A052551 -id:A052551 - cp's OEIS frontend

A305750 Number of achiral color patterns (set partitions) in a row or cycle of length n with 4 or fewer colors (subsets).

1, 1, 2, 3, 7, 11, 27, 43, 107, 171, 427, 683, 1707, 2731, 6827, 10923, 27307, 43691, 109227, 174763, 436907, 699051, 1747627, 2796203, 6990507, 11184811, 27962027, 44739243, 111848107, 178956971, 447392427, 715827883, 1789569707, 2863311531, 7158278827

Offset: 0

Robert A. Russell, Jun 09 2018

An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABCD are equivalent, as are AABCD and BBCDA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a cycle are equivalent, so for cycles AABBCD = BBCDAA = CDAABB.

			For a(4) = 7, the achiral row patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD. The cycle patterns are AAAA, AAAB, AABB, ABAB, AABC, ABAC, and ABCD.

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

Fourth column of A305749.
Cf. A124303 (oriented), A056323 (unoriented), A320934 (chiral), for rows.
Cf. A056292 (oriented), A056354 (unoriented), A320744 (chiral), for cycles.

a:=[1,2,3];; for n in [4..40] do a[n]:=a[n-1]+4*a[n-2]-4*a[n-3]; od; Concatenation([1],a); # Muniru A Asiru, Oct 28 2018

seq(coeff(series((1-3*x^2+x^3)/((1-x)*(1-4*x^2)),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 28 2018

Table[If[EvenQ[n], StirlingS2[(n+8)/2, 4] - 8 StirlingS2[(n+6)/2, 4] + 22 StirlingS2[(n+4)/2, 4] - 23 StirlingS2[(n+2)/2, 4] + 6 StirlingS2[n/2, 4], StirlingS2[(n+7)/2, 4] - 7 StirlingS2[(n+5)/2, 4] + 16 StirlingS2[(n+3)/2, 4] - 12 StirlingS2[(n+1)/2, 4]], {n, 0, 40}]
Ach[n_, k_] := Ach[n,k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]]; (* A304972 *)
k=4; Table[Sum[Ach[n, j], {j, 0, k}], {n, 0, 40}]
(* or *)
CoefficientList[Series[(1-3x^2+x^3)/((1-x)(1-4x^2)), {x,0,40}], x]
(* or *)
Join[{1},LinearRecurrence[{1,4,-4},{1,2,3},40]]
(* or *)
Join[{1},Table[If[EvenQ[n], (4+5 4^(n/2))/12, (2+4^((n+1)/2))/6], {n,40}]]

a(n) = Sum_{j=0..4} Ach(n,j), where Ach(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [0<=n<2 & n==k].
G.f.: (1 - 3x^2 + x^3) / ((1-x) * (1-4x^2)).
a(2m) = S2(m+4,4) - 8*S2(m+3,4) + 22*S2(m+2,4) - 23*S2(m+1,4) + 6*S2(m,4);
a(2m-1) = S2(m+3,4) - 7*S2(m+2,4) + 16*S2(m+1,4) - 12*S2(m,4), where S2(n,k) is the Stirling subset number A008277.
For m>0, a(2m) = (4 + 5*4^m) / 12.
a(2m-1) = (2 + 4^m) / 6.
a(n) = 2*A056323(n) - A124303(n) = A124303(n) - 2*A320934(n) = A056323(n) - A320934(n).
a(n) = 2*A056354(n) - A056292(n) = A056292(n) - 2*A320744(n) = A056354(n) - A320744(n).
a(n) = A057427(n) + A052551(n-2) + A304973(n) + A304974(n).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3). - Muniru A Asiru, Oct 28 2018

A289404 Binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 566", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 1, 11, 11, 111, 111, 1111, 1111, 11111, 11111, 111111, 111111, 1111111, 1111111, 11111111, 11111111, 111111111, 111111111, 1111111111, 1111111111, 11111111111, 11111111111, 111111111111, 111111111111, 1111111111111, 1111111111111, 11111111111111

Offset: 0

Robert Price, Jul 05 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A289405, A052551, A032085.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 566; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Table[ca[[i, j, j]], {j, 1, i}], 10], {i, 1, stages - 1}]

Conjectures from Colin Barker, Jul 05 2017: (Start)
G.f.: 1 / ((1 - x)*(1 - 10*x^2)).
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n>2.
(End)
Conjectures from Federico Provvedi, Nov 21 2018: (Start)
a(n) = (10^(1 + floor(n/2)) - 1)/9.
a(n) = (sqrt(10)^(n+1)*((sqrt(10)-1)*(-1)^n+(sqrt(10)+1))-2)/18.
(End)

A305751 Number of achiral color patterns (set partitions) in a row or cycle of length n with 5 or fewer colors (subsets).

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 30, 55, 141, 266, 688, 1313, 3407, 6532, 16970, 32595, 84721, 162846, 423348, 813973, 2116227, 4069352, 10580110, 20345735, 52898501, 101726626, 264488408, 508629033, 1322433847, 2543136972, 6612152850, 12715668475

Offset: 0

Robert A. Russell, Jun 09 2018

An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABCDE are equivalent, as are AABCDE and BBCDEA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a cycle are equivalent, so for cycles AABBCDE = BBCDEAA = CDEAABB.

			For a(5) = 12, the achiral patterns for both rows and cycles are AAAAA, AABAA, ABABA, ABBBA, AABCC, ABACA, ABBBC, ABCAB, ABCBA, ABCBD, ABCDA, and ABCDE.

Index entries for linear recurrences with constant coefficients, signature (1,7,-7,-10,10).

Fifth column of A305749.
Cf. A056272 (oriented), A056324 (unoriented), A320935 (chiral), for rows.
Cf. A056293 (oriented), A056355 (unoriented), A320745 (chiral), for cycles.

seq(coeff(series((1-2*x)*(1+2*x-2*x^2-3*x^3+x^4)/((1-x)*(1-2*x^2)*(1-5*x^2)),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 30 2018

Table[If[EvenQ[n], StirlingS2[(n+10)/2, 5] - 13 StirlingS2[(n+8)/2, 5] + 62 StirlingS2[(n+6)/2, 5] - 130 StirlingS2[(n+4)/2, 5] + 110 StirlingS2[(n+2)/2, 5] - 24 StirlingS2[n/2, 5], StirlingS2[(n+9)/2, 5] - 12 StirlingS2[(n+7)/2, 5] + 52 StirlingS2[(n+5)/2, 5] - 95 StirlingS2[(n+3)/2, 5] + 60 StirlingS2[(n+1)/2, 5]], {n, 0, 40}]
Ach[n_, k_] := Ach[n,k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]]; (* A304972 *)
k=5; Table[Sum[Ach[n, j], {j, 0, k}], {n, 0, 40}]
CoefficientList[Series[(1-2x)(1+2x-2x^2-3x^3+x^4) / ((1- x)(1-2x^2)(1-5x^2)), {x,0,40}], x]
Join[{1},LinearRecurrence[{1,7,-7,-10,10},{1,2,3,7,12},40]]
Join[{1}, Table[If[EvenQ[n], (15 + 20 2^(n/2) + 13 5^(n/2)) / 60, (3 + 2 2^((n+1)/2) + 5^((n+1)/2)) / 12], {n,40}]]

a(n) = Sum_{j=0..5} Ach(n,j), where Ach(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [0<=n<2 & n==k].
G.f.: (1 - 2x)*(1+2x-2x^2-3x^3+x^4) / ((1-x)*(1-2x^2)*(1-5x^2)).
a(2m) = S2(m+5,5) - 13*S2(m+4,5) + 62*S2(m+3,5) - 130*S2(m+2,5) + 110*S2(m+1,5) - 24*S2(m,5);
a(2m-1) = S2(m+4,5) - 12*S2(m+3,5) + 52*S2(m+2,5) - 95*S2(m+1,5) + 60*S2(m,5), where S2(n,k) is the Stirling subset number A008277.
For n>0, a(2m) = (15 + 20*2^m + 13*5^m) / 60.
a(2m-1) = (3 + 2*2^m + 5^m) / 12.
a(n) = 2*A056324(n) - A056272(n) = A056272(n) - 2*A320935(n) = A056324(n) - A320935(n).
a(n) = 2*A056355(n) - A056293(n) = A056293(n) - 2*A320745(n) = A056355(n) - A320745(n).
a(n) = A057427(n) + A052551(n-2) + A304973(n) + A304974(n) + A304975(n).
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - 10*a(n-4) + 10*a(n-5). - Muniru A Asiru, Oct 30 2018

A305752 Number of achiral color patterns (set partitions) in a row or cycle of length n with 6 or fewer colors (subsets).

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 31, 58, 159, 312, 883, 1774, 5103, 10368, 30067, 61414, 178815, 366168, 1068259, 2190190, 6395919, 13120944, 38335123, 78665590, 229890591, 471814344, 1378985155, 2830350526, 8272839855, 16980500640, 49633834099, 101878204486

Offset: 0

Robert A. Russell, Jun 09 2018

An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABCDE are equivalent, as are AABCDEF and BBCDEFA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a cycle are equivalent, so for cycles AABCCDEF = BCCDEFAA = CCDEFAAB.

			For a(5) = 12, the achiral patterns for both rows and cycles are AAAAA, AABAA, ABABA, ABBBA, AABCC, ABACA, ABBBC, ABCAB, ABCBA, ABCBD, ABCDA, and ABCDE.

Index entries for linear recurrences with constant coefficients, signature (1,11,-11,-36,36,36,-36).

Sixth column of A305749.
Cf. A056273 (oriented), A056325 (unoriented), A320936 (chiral), for rows.
Cf. A056294 (oriented), A056356 (unoriented), A320746 (chiral), for cycles.

seq(coeff(series((1-10*x^2+x^3+29*x^4-6*x^5-25*x^6+8*x^7)/((1-x)*(1-2*x^2)*(1-3*x^2)*(1-6*x^2)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 30 2018

Table[If[EvenQ[n], StirlingS2[(n+12)/2, 6] - 19 StirlingS2[(n+10)/2, 6] + 140 StirlingS2[(n+8)/2, 6] - 501 StirlingS2[(n+6)/2, 6] + 887 StirlingS2[(n+4)/2, 6] - 692 StirlingS2[(n+2)/2, 6] + 160 StirlingS2[n/2, 6], StirlingS2[(n+11)/2, 6] - 18 StirlingS2[(n+9)/2, 6] + 124 StirlingS2[(n+7)/2, 6] - 404 StirlingS2[(n+5)/2, 6] + 613 StirlingS2[(n+3)/2, 6] - 340 StirlingS2[(n+1)/2, 6]], {n, 0, 40}]
Ach[n_, k_] := Ach[n,k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]]; (* A304972 *)
k=6; Table[Sum[Ach[n, j], {j, 0, k}], {n, 0, 40}]
CoefficientList[Series[(1-10x^2+x^3+29x^4-6x^5-25x^6+8x^7) / ((1-x)(1-2x^2)(1-3x^2)(1-6 x^2)), {x, 0, 40}], x]
LinearRecurrence[{1,11,-11,-36,36,36,-36},{1,1,2,3,7,12,31,58},40]
Join[{1}, Table[If[EvenQ[n], (36 + 45 2^(n/2) + 40 3^(n/2) + 19 6^(n/2)) / 180, (72 + 45 2^((n+1)/2) + 40 3^((n+1)/2) + 13 6^((n+1)/2)) / 360], {n,40}]]

a(n) = Sum_{j=0..6} Ach(n,j), where Ach(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [0<=n<2 & n==k].
G.f.: (1-10x^2+x^3+29x^4-6x^5-25x^6+8x^7) / ((1-x)*(1-2x^2)*(1-3x^2)*(1-6x^2)).
a(2m) = S2(m+6,6) - 19*S2(m+5,6) + 140*S2(m+4,6) - 501*S2(m+3,6) + 887*S2(m+2,6) - 692*S2(m+1,6) + 160*S2(m,6);
a(2m-1) = S2(m+5,6) - 18*S2(m+4,6) + 124*S2(m+3,6) - 404*S2(m+2,6) + 613*S2(m+1,6) - 340*S2(m,6), where S2(n,k) is the Stirling subset number A008277.
For n>0, a(2m) = (36 + 45*2^m + 40*3^m + 19*6^m) / 180.
a(2m-1) = (72 + 45*2^m + 40*3^m + 13*6^m) / 360.
a(n) = 2*A056325(n) - A056273(n) = A056273(n) - 2*A320936(n) = A056325(n) - A320936(n).
a(n) = 2*A056356(n) - A056294(n) = A056294(n) - 2*A320746(n) = A056356(n) - A320936(n).
a(n) = A057427(n) + A052551(n-2) + A304973(n) + A304974(n) + A304975(n) + A304976(n).
a(n) = a(n-1) + 11*a(n-2) - 11*a(n-3) - 36*a(n-4) + 36*a(n-5) + 36*a(n-6) - 36*a(n-7). - Muniru A Asiru, Oct 30 2018

A030111 Triangular array in which k-th entry in n-th row is C([ (n+k)/2 ],k) (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 3, 4, 1, 1, 3, 6, 4, 5, 1, 1, 4, 6, 10, 5, 6, 1, 1, 4, 10, 10, 15, 6, 7, 1, 1, 5, 10, 20, 15, 21, 7, 8, 1, 1, 5, 15, 20, 35, 21, 28, 8, 9, 1, 1, 6, 15, 35, 35, 56, 28, 36, 9, 10, 1, 1, 6, 21, 35, 70, 56, 84, 36, 45, 10, 11, 1, 1, 7, 21, 56, 70, 126, 84, 120, 45, 55, 11, 12, 1

Offset: 1

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Same as A046854, but missing the initial column of ones.

Riordan array (1/((1-x)(1-x^2)),x/(1-x^2)). Diagonal sums are A052551. - Paul Barry, Sep 30 2006

			1;
1 1;
2 1 1;
2 3 1 1;
3 3 4 1 1;
3 6 4 5 1 1;
...

Indranil Ghosh, Rows 0..125, flattened

Cf. A066170.

Flatten[Table[Binomial[Floor[(n+k)/2],k],{n,20},{k,n}]] (* Harvey P. Dale, Jun 03 2014 *)

{T(n, k) = binomial((n+k)\2, k)}; /* Michael Somos, Jul 23 1999 */

printp(matrix(8,8,n,k,binomial((n+k)\2,k)))

for(n=1,7, for(k=1,n,print1(binomial((n+k)\2,k)); if(k==n,print1("; ")); print1(" ")))

G.f.: 1 / (1 - x - xy - x^2 + x^2y + x^3). - Ralf Stephan, Feb 13 2005

Sum(k=1, n, T(n, k)) = F(n+2)-1 where F(n) is the n-th Fibonacci number. - Benoit Cloitre, Oct 07 2002

Description corrected by Michael Somos, Jul 23 1999

Corrected and extended by Harvey P. Dale, Jun 03 2014

A289405 Binary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 566", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 10, 110, 1100, 11100, 111000, 1111000, 11110000, 111110000, 1111100000, 11111100000, 111111000000, 1111111000000, 11111110000000, 111111110000000, 1111111100000000, 11111111100000000, 111111111000000000, 1111111111000000000, 11111111110000000000

Offset: 0

Robert Price, Jul 05 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A289404, A052551, A032085.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 566; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Conjectures from Colin Barker, Jul 05 2017: (Start)
G.f.: 1 / ((1 - 10*x)*(1 - 10*x^2)).
a(n) = 10*a(n-1) + 10*a(n-2) - 100*a(n-3) for n>2.
(End)

A056508 Number of periodic palindromic structures of length n using exactly two different symbols.

Original entry on oeis.org

0, 1, 1, 3, 3, 6, 7, 13, 15, 25, 31, 50, 63, 99, 127, 197, 255, 391, 511, 777, 1023, 1551, 2047, 3090, 4095, 6175, 8191, 12323, 16383, 24639, 32767, 49221, 65535, 98431, 131071, 196743, 262143, 393471, 524287, 786697, 1048575, 1573375, 2097151, 3146255, 4194303

Offset: 1

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.

For odd n, a palindrome cannot be the complement of itself, so a(n) is given by A284855(n,2)/2 - 1. - Andrew Howroyd, Apr 08 2017

			From _Andrew Howroyd_, Apr 07 2017: (Start)
Example for n=6:
Periodic symmetry means results are either in the form abccba or abcdcb.
There are 3 binary words in the form abccba that start with 0 and contain a 1 which are 001100, 010010, 011110. Of these, 011110 is equivalent to 001100 after rotation.
There are 7 binary words in the form abcdcb that start with 0 and contain a 1 which are 000100, 001010, 001110, 010001, 010101, 011011, 011111. Of these, 011111 is equivalent to 000100, 010001 is equivalent to 001010 and 011011 is equivalent to 010010 from the first set.
There are therefore a total of 7 + 3 - 4 = 6 equivalence classes so a(6) = 6.
(End)

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Giovanni Resta, Table of n, a(n) for n = 1..1000

Column 2 of A285012.

Cf. A052551.

(* b = A164090, c = A045674 *)
b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1;
c[n_] := c[n] = If[EvenQ[n], 2^(n/2 - 1) + c[n/2], 2^((n - 1)/2)];
a[n_] := If[OddQ[n], b[n]/2, (1/2)*(b[n] + c[n/2])] - 1;
Array[a, 45] (* Jean-François Alcover, Jun 29 2018, after Andrew Howroyd *)

a(n) = A056503(n) - 1.

a(2n + 1) = 2^n - 1. - Andrew Howroyd, Apr 07 2017

a(17)-a(45) from Andrew Howroyd, Apr 07 2017

A060030 a(1) = 1, a(2) = 2; thereafter a "hole" is defined to be any positive number not in the sequence a(1)..a(n-1) and less than the largest term; if there exists at least one hole, then a(n) is the largest hole, otherwise a(n) = a(n-2) + a(n-1).

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 8, 7, 6, 13, 12, 11, 10, 21, 20, 19, 18, 17, 16, 15, 14, 29, 28, 27, 26, 25, 24, 23, 22, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84, 83

Offset: 1

William Nelles (wnelles(AT)flashmail.com), Mar 17 2001

A self-inverse permutation of the natural numbers: a(a(n)) = n and a(n) <> n for n > 3. [Reinhard Zumkeller, Apr 29 2012]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

See A060482 for successive records, A027383 for the final hole-filling values, A016116 for the difference between top and bottom of downward subsequences, A052551 for number of terms in downward subsequences.

Cf. A060000, A000045.

import Data.List (delete)
a060030 n = a060030_list !! (n-1)
a060030_list = 1 : 2 : f 1 2 [3..] where
   f u v ws = y : f v y (delete y ws) where
     y = if null xs then u + v else last xs
     xs = takeWhile (< v) ws
-- Reinhard Zumkeller, Apr 29 2012

a[1] = 1; a[2] = 2;
a[n_] := a[n] = Module[{A, H}, A = Array[a, n-1]; H = Complement[ Range[a[n-1]], A]; If[H != {}, H[[-1]], a[n-2] + a[n-1]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 23 2024 *)

Offset corrected by Reinhard Zumkeller, Apr 29 2012

Name made more explicit by Jean-François Alcover, Apr 23 2024

A132890 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n that have height k (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 1, 7, 5, 5, 1, 1, 1, 7, 13, 6, 6, 1, 1, 1, 15, 18, 20, 7, 7, 1, 1, 1, 15, 39, 26, 27, 8, 8, 1, 1, 1, 31, 57, 73, 35, 35, 9, 9, 1, 1, 1, 31, 112, 99, 109, 44, 44, 10, 10, 1, 1, 1, 63, 169, 253, 152, 154, 54, 54, 11, 11, 1, 1

Offset: 1

Emeric Deutsch, Sep 08 2007

Sum of terms in row n = binomial(n, floor(n/2)) = A001405(n).
T(n,2) = A052551(n-2) (n >= 2).
T(n,3) = A005672(n) = Fibonacci(n+1) - 2^floor(n/2).
Sum_{k=1..n} k*T(n,k) = A132891(n).

			T(5,3)=4 because we have UDUUU, UUDUU, UUUDD and UUUDU, where U=(1,1) and D=(1,-1).
Triangle starts:
  1;
  1, 1;
  1, 1, 1;
  1, 3, 1, 1;
  1, 3, 4, 1; 1;
  1, 7, 5, 5, 1, 1;

Alois P. Heinz, Rows n = 1..141, flattened
Steven R. Finch, How far might we walk at random?, arXiv:1802.04615 [math.HO], 2018.
R. Kemp, On the average depth of a prefix of the Dycklanguage D_1, Discrete Math., 36, 1981, 155-170.
Toufik Mansour, Gokhan Yilidirim, Longest increasing subsequences in involutions avoiding patterns of length three, Turkish Journal of Mathematics (2019), Section 2.2

Cf. A001405, A052551, A005672, A132891, A068914.

v := ((1-sqrt(1-4*z^2))*1/2)/z: g := proc (k) options operator, arrow: v^k*(1+v)*(1+v^2)/((1+v^(k+1))*(1+v^(k+2))) end proc: T := proc (n, k) options operator, arrow: coeff(series(g(k), z = 0, 50), z, n) end proc: for n from 0 to 12 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, k) option remember; `if`(x=0, z^k, `if`(y>0,
      b(x-2, y-1, k), 0)+ b(x-2, y+1, max(y+1, k)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..n))(b(2*n, 0$2)):
seq(T(n), n=1..16);  # Alois P. Heinz, Sep 05 2017

b[x_, y_, k_] := b[x, y, k] = If[x == 0, z^k, If[y > 0, b[x - 2, y - 1, k], 0] + b[x - 2, y + 1, Max[y + 1, k]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 1, n}]][b[2n, 0, 0]];
Table[T[n], {n, 1, 16}] // Flatten (* Jean-François Alcover, Apr 01 2018, after Alois P. Heinz *)

The g.f. of column k is g(k, z) = v^k*(1+v)*(1+v^2)*/((1+v^(k+1))*(1+v^(k+2))), where v = (1-sqrt(1-4*z^2))/(2*z). (Obtained as the difference G(k,z)-G(k-1,z), where G(k,z) is given in the R. Kemp reference (p. 159).)

Keyword tabl added by Michel Marcus, Apr 09 2013

A210552 Triangle of coefficients of polynomials u(n,x) jointly generated with A210553; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 5, 5, 1, 5, 7, 10, 8, 1, 6, 9, 16, 18, 13, 1, 7, 11, 23, 31, 33, 21, 1, 8, 13, 31, 47, 62, 59, 34, 1, 9, 15, 40, 66, 101, 119, 105, 55, 1, 10, 17, 50, 88, 151, 205, 227, 185, 89, 1, 11, 19, 61, 113, 213, 321, 414, 426, 324, 144, 1, 12, 21, 73

Offset: 1

Clark Kimberling, Mar 22 2012

Let T(n,k) denote the term in row n, column k.
T(n,n):  A000045 (Fibonacci numbers)
T(n,n-1): A010049 (second-order Fibonacci numbers)
T(n,1): 1,1,1,1,1,1,1,1,1,1,1,,...
T(n,2): 2,3,4,5,6,7,8,9,10,11,...
T(n,3): 3,5,7,9,11,13,15,17,19,...
T(n,4): A052905
Row sums: A000225
Alternating row sums: A094024 (signed)
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
1...3...3
1...4...5...5
1...5...7...10...8
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 3x^2.

Cf. A210553, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + 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[%]   (* A210552 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210553 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)

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

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A305750 Number of achiral color patterns (set partitions) in a row or cycle of length n with 4 or fewer colors (subsets).

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A289404 Binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 566", based on the 5-celled von Neumann neighborhood.

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A305751 Number of achiral color patterns (set partitions) in a row or cycle of length n with 5 or fewer colors (subsets).

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A305752 Number of achiral color patterns (set partitions) in a row or cycle of length n with 6 or fewer colors (subsets).

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A030111 Triangular array in which k-th entry in n-th row is C([ (n+k)/2 ],k) (1<=k<=n).

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A289405 Binary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 566", based on the 5-celled von Neumann neighborhood.

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A056508 Number of periodic palindromic structures of length n using exactly two different symbols.

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