A169985 -id:A169985 - cp's OEIS frontend

A323955 Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} with no block containing k cyclically successive vertices, n >= 1, 2 <= k <= n + 1.

1, 1, 2, 1, 4, 5, 4, 10, 14, 15, 11, 36, 46, 51, 52, 41, 145, 184, 196, 202, 203, 162, 631, 806, 855, 869, 876, 877, 715, 3015, 3847, 4059, 4115, 4131, 4139, 4140, 3425, 15563, 19805, 20813, 21056, 21119, 21137, 21146, 21147, 17722, 86144, 109339, 114469

Offset: 1

Gus Wiseman, Feb 10 2019

Cyclically successive means 1 is a successor of n.

			Triangle begins:
    1
    1    2
    1    4    5
    4   10   14   15
   11   36   46   51   52
   41  145  184  196  202  203
  162  631  806  855  869  876  877
  715 3015 3847 4059 4115 4131 4139 4140
Row 4 counts the following partitions:
  {{13}{24}}      {{12}{34}}      {{1}{234}}      {{1234}}
  {{1}{24}{3}}    {{13}{24}}      {{12}{34}}      {{1}{234}}
  {{13}{2}{4}}    {{14}{23}}      {{123}{4}}      {{12}{34}}
  {{1}{2}{3}{4}}  {{1}{2}{34}}    {{124}{3}}      {{123}{4}}
                  {{1}{23}{4}}    {{13}{24}}      {{124}{3}}
                  {{12}{3}{4}}    {{134}{2}}      {{13}{24}}
                  {{1}{24}{3}}    {{14}{23}}      {{134}{2}}
                  {{13}{2}{4}}    {{1}{2}{34}}    {{14}{23}}
                  {{14}{2}{3}}    {{1}{23}{4}}    {{1}{2}{34}}
                  {{1}{2}{3}{4}}  {{12}{3}{4}}    {{1}{23}{4}}
                                  {{1}{24}{3}}    {{12}{3}{4}}
                                  {{13}{2}{4}}    {{1}{24}{3}}
                                  {{14}{2}{3}}    {{13}{2}{4}}
                                  {{1}{2}{3}{4}}  {{14}{2}{3}}
                                                  {{1}{2}{3}{4}}

First column (k = 2) is A000296. Second column (k = 3) is A323949. Rightmost terms are A000110. Second to rightmost terms are A058692.

Cf. A000126, A000325, A001610, A001644, A169985, A306351, A306357, A323950, A323954.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Select[Partition[Range[n],k,1,1],Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&],Range[n]]],{n,7},{k,2,n+1}]

A323956 Triangle read by rows: T(n, k) = 1 + n * (n - k) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 13, 9, 5, 1, 21, 16, 11, 6, 1, 31, 25, 19, 13, 7, 1, 43, 36, 29, 22, 15, 8, 1, 57, 49, 41, 33, 25, 17, 9, 1, 73, 64, 55, 46, 37, 28, 19, 10, 1, 91, 81, 71, 61, 51, 41, 31, 21, 11, 1, 111, 100, 89, 78, 67, 56, 45, 34, 23, 12, 1

Offset: 1

Gus Wiseman, Feb 10 2019

			Triangle begins:
  n\k:   1   2   3   4   5   6   7   8   9  10  11  12
  ====================================================
    1:   1
    2:   3   1
    3:   7   4   1
    4:  13   9   5   1
    5:  21  16  11   6   1
    6:  31  25  19  13   7   1
    7:  43  36  29  22  15   8   1
    8:  57  49  41  33  25  17   9   1
    9:  73  64  55  46  37  28  19  10   1
   10:  91  81  71  61  51  41  31  21  11   1
   11: 111 100  89  78  67  56  45  34  23  12   1
   12: 133 121 109  97  85  73  61  49  37  25  13   1
  etc.

G. C. Greubel, Rows n = 1..100 of triangle, flattened

First column is A002061. Second column is A000290. Third column is A028387.

Cf. A000126, A001610, A001644, A006000, A169985, A306351, A323952, A323955.

[[1+n*(n-k): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Apr 22 2019

Mathematica
```
Table[1+n*(n-k),{n,12},{k,n}]//Flatten
```

{T(n,k) = 1+n*(n-k)}; \\ G. C. Greubel, Apr 22 2019

[[1+n*(n-k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Apr 22 2019

From Werner Schulte, Feb 12 2019: (Start)
G.f.: Sum_{n>0,k=1..n} T(n,k)*x^k*t^n = x*t*((1-t+2*t^2)*(1-x*t) + (1-t)*t)/((1-t)^3*(1-x*t)^2).
Row sums: Sum_{k=1..n} T(n,k) = A006000(n-1) for n > 0.
Recurrence: T(n,k) = T(n,k-1) - n for 1 < k <= n with initial values T(n,1) = n^2-n+1 for n > 0.
Recurrence: T(n,k) = T(n-1,k) + 2*n-k-1 for 1 <= k < n with initial values T(n,n) = 1 for n > 0.
(End)

A271355 Triangular array: T(n,k) = |round((r^n)*(s^k))|, where r = golden ratio = (1+sqrt(5))/2, s = (1-sqrt(5))/2, 1 <= k <= n, n >= 1.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 7, 4, 3, 2, 1, 11, 7, 4, 3, 2, 1, 18, 11, 7, 4, 3, 2, 1, 29, 18, 11, 7, 4, 3, 2, 1, 47, 29, 18, 11, 7, 4, 3, 2, 1, 76, 47, 29, 18, 11, 7, 4, 3, 2, 1, 123, 76, 47, 29, 18, 11, 7, 4, 3, 2, 1, 199, 123, 76, 47, 29, 18, 11, 7, 4, 3

Offset: 1

Clark Kimberling, May 01 2016

Row n consists of the first n numbers of A169985 = (1,2,3,4,7,... ) in reverse order; these are the Lucas numbers, A000032, with order of initial two terms reversed. Every column of the triangle is A169985.

			First six rows:
  1
  2   1
  3   2   1
  4   3   2   1
  7   4   3   2   1
  11  7   4   3   2   1

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A169985, A000032, A000045, A104762

r = N[(1 + Sqrt[5])/2, 100]; s = N[(1 - Sqrt[5])/2, 100];
t = Table[Abs[Round[(r^n)*(s^k)]], {n, 0, 15}, {k, 1, n}];
Flatten[t]  (* A271355, sequence *)
TableForm[t]  (* A271355, array *)

T(n,k) = |round((r^n)*(s^k))|, where r = golden ratio = (1+sqrt(5))/2, s = (1-sqrt(5))/2, 1 <= k <= n, n >= 1.

T(k+j-1,j) = A000032(k) = k-th Lucas number, for k >= 2.

A306416 Number of ordered set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 26, 84, 950, 6000, 62522, 556116, 6259598, 69319848, 874356338, 11384093196, 161462123894, 2397736692144, 37994808171962, 631767062124564, 11088109048500158, 203828700127054008, 3928762035148317314, 79079452776283889820, 1661265965479375937030, 36332908076071038467520, 826376466514358722894154

Offset: 0

Gus Wiseman, Feb 14 2019

nonn

			The a(4) = 2 ordered set partitions are: {{1,3},{2,4}}, {{2,4},{1,3}}.

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000126, A000296, A000670, A001610, A032032 (adjacencies allowed), A052841 (singletons allowed), A124323, A169985, A306417, A324011 (orderless case), A324012, A324015.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Length[stn]!,{stn,Select[sps[Range[n]],And[Count[#,{_}]==0,Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]}],{n,0,10}]

a(12)-a(26) from Alois P. Heinz, Feb 14 2019

A254729 Number of numbers j + ksqrt(2) of length n, where the length is the least number of steps to reach 0, the allowable steps being x -> x + 1 and x -> xsqrt(2).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282, 54018521, 87403803

Offset: 0

Clark Kimberling, Feb 06 2015

See the MathOverflow link for a proof that the sequence coincides with the Lucas sequence, A000032, beginning at 4.

Therefore also the same as A080023 (beginning at 2). - Georg Fischer, Oct 09 2018

			One can view the minimal paths in a tree having generation g(0) = {0} followed by generations g(1) = {1}, g(2) = {2, sqrt(2)}, g(3) = {3, 2*sqrt(2), 1+sqrt(2)}, and so on. Duplicates are removed as they occur. Also, a(n) = |g(n)| for n >= 0.

Clark Kimberling, Table of n, a(n) for n = 0..1000
MathOverflow, A possibly surprising appearance of Lucas numbers
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A080023, A169985, A252864.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x^4)/(1-x-x^2))); // G. C. Greubel, Sep 30 2018

t = NestList[DeleteDuplicates[Flatten[Map[{# + {0, 1}, {Last[#], 2*First[#]}} &, #], 1]] &, {{0, 0}}, 25] ; s[0] = t[[1]]; s[n_] := s[n] = Union[t[[n + 1]], s[n - 1]]; g[n_] := Complement[s[n], s[n - 1]]; g[0] = {{0, 0}}; Table[Length[g[z]], {z, 0, 25}]
CoefficientList[Series[(-1 + x^4)/(-1 + x + x^2), {x, 0, 39}], x] (* Robert G. Wilson v, Feb 28 2015 *)

x='x+O('x^40); Vec((1-x^4)/(1-x-x^2)) \\ G. C. Greubel, Sep 30 2018

a(n) = a(n-1) + a(n-2) for n >= 6.

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

A382641 a(n) = round(c^n), where c is the supergolden ratio A092526.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 21, 31, 46, 67, 98, 144, 211, 309, 453, 664, 973, 1426, 2090, 3063, 4489, 6579, 9642, 14131, 20710, 30352, 44483, 65193, 95545, 140028, 205221, 300766, 440794, 646015, 946781, 1387575, 2033590, 2980371, 4367946, 6401536, 9381907, 13749853

Offset: 0

Jwalin Bhatt, Apr 01 2025

Wikipedia, Supergolden ratio
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Cf. A000930, A001609, A092526, A169985, A205579.

r = Root[x^3-x^2-1, 1]; Table[Round[r^i], {i,0,120 }]
CoefficientList[Series[(1+x^2+x^4-x^8)/(1-x-x^3), {x,0,120}], x]

G.f.: (1 + x^2 + x^4 - x^8)/(1 - x - x^3).
a(n) = a(n-1) + a(n-3) for n>=9.
a(n) = round(((2/3)*cos((1/3)*arccos(29/2))+1/3)^n) = round(A092526^n).
a(n) = A001609(n) for n>=6.

cp's OEIS Frontend

A323955 Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} with no block containing k cyclically successive vertices, n >= 1, 2 <= k <= n + 1.

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A323956 Triangle read by rows: T(n, k) = 1 + n * (n - k) for 1 <= k <= n.

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A271355 Triangular array: T(n,k) = |round((r^n)*(s^k))|, where r = golden ratio = (1+sqrt(5))/2, s = (1-sqrt(5))/2, 1 <= k <= n, n >= 1.

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A306416 Number of ordered set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

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A254729 Number of numbers j + k*sqrt(2) of length n, where the length is the least number of steps to reach 0, the allowable steps being x -> x + 1 and x -> x*sqrt(2).

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Magma

Mathematica

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Formula

A382641 a(n) = round(c^n), where c is the supergolden ratio A092526.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A254729 Number of numbers j + ksqrt(2) of length n, where the length is the least number of steps to reach 0, the allowable steps being x -> x + 1 and x -> xsqrt(2).