219988, A219989, A219990, A219991, A219992, A219993.
Main diagonal gives: A219994.
Cf. A219866, A364457.

b:= proc(n, l) option remember; local k, t;
      if max(l[])>n then 0 elif n=0 or l=[] then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od;
         b(n, subsop(k=2, l))+
         `if`(k>1 and l[k-1]=1, b(n, subsop(k=2, k-1=2, l)), 0)+
         `if`(k `if`(n>=k, b(n, [0$k]), b(k, [0$n])):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, l_] := b[n, l] = Module[{k, t}, If[Max[l] > n, 0, If[n == 0 || l == {}, 1, If[Min[l] > 0, t = Min[l]; b[n-t, l-t], For[k = 1, True, k++, If[l[[k]] == 0, Break[]]]; b[n, ReplacePart[l, k -> 2]] + If[k > 1 && l[[k-1]] == 1, b[n, ReplacePart[l, {k -> 2, k-1 -> 2}]], 0] + If[k < Length[l] && l[[k+1]] == 1, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1}]] + b[n, ReplacePart[l, {k -> 1, k+1 -> 2}]] + b[n, ReplacePart[l, {k -> 2, k+1 -> 1}]], 0] + If[k+1 < Length[l] && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, ReplacePart[l, {k -> 2, k+1 -> 2, k+2 -> 2}]] + b[n, ReplacePart[l, {k -> 2, k+1 -> 2, k+2 -> 1}]], 0]]]]]; a[n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 05 2013, translated from Alois P. Heinz's Maple program *)

from sage.combinat.tiling import TilingSolver, Polyomino
def A(n,k):
    p = Polyomino([(0,0), (0,1)])
    q = Polyomino([(0,0), (0,1), (1,0)])
    T = TilingSolver([p,q], box=[n,k], reusable=True, reflection=True)
    return T.number_of_solutions()
# Ralf Stephan, May 21 2014

A124182 A skewed version of triangular array A081277.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 3, 4, 0, 0, 1, 8, 8, 0, 0, 0, 5, 20, 16, 0, 0, 0, 1, 18, 48, 32, 0, 0, 0, 0, 7, 56, 112, 64, 0, 0, 0, 0, 1, 32, 160, 256, 128, 0, 0, 0, 0, 0, 9, 120, 432, 576, 256, 0, 0, 0, 0, 0, 1, 50, 400, 1120, 1280, 512

Offset: 0

Philippe Deléham, Dec 05 2006

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0,...] where DELTA is the operator defined in A084938. Falling diagonal sums in A052980.

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 3, 4;
  0, 0, 1, 8,  8;
  0, 0, 0, 5, 20, 16;
  0, 0, 0, 1, 18, 48,  32;
  0, 0, 0, 0,  7, 56, 112,  64;
  0, 0, 0, 0,  1, 32, 160, 256,  128;
  0, 0, 0, 0,  0,  9, 120, 432,  576,  256;
  0, 0, 0, 0,  0,  1,  50, 400, 1120, 1280, 512;

Cf. A025192 (column sums). Diagonals include A011782, A001792, A001793, A001794, A006974, A006975, A006976.

T(0,0)=T(1,1)=1, T(n,k)=0 if n < k or if k < 0, T(n,k) = T(n-2,k-1) + 2*T(n-1,k-1).
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A090965(n), (-1)^n*A084120(n), (-1)^n*A006012(n), A033999(n), A000007(n), A001333(n), A084059(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively.
Sum_{k=0..floor(n/2)} T(n-k,k) = Fibonacci(n-1) = A000045(n-1).
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 respectively. - Philippe Deléham, Dec 26 2007
Sum_{k=0..n} T(n,k)*(-x)^(n-k) = A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x= 0,1,2,3,4,5,6 respectively. - Philippe Deléham, Nov 14 2008
G.f.: (1-y*x)/(1-2y*x-y*x^2). - Philippe Deléham, Dec 04 2011
Sum_{k=0..n} T(n,k)^2 = A002002(n) for n > 0. - Philippe Deléham, Dec 04 2011

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 3, 0, 0, 8, 8, 1, 0, 0, 16, 20, 5, 0, 0, 0, 32, 48, 18, 1, 0, 0, 0, 64, 112, 56, 7, 0, 0, 0, 0, 128, 256, 160, 32, 1, 0, 0, 0, 0, 256, 576, 432, 120, 9, 0, 0, 0, 0, 0, 512, 1280, 1120, 400, 50, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 03 2011

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
Skewed version of triangle in A200139.
Triangle without zeros: A207537.
For the version with negative odd numbered columns, which is Riordan ((1-x)/(1-2*x), -x^2/(1-2*x)) see comments on A028297 and A039991. - Wolfdieter Lang, Aug 06 2014
This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - Peter Bala, Jul 14 2015

			The triangle T(n,k) begins:
  n\k      0     1     2     3     4    5   6  7 8 9 10 11 ...
  0:       1
  1:       1     0
  2:       2     1     0
  3:       4     3     0     0
  4:       8     8     1     0     0
  5:      16    20     5     0     0    0
  6:      32    48    18     1     0    0   0
  7:      64   112    56     7     0    0   0  0
  8:     128   256   160    32     1    0   0  0 0
  9:     256   576   432   120     9    0   0  0 0 0
  10:    512  1280  1120   400    50    1   0  0 0 0  0
  11:   1024  2816  2816  1232   220   11   0  0 0 0  0  0
  ...  reformatted and extended. - _Wolfdieter Lang_, Aug 06 2014

C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Columns include A011782, A001792, A001793, A001794, A006974, A006975, A006976.
Diagonals sums are in A052980.
Cf. A028297, A081265, A124182, A131577, A039991 (zero-columns deleted, unsigned and zeros appended).
Cf. A098158, A200139, A207537.
Cf. A028297 (signed version, zeros deleted). Cf. A034839.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - #)/(1 - 2 #)&, #^2/(1 - 2 #)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 for k<0 or for n

Sum_{k=0..n} T(n,k)^2 = A002002(n) for n>0.

Sum_{k=0..n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n), A087455(n), A146559(n), A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 respectively.

G.f.: (1-x)/(1-2*x-y*x^2). - Philippe Deléham, Mar 03 2012

From Peter Bala, Jul 14 2015: (Start)

Factorizes as A034839 * A007318 = (1/(1 - x), x^2/(1 - x)^2) * (1/(1 - x), x/(1 - x)) as a product of Riordan arrays.

T(n,k) = Sum_{i = k..floor(n/2)} binomial(n,2*i) *binomial(i,k). (End)

Name changed, keyword:easy added, crossrefs A028297 and A039991 added, and g.f. corrected by Wolfdieter Lang, Aug 06 2014

A110513 Expansion of (1 + x)/(1 + 2x + x^3).

Original entry on oeis.org

1, -1, 2, -5, 11, -24, 53, -117, 258, -569, 1255, -2768, 6105, -13465, 29698, -65501, 144467, -318632, 702765, -1549997, 3418626, -7540017, 16630031, -36678688, 80897393, -178424817, 393528322, -867954037, 1914332891, -4222194104, 9312342245, -20539017381, 45300228866, -99912799977

Offset: 0

Paul Barry, Jul 24 2005

Diagonal sums of A110511.

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

A minor variation of A052980.

CoefficientList[Series[(1+x)/(1+2x+x^3),{x,0,40}],x] (* or *) LinearRecurrence[ {-2,0,-1},{1,-1,2},40] (* Harvey P. Dale, Jun 27 2012 *)

my(x='x+O('x^50)); Vec((1+x)/(1+2*x+x^3)) \\ G. C. Greubel, Aug 29 2017

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..(n-k)} (-1)^(n-k-j)*C(n-k, j)*(-2)^(j-k)*C(k, j-k).

a(0)=1, a(1)=-1, a(2)=2, a(n) = -2*a(n-1) - a(n-3). - Harvey P. Dale, Jun 27 2012

A205575 Triangle read by rows, related to Pascal's triangle, starting with rows 1; 1,0.

Original entry on oeis.org

1, 1, 0, 2, 2, 1, 3, 5, 4, 1, 5, 12, 14, 8, 2, 8, 25, 38, 32, 15, 3, 13, 50, 94, 104, 71, 28, 5, 21, 96, 215, 293, 260, 149, 51, 8, 34, 180, 468, 756, 822, 612, 304, 92, 13, 55, 331, 980, 1828, 2346, 2136, 1376, 604, 164, 21

Offset: 0

Philippe Deléham, Jan 29 2012

Antidiagonal sums are in A052980, row sums are in A046717.

Similar to A091533 and to A091562. Triangle satisfying the same recurrence as A091533 and A091562, but with the initial values T(0,0) = 1, T(0,1) = 1, T(1,1) = 0.

			Triangle begins :
1
1, 0
2, 2, 1
3, 5, 4, 1
5, 12, 14, 8, 2
8, 25, 38, 32, 15, 3
13, 50, 94, 104, 71, 28, 5

Cf. Column 0: A000045, Diagonals : A000045, A029907, A036681.

Cf. A090171, A090172, A090173, A090174, A091533, A091562 (same recurrence).

T(n,k) = {if(n<0, return(0)); if (n==0, if (k<0, return(0)); if (k==0, return(1))); if (n==1, if (k<0, return(0)); if (k==0, return(1)); if (k==1, return(0))); T(n-1,k)+T(n-1,k-1)+T(n-2,k)+T(n-2,k-1)+T(n-2,k-2);} \\ Michel Marcus, Oct 27 2021

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2) for n>=2, k>=0, with initial conditions specified by first two rows. T(0,0) = 1, T(1,0) = 1, T(1,1) = 0.

a(46), a(48) corrected by Georg Fischer, Oct 27 2021

A207537 Triangle of coefficients of polynomials u(n,x) jointly generated with A207538; see Formula section.

Original entry on oeis.org

1, 2, 1, 4, 3, 8, 8, 1, 16, 20, 5, 32, 48, 18, 1, 64, 112, 56, 7, 128, 256, 160, 32, 1, 256, 576, 432, 120, 9, 512, 1280, 1120, 400, 50, 1, 1024, 2816, 2816, 1232, 220, 11, 2048, 6144, 6912, 3584, 840, 72, 1, 4096, 13312, 16640, 9984, 2912, 364, 13

Offset: 1

Clark Kimberling, Feb 18 2012

Another version in A201701. - Philippe Deléham, Mar 03 2012
Subtriangle of the triangle given by (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012
Columns: A011782, A001792, A001793, A001794, A006974, A006975, A006976. - Philippe Deléham, Mar 03 2012
Diagonal sums: A052980. - Philippe Deléham, Mar 03 2012

			First seven rows:
   1;
   2,   1;
   4,   3;
   8,   8,  1;
  16,  20,  5,
  32,  48, 18, 1;
  64, 112, 56, 7;
From _Philippe Deléham_, Mar 03 2012: (Start)
Triangle A201701 begins:
   1;
   1,   0;
   2,   1,  0;
   4,   3,  0, 0;
   8,   8,  1, 0, 0;
  16,  20,  5, 0, 0, 0;
  32,  48, 18, 1, 0, 0, 0;
  64, 112, 56, 7, 0, 0, 0, 0;
  ... (End)

Cf. A028297, A207538, A133156.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207537, |A028297| *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207538, |A133156| *)
(* Prepending 1 and with offset 0: *)
Tpoly[n_] := HypergeometricPFQ[{-n/2, -n/2 + 1/2}, {1/2}, x + 1];
Table[CoefficientList[Tpoly[n], x], {n, 0, 12}] // Flatten (* Peter Luschny, Feb 03 2021 *)

u(n,x) = u(n-1,x) + (x+1)*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x), where u(1,x)=1, v(1,x)=1.  Also, A207537 = |A028297|.
T(n,k) = 2*T(n-1,k) + T(n-2,k-1). - Philippe Deléham, Mar 03 2012
G.f.: -(1+x*y)*x*y/(-1+2*x+x^2*y). - R. J. Mathar, Aug 11 2015
T(n, k) = [x^k] hypergeom([-n/2, -n/2 + 1/2], [1/2], x + 1) provided offset is set to 0 and 1 prepended. - Peter Luschny, Feb 03 2021

A277731 Fixed point of the morphism 0 -> 01, 1 -> 012, 2 -> 0; starting with a(1) = 0.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2

Offset: 1

N. J. A. Sloane, Nov 07 2016

nonn

After k = 0,1,2,3,... applications of the morphism we have 0, 01, 01012, 01012010120, ... which have lengths 1, 2, 5, 11, 24, 53, 117, ..., satisfying b(n) = 2*b(n-1) + b(n-3) (cf. A052980).

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A052980, A277732, A277733, A277734.

with(ListTools);
T:=proc(S) Flatten(subs( {0=[0,1], 1=[0,1,2], 2=[0]}, S)); end;
S:=[0];
for n from 1 to 10 do S:=T(S); od:
S;

m = 100; (* number of terms required *)
S[1] = {0};
S[n_] := S[n] = SubstitutionSystem[{0 -> {0, 1}, 1 -> {0, 1, 2}, 2 -> {0}}, S[n-1]];
For[n = 2, True, n++, If[PadRight[S[n], m] == PadRight[S[n-1], m], Print["n = ", n]; Break[]]];
Take[S[n], m] (* Jean-François Alcover, Mar 20 2023 *)

A277732 Positions of 0's in A277731.

Original entry on oeis.org

1, 3, 6, 8, 11, 12, 14, 17, 19, 22, 23, 25, 27, 30, 32, 35, 36, 38, 41, 43, 46, 47, 49, 51, 54, 56, 59, 61, 64, 65, 67, 70, 72, 75, 76, 78, 80, 83, 85, 88, 89, 91, 94, 96, 99, 100, 102, 104, 107, 109, 112, 114, 117, 118, 120, 123, 125, 128, 129, 131, 134, 136, 139, 140, 142, 144, 147, 149, 152, 153

Offset: 1

N. J. A. Sloane, Nov 07 2016

nonn

{A277732, A277733, A277734} forms a three-way partition of the positive integers, similar to {A003144, A003145, A003146}.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A052980, A277731, A277733, A277734.

Cf. also A003144, A003145, A003146.

with(ListTools);
T:=proc(S) Flatten(subs( {0=[0,1], 1=[0,1,2], 2=[0]}, S)); end;
S:=[0];
for n from 1 to 14 do S:=T(S); od:
S; # A277731
p0:=[]: p1:=[]: p2:=[]:
for i from 1 to nops(S) do
j:=S[i];
if j=0 then p0:=[op(p0),i];
elif j=1 then p1:=[op(p1),i];
else p2:=[op(p2),i]; fi: od:
p0; # A277732
p1; # A277733
p2; # A277734

m = 1000; (* number of terms of A277731 *)
S[1] = {0};
S[n_] := S[n] = SubstitutionSystem[{0 -> {0, 1}, 1 -> {0, 1, 2}, 2 -> {0}}, S[n - 1]];
For[n = 2, True, n++, If[PadRight[S[n], m] == PadRight[S[n - 1], m], Print["n = ", n]; Break[]]];
A277731 = Take[S[n], m];
Position[A277731, 0] // Flatten (* Jean-François Alcover, Mar 20 2023 *)

Showing 1-10 of 22 results. Next

cp's OEIS Frontend

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