cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A179749 Subsequence of A179748. Equals A179748 beginning at the second column.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 5, 4, 1, 1, 2, 6, 9, 5, 1, 1, 2, 6, 15, 14, 6, 1, 1, 2, 6, 20, 29, 20, 7, 1, 1, 2, 6, 23, 49, 49, 27, 8, 1, 1, 2, 6, 24, 71, 98, 76, 35, 9, 1, 1, 2, 6, 24, 91, 169, 174, 111, 44, 10, 1, 1, 2, 6, 24, 106, 259, 343, 285, 155, 54, 11, 1, 1, 2, 6, 24, 115, 360
Offset: 1

Views

Author

Mats Granvik, Jul 26 2010

Keywords

Comments

Matrix inverse is A179750.

Examples

			Table begins:
1,
1,1,
1,2,1,
1,2,3,1,
1,2,5,4,1,
1,2,6,9,5,1,
1,2,6,15,14,6,1,
1,2,6,20,29,20,7,1,
1,2,6,23,49,49,27,8,1,
1,2,6,24,71,98,76,35,9,1,
1,2,6,24,91,169,174,111,44,10,1,
1,2,6,24,106,259,343,285,155,54,11,1,
1,2,6,24,115,360,602,628,440,209,65,12,1,
1,2,6,24,119,461,961,1230,1068,649,274,77,13,1,

Crossrefs

Cf. A179748, A179750.

A186425 Antidiagonal sums of A179748.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 10, 14, 20, 30, 45, 68, 104, 161, 251, 393, 618, 976, 1547, 2459, 3917, 6251, 9993, 15999, 25647, 41157, 66108, 106272, 170961, 275202, 443250, 714265, 1151486, 1857057, 2995991, 4834907, 7804653, 12601553, 20351114, 32872743, 53107823, 85811996, 138674777, 224130364, 362286475
Offset: 1

Views

Author

Mats Granvik, Feb 21 2011

Keywords

Comments

a(n+1)/a(n) tends to the golden ratio. [Note added by Joerg Arndt, Mar 16 2013: this is only a conjecture so far!]
Grows slower than the Fibonacci sequence. More complicated than the Fibonacci sequence.
The divisibility related table A051731 can be described by the recurrence:
T(n,1) = 1, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) - 1*(Sum_{i=1..k-1} T(n-i,k)).
The silver means can be found as limiting ratios of the antidiagonal sums of the tables described by the following similar recurrences:
T(n,1) = 1, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) + 0*(Sum_{i=1..k-1} T(n-i,k)). --> antidiagonal sums limiting ratio tends to the golden ratio, A001622.
T(n,1) = 1, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) + 1*(Sum_{i=1..k-1} T(n-i,k)). --> antidiagonal sums limiting ratio tends to the silver ratio, A014176.
T(n,1) = 1, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) + 2*(Sum_{i=1..k-1} T(n-i,k)). --> antidiagonal sums limiting ratio tends to the bronze ratio, A098316
The limiting ratio becomes apparent after the first 275 terms or so of the antidiagonal sums.
The empirical observation that the ratio a(n+1)/a(n) tends to the golden ratio 1.6180339887498... has been verified up to a(1500)/a(1499) which gives the first 65 digits of A001622. - Mats Granvik, Sep 16 2017

Links

Crossrefs

Cf. A001622, A179748, cumulative sums of A186426.

Programs

  • Mathematica
    Clear[a,t]; nn = 58; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}], 0]; a = Table[Total[Table[t[n - k + 1, k], {k, 1, nn}]], {n, 1, nn}]; a (* Mats Granvik, Apr 27 2013 *)

A177510 Number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i and pi >= p0.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 25, 46, 87, 167, 324, 634, 1248, 2466, 4887, 9706, 19308, 38455, 76659, 152925, 305232, 609488, 1217429, 2432399, 4860881, 9715511, 19421029, 38826059, 77626471, 155211785, 310357462, 620608652, 1241046343, 2481817484, 4963191718, 9925669171, 19850186856, 39698516655, 79394037319
Offset: 0

Views

Author

Mats Granvik, Dec 11 2010

Keywords

Comments

a(0)=1, otherwise row sums of A179748.
For n>=1 cumulative sums of A008930.
a(n) is proportional to A048651*A000079. The error (a(n)-A048651*A000079) divided by sequence A186425 tends to the golden ratio A001622. This can be seen when using about 1000 decimals of the constant A048651 = 0.2887880950866024212... - [Mats Granvik, Jan 01 2015]
From Gus Wiseman, Mar 31 2022: (Start)
Also the number of integer compositions of n with exactly one part on or above the diagonal. For example, the a(1) = 1 through a(5) = 8 compositions are:
(1) (2) (3) (4) (5)
(11) (21) (31) (41)
(111) (112) (212)
(211) (311)
(1111) (1112)
(1121)
(2111)
(11111)
(End)

Examples

			From _Joerg Arndt_, Mar 24 2014: (Start)
The a(7) = 25 such compositions are:
01:  [ 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 3 ]
05:  [ 1 1 1 2 1 1 ]
06:  [ 1 1 1 2 2 ]
07:  [ 1 1 1 3 1 ]
08:  [ 1 1 1 4 ]
09:  [ 1 1 2 1 1 1 ]
10:  [ 1 1 2 1 2 ]
11:  [ 1 1 2 2 1 ]
12:  [ 1 1 2 3 ]
13:  [ 1 1 3 1 1 ]
14:  [ 1 1 3 2 ]
15:  [ 1 2 1 1 1 1 ]
16:  [ 1 2 1 1 2 ]
17:  [ 1 2 1 2 1 ]
18:  [ 1 2 1 3 ]
19:  [ 1 2 2 1 1 ]
20:  [ 1 2 2 2 ]
21:  [ 1 2 3 1 ]
22:  [ 2 2 3 ]
23:  [ 2 3 2 ]
24:  [ 3 4 ]
25:  [ 7 ]
(End)

Crossrefs

Cf. A238859 (compositions with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
The version for partitions is A001477, strong A002620.
The version for permutations is A057427, strong A000295.
The opposite version is A238874, first column of A352522.
The version for fixed points is A240736, nonfixed A352520.
The strong version is A351983, column k=1 of A352524.
This is column k = 1 of A352525.
A238349 counts compositions by fixed points, first col A238351.
A352517 counts weak excedances of standard compositions.
Cf. A008930, A010054, A088218, A098825, A114088, A219282, A238352, A319005, A350839, A352488, A352489, A352514, A352515, A352516.

Programs

  • Maple
    A179748 := proc(n,k) option remember; if k= 1 then 1; elif k> n then 0 ; else add( procname(n-i,k-1),i=1..k-1) ; end if; end proc:
A177510 := proc(n) add(A179748(n,k),k=1..n) ;end proc:
seq(A177510(n),n=1..20) ; # R. J. Mathar, Dec 14 2010
  • Mathematica
    Clear[t, nn]; nn = 39; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}], 0]; Table[Sum[t[n, k], {k, 1, n}], {n, 1, nn}] (* Mats Granvik, Jan 01 2015 *)
pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]]; Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pdw[#]==1&]],{n,0,10}] (* Gus Wiseman, Mar 31 2022 *)
  • PARI
    N=66; q='q+O('q^N); Vec( 1 + q/(1-q) * sum(n=0, N, q^n * prod(k=1, n, (1-q^k)/(1-q) ) ) ) \\ Joerg Arndt, Mar 24 2014
  • Sage
    @CachedFunction
def T(n, k): # A179748
    if n == 0:  return int(k==0);
    if k == 1:  return int(n>=1);
    return sum( T(n-i, k-1) for i in [1..k-1] );
# to display triangle A179748 including column zero = [1,0,0,0,...]:
#for n in [0..10]: print([ T(n,k) for k in [0..n] ])
def a(n): return sum( T(n,k) for k in [0..n] )
print([a(n) for n in [0..66]])
# Joerg Arndt, Mar 24 2014

Formula

G.f.: 1 + q/(1-q) * sum(n>=0, q^n * prod(k=1..n, (1-q^k)/(1-q) ) ). [Joerg Arndt, Mar 24 2014]

Extensions

New name and a(0) = 1 prepended, Joerg Arndt, Mar 24 2014

A175105 Triangle T(n,k) read by rows. T(n,1)=1; T(n,k) = Sum_{i=1..k-1} ( T(n-i,k-1) + T(n-i,k) ), k>1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 6, 1, 1, 5, 21, 22, 8, 1, 1, 6, 40, 64, 38, 10, 1, 1, 7, 72, 163, 140, 58, 12, 1, 1, 8, 125, 382, 442, 256, 82, 14, 1, 1, 9, 212, 846, 1259, 954, 420, 110, 16, 1, 1, 10, 354, 1800, 3334, 3166, 1794, 640, 142, 18, 1, 1, 11, 585, 3719, 8366, 9657, 6754, 3074, 924, 178, 20, 1
Offset: 1

Views

Author

Mats Granvik, Feb 10 2010

Keywords

Comments

Are there closed forms for diagonals and columns?
With the definition of the array, I note that the sequence (phi(k)) (phi(k)= g.f. of the column number k) is given by the recurrence relation: phi(k+1)=phi(k)*(1-z^k)/(1-2*z+z^(k+1)). The consequence is: the sequence number k+1 column is the convolution of the k-one and a "-acci like" sequence whose g.f. is given by (1-z^k)/(1-2*z+z^(k+1)). E.g., the 2-column is the convolution of the 1-column and the sequence 1, 2, 3, 5, ... classical Fibonacci sequence without the first 1. The 3-column is the convolution of the 2-column and 1, 2, 4, 7, 13, ... tribonacci like-sequence (exactly: A000073 without beginning 0, 0, 1). - Richard Choulet, Feb 19 2010
Relation to metallic means:
T(n,1)=1, k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) + 0*Sum_{i=1..k-1} T(n-i,k)
has antidiagonal sums for which the limiting ratio tends to the golden ratio, A001622.
T(n,1)=1, k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) + 1*Sum_{i=1..k-1} T(n-i,k)
has antidiagonal sums for which the limiting ratio tends to the silver ratio, A014176.
T(n,1)=1, k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) + 2*Sum_{i=1..k-1} T(n-i,k)
has antidiagonal sums for which the limiting ratio tends to the bronze ratio, A098316.
A similar point can be made about variations of the Pascal triangle.

Examples

			Table begins:
  n/k| 1    2    3    4    5    6    7    8    9   10   11
  ---+-----------------------------------------------------
   1 | 1
   2 | 1    1
   3 | 1    2    1
   4 | 1    3    4    1
   5 | 1    4   10    6    1
   6 | 1    5   21   22    8    1
   7 | 1    6   40   64   38   10    1
   8 | 1    7   72  163  140   58   12    1
   9 | 1    8  125  382  442  256   82   14    1
  10 | 1    9  212  846 1259  954  420  110   16    1
  11 | 1   10  354 1800 3334 3166 1794  640  142   18    1
Example: T(8,4) = 163 because it is the sum of the numbers:
  10    6
  21   22
  40   64
For k=1, we obtain phi(k)(z)=1/(1-z) which is clear; for k=2, we obtain phi(k)(z)=1/(1-z)^2. For k=3, we obtain phi(3)(z)=(1+z)/((1-2*z+z^3)*(1-z)); this is A001891 without the beginning zero. - _Richard Choulet_, Feb 19 2010

Links

Crossrefs

Cf. A172119, A051731, A001891 (column k=3), A176084 (row sums).
(1-((-1)^T(n, k)))/2 = T(n, k) mod 2 = A051731.
Cf. A179807=antidiagonal sums. A179748 has simpler recurrence.

Programs

  • Excel
    =if(column()=1;1;if(row()>=column();sum(indirect(address(row()-column()+1;column()-1;4)&":"&address(row()-column()+column()-1;column()-1;4);4))+sum(indirect(address(row()-column()+1;column();4)&":"&address(row()-column()+column()-1;column();4);4));0)) ' Mats Granvik, Mar 28 2010
  • Maple
    A175105 := proc(n,k) if k =1 then 1; elif k > n or k< 1 then 0 ; else    add(procname(n-i,k-1)+procname(n-i,k),i=1..k-1) ; end if; end proc; # R. J. Mathar, Feb 16 2011
  • Mathematica
    T[_, 1] = 1;
T[n_, k_] /; 1, ] = 0;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 19 2019 *)

Formula

The g.f of the number k column is phi(k)(z) = (1/(1-z))*Product_{i=1..k-1}(1-z^i)/(1-2*z+z^(i+1)). - Richard Choulet, Feb 19 2010

Extensions

Corrected and edited by Mats Granvik, Jul 28 2010, Dec 09 2010
Choulet formulas indices shifted (to adapt to the new column index) by R. J. Mathar, Dec 13 2010

A179750 Triangle T(n,k) read by rows. Matrix inverse of A179749.

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -2, 4, -3, 1, 4, -8, 7, -4, 1, -7, 14, -14, 11, -5, 1, 11, -22, 25, -25, 16, -6, 1, -18, 36, -44, 51, -41, 22, -7, 1, 35, -70, 83, -99, 92, -63, 29, -8, 1, -76, 152, -166, 188, -190, 155, -92, 37, -9, 1, 166, -332, 337, -354, 373, -345, 247, -129, 46
Offset: 1

Views

Author

Mats Granvik, Jul 26 2010

Keywords

Comments

First column of this triangle is A127926.

Examples

			Table begins:
1,
-1,1,
1,-2,1,
-2,4,-3,1,
4,-8,7,-4,1,
-7,14,-14,11,-5,1,
11,-22,25,-25,16,-6,1,
-18,36,-44,51,-41,22,-7,1,
35,-70,83,-99,92,-63,29,-8,1,
-76,152,-166,188,-190,155,-92,37,-9,1,
166,-332,337,-354,373,-345,247,-129,46,-10,1,
-358,716,-693,678,-717,719,-592,376,-175,56,-11,1,

Crossrefs

Cf. A179748, A179749, A127926.
Showing 1-5 of 5 results.

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