A003116 -id:A003116 - cp's OEIS frontend

A002951 Continued fraction for fifth root of 5.

1, 2, 1, 1, 1, 2, 1, 2, 8, 1, 25, 1, 5, 1, 22, 1, 8, 1, 1, 9, 1, 1, 4, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 6, 2, 46, 1, 12, 1, 32, 1, 2, 3, 2, 3, 55, 1, 12, 3, 8, 1, 1, 11, 1, 4, 1, 1, 1, 2, 1, 1, 7, 1, 1, 4, 3, 3, 3218, 1, 3, 1, 2, 2, 3, 1, 1, 2, 11, 1, 7, 57, 2, 2, 2, 2, 1, 1, 67, 1, 2, 3, 1, 1, 13, 3

Offset: 0

N. J. A. Sloane

Fifth root of 5 = 5^(1/5). - Harry J. Smith, May 10 2009

			1.379729661461214832390063464... = 1 + 1/(2 + 1/(1 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, May 10 2009

H. P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..19999
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
J. Shallit & N. J. A. Sloane, Correspondence 1974-1975
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A005534 (decimal expansion).

Cf. A002363, A002364 (convergents).

SetDefaultRealField(RealField(100)); ContinuedFraction(5^(1/5)); // G. C. Greubel, Nov 02 2018

with(numtheory): cfrac(5^(1/5),100,'quotients'); # Muniru A Asiru, Nov 02 2018

ContinuedFraction[5^(1/5), 100] (* G. C. Greubel, Nov 02 2018 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(5^(1/5)); for (n=1, 20000, write("b002951.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 10 2009

More terms copied from Smith's b-file by Hagen von Eitzen, Jul 20 2009

Offset changed by Andrew Howroyd, Jul 05 2024

A128742 Number of compositions of n which avoid the pattern 112.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 24, 43, 78, 142, 256, 463, 838, 1513, 2735, 4944, 8931, 16139, 29164, 52693, 95213, 172042, 310855, 561682, 1014898, 1833794, 3313454, 5987026, 10817836, 19546558, 35318325, 63816013, 115307993, 208347899, 376459955, 680218580, 1229074432

Offset: 0

Ralf Stephan, May 08 2007

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
S. Heubach and T. Mansour, Enumeration of 3-letter patterns in combinations, arXiv:math/0603285 [math.CO], 2006.

Cf. A003116, A003475.

b:= proc(n, t, l) option remember; `if`(n=0, 1, add(
      b(n-j, is(j=l), j), j=1..min(n, `if`(t, l, n))))
    end:
a:= n-> b(n, false, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 24 2017

b[n_, t_, l_] := b[n, t, l] = If[n == 0, 1, Sum[b[n - j, j == l, j], {j, 1, Min[n, If[t, l, n]]}]];
a[n_] := b[n, False, 0];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

my(N=40, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, x^k*prod(j=1, k-1, 1-x^(2*j))))) \\ Seiichi Manyama, Jan 13 2022

my(N=40, x='x+O('x^N)); Vec(1/sum(k=0, N, (-1)^k*x^k^2/prod(j=1, k, 1-x^(2*j-1)))) \\ Seiichi Manyama, Jan 13 2022

G.f.: 1/( 1 - Sum_{j>=1} x^j*Product_{i=1..j-1} (1-x^(2*i)) ).

G.f.: 1/( Sum_{k>=0} (-1)^k * x^(k^2) / Product_{j=1..k} (1-x^(2*j-1)) ). - Seiichi Manyama, Jan 13 2022

A168396 Triangle, T(n,k) = number of compositions a(1),...,a(j) of n with a(1) = k, such that a(i+1) <= a(i) + 1 for 1 <= i < j.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 4, 2, 1, 1, 9, 6, 4, 2, 1, 1, 15, 11, 7, 4, 2, 1, 1, 26, 19, 12, 7, 4, 2, 1, 1, 45, 33, 21, 13, 7, 4, 2, 1, 1, 78, 57, 37, 22, 13, 7, 4, 2, 1, 1, 135, 99, 64, 39, 23, 13, 7, 4, 2, 1, 1, 234, 172, 112, 68, 40, 23, 13, 7, 4, 2, 1, 1, 406, 298, 194, 119, 70, 41, 23, 13, 7, 4, 2, 1, 1

Offset: 1

Franklin T. Adams-Watters, Nov 24 2009

The definition is a replica of the recursion formula in A005169: T(n,1) = A005169(n). Row sums, central terms and A003116 coincide: sum(T(n,k): k=1..n) = A003116(n); T(2*n-1,n) = A003116(n-1). - Reinhard Zumkeller, Sep 13 2013

			First 16 rows of triangle:
.   1:     1
.   2:     1    1
.   3:     2    1    1
.   4:     3    2    1   1
.   5:     5    4    2   1   1
.   6:     9    6    4   2   1   1
.   7:    15   11    7   4   2   1   1
.   8:    26   19   12   7   4   2   1  1
.   9:    45   33   21  13   7   4   2  1  1
.  10:    78   57   37  22  13   7   4  2  1  1
.  11:   135   99   64  39  23  13   7  4  2  1  1
.  12:   234  172  112  68  40  23  13  7  4  2  1 1
.  13:   406  298  194 119  70  41  23 13  7  4  2 1 1
.  14:   704  518  337 207 123  71  41 23 13  7  4 2 1 1
.  15:  1222  898  586 360 214 125  72 41 23 13  7 4 2 1 1
.  16:  2120 1559 1017 626 373 218 126 72 41 23 13 7 4 2 1 1

Reinhard Zumkeller, Rows n=1..120 of triangle, flattened

Cf. A005169 (first column), A003116 (apparently row sums).

a168396 n k = a168396_tabl !! (n-1) !! (k-1)
a168396_row n = a168396_tabl !! (n-1)
a168396_tabl = [1] : f [[1]] where
   f xss = ys : f (ys : xss) where
     ys = (map sum $ zipWith take [2..] xss) ++ [1] -- Reinhard Zumkeller, Sep 13 2013

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, j+1), j=1..min(n, k)))
    end:
T:= (n, k)-> b(n-k, k+1):
seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Sep 19 2013

t[n_, k_] /; k > n = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = Sum[ t[n-k, j], {j, 1, k+1}]; Flatten[ Table[ t[n, k], {n, 1, 13}, {k, 1, n}] ](* Jean-François Alcover, Feb 17 2012, after Pari *)

T(n,k)=if(k>=n,k==n,sum(j=1,k+1,T(n-k,j)))

Tm(n)=local(m);m=matrix(n,n);for(i=1,n,for(j=1,i,m[i,j]=if(i==j,1,sum(k=1,j+1,m[i-j,k]))));m

A224959 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) - p(j-1) <= 2.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 29, 55, 105, 199, 378, 716, 1358, 2572, 4873, 9229, 17480, 33102, 62688, 118709, 224795, 425676, 806068, 1526371, 2890338, 5473125, 10363871, 19624925, 37161558, 70368705, 133249369, 252319408, 477788980, 904735349, 1713195705, 3244086145

Offset: 0

Joerg Arndt, Apr 21 2013

nonn

			There are a(5) = 15 such compositions of 5:
01:  [ 1 1 1 1 1 ]
02:  [ 1 1 1 2 ]
03:  [ 1 1 2 1 ]
04:  [ 1 1 3 ]
05:  [ 1 2 1 1 ]
06:  [ 1 2 2 ]
07:  [ 1 3 1 ]
08:  [ 2 1 1 1 ]
09:  [ 2 1 2 ]
10:  [ 2 2 1 ]
11:  [ 2 3 ]
12:  [ 3 1 1 ]
13:  [ 3 2 ]
14:  [ 4 1 ]
15:  [ 5 ]
(the single forbidden composition is [ 1 4 ]).

Alois P. Heinz, Table of n, a(n) for n = 0..3607

Cf. A003116 (compositions such that p(j) - p(j-1) <= 1).
Cf. A225084 (triangle: compositions of n such that max(p(j) - p(j-1)) = k).
Cf. A225085 (triangle: compositions of n such that max(p(j) - p(j-1)) <= k).

b:= proc(n, i) option remember;
      `if`(n=0, 1, add(b(n-j, max(1, j-2)), j=i..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..40);  # Alois P. Heinz, May 02 2013

b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, Max[1, j-2]], {j, i, n}]];
a[n_] := b[n, 1];
a /@ Range[0, 40] (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)

a(n) ~ c * d^n, where d=1.893587506319686491635881459546948770530553555112342985931092896452453511... and c=0.6398882559654423774981963082429746674258714212085034829366885993226... - Vaclav Kotesovec, May 01 2014

A002946 Continued fraction for cube root of 3.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 5, 1, 1, 6, 2, 5, 8, 3, 3, 4, 2, 6, 4, 4, 1, 3, 2, 3, 4, 1, 4, 9, 1, 8, 4, 3, 1, 3, 2, 6, 1, 6, 1, 3, 1, 1, 1, 1, 12, 3, 1, 3, 1, 1, 4, 1, 6, 1, 5, 1, 2, 1, 3, 3, 11, 8, 1, 139, 8, 2, 8, 5, 1, 2, 2, 2, 2, 3, 1, 1, 2, 1, 1, 1, 52, 2, 46, 2, 2, 3

Offset: 0

N. J. A. Sloane

			3^(1/3) = 1.44224957030740838... = 1 + 1/(2 + 1/(3 + 1/(1 + 1/(4 + ...)))). - _Harry J. Smith_, May 08 2009

H. P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..19999
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134.
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134. [Annotated scanned copy]
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A002581 (decimal expansion).

Cf. A002353, A002354 (convergents).

SetDefaultRealField(RealField(100)); ContinuedFraction(3^(1/3)); // G. C. Greubel, Nov 02 2018

with(numtheory): cfrac(3^(1/3),80,'quotients'); # Muniru A Asiru, Nov 02 2018

ContinuedFraction[Power[3, (3)^-1],120] (* Harvey P. Dale, May 11 2011 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(3^(1/3)); for (n=1, 20000, write("b002946.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 08 2009

Offset changed by Andrew Howroyd, Jul 04 2024

A002947 Continued fraction for cube root of 4.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 2, 3, 1, 3, 1, 30, 1, 4, 1, 2, 9, 6, 4, 1, 1, 2, 7, 2, 3, 2, 1, 6, 1, 1, 1, 25, 1, 7, 7, 1, 1, 1, 1, 266, 1, 3, 2, 1, 3, 60, 1, 5, 1, 8, 5, 6, 1, 4, 20, 1, 4, 1, 1, 14, 1, 4, 4, 1, 1, 1, 1, 7, 3, 1, 1, 2, 1, 3, 1, 4, 4, 1, 1, 1, 3, 1, 34, 8, 2, 10, 6, 3, 1, 2, 31, 1, 1, 1, 4, 3, 44, 1, 45

Offset: 0

N. J. A. Sloane

			4^(1/3) = 1.58740105196819947... = 1 + 1/(1 + 1/(1 + 1/(2 + 1/(2 + ...)))). - _Harry J. Smith_, May 08 2009

H. P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..19999
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134.
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134. [Annotated scanned copy]
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
Gang Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A005480 (decimal expansion). - Harry J. Smith, May 08 2009

Cf. A002355, A002356 (convergents).

[ContinuedFraction(4^(1/3))]; // Vincenzo Librandi, Aug 02 2015

ContinuedFraction[4^(1/3), 80] (* Alonso del Arte, Jul 24 2015 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(4^(1/3)); for (n=1, 20000, write("b002947.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 08 2009

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

Offset changed by Andrew Howroyd, Jul 04 2024

A002948 Continued fraction for cube root of 5.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 3, 1, 5, 1, 1, 4, 10, 17, 1, 14, 1, 1, 3052, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3, 2, 1, 13, 5, 1, 1, 1, 13, 2, 41, 1, 4, 12, 1, 5, 2, 7, 1, 1, 3, 33, 2, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 15, 12, 8, 10, 48, 1, 2, 1, 1, 3, 4, 1, 474, 1, 13, 2, 4, 1, 1, 49

Offset: 0

N. J. A. Sloane

			5^(1/3) = 1.70997594667669698... = 1 + 1/(1 + 1/(2 + 1/(2 + 1/(4 + ...)))). - _Harry J. Smith_, May 08 2009

H. P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..19999
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134.
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134. [Annotated scanned copy]
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A005481 (decimal expansion).

Cf. A002357, A002358 (convergents).

SetDefaultRealField(RealField(100)); ContinuedFraction(5^(1/3)); // G. C. Greubel, Nov 02 2018

with(numtheory): cfrac(5^(1/3),80,'quotients'); # Muniru A Asiru, Nov 02 2018

ContinuedFraction[5^(1/3), 100] (* G. C. Greubel, Nov 02 2018 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(5^(1/3)); for (n=1, 20000, write("b002948.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 08 2009

Offset changed by Andrew Howroyd, Jul 04 2024

A225085 Triangle read by rows: T(n,k) is the number of compositions of n with maximal up-step <= k; n>=1, 0<=k

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 7, 8, 8, 7, 13, 15, 16, 16, 11, 23, 29, 31, 32, 32, 15, 41, 55, 61, 63, 64, 64, 22, 72, 105, 119, 125, 127, 128, 128, 30, 127, 199, 233, 247, 253, 255, 256, 256, 42, 222, 378, 455, 489, 503, 509, 511, 512, 512, 56, 388, 716, 889, 967, 1001, 1015, 1021, 1023, 1024, 1024

Offset: 1

Joerg Arndt, Apr 27 2013

T(n,k) is the number of compositions [p(1), p(2), ..., p(k)] of n such that max(p(j) - p(j-1)) <= k.
Rows are partial sums of rows of A225084.
The first column is A000041 (partition numbers), the second column is A003116, and the third column is A224959.
The diagonal is A011782.

			Triangle begins
01: 1,
02: 2, 2,
03: 3, 4, 4,
04: 5, 7, 8, 8,
05: 7, 13, 15, 16, 16,
06: 11, 23, 29, 31, 32, 32,
07: 15, 41, 55, 61, 63, 64, 64,
08: 22, 72, 105, 119, 125, 127, 128, 128,
09: 30, 127, 199, 233, 247, 253, 255, 256, 256,
10: 42, 222, 378, 455, 489, 503, 509, 511, 512, 512,
...
The fifth row corresponds to the following statistics:
#:  M   composition
01:  0  [ 1 1 1 1 1 ]
02:  1  [ 1 1 1 2 ]
03:  1  [ 1 1 2 1 ]
04:  2  [ 1 1 3 ]
05:  1  [ 1 2 1 1 ]
06:  1  [ 1 2 2 ]
07:  2  [ 1 3 1 ]
08:  3  [ 1 4 ]
09:  0  [ 2 1 1 1 ]
10:  1  [ 2 1 2 ]
11:  0  [ 2 2 1 ]
12:  1  [ 2 3 ]
13:  0  [ 3 1 1 ]
14:  0  [ 3 2 ]
15:  0  [ 4 1 ]
16:  0  [ 5 ]
There are 7 compositions with no up-step (M<=0), 13 with M<=1, 15 with M<=2, 16 with M<=3, and 16 with M<=4.

Joerg Arndt and Alois P. Heinz, Rows n = 1..141, flattened

A003117 Continued fraction for fifth root of 3.

Original entry on oeis.org

1, 4, 14, 2, 1, 1, 3, 2, 29, 2, 1, 7, 1, 5, 2, 1, 1, 19, 12, 77, 2, 16, 2, 1, 1, 15, 1, 1, 3, 14, 5, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 463, 1, 379, 3, 5, 3, 11, 1, 7, 7, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 46, 17, 44, 1, 1, 1, 2, 24, 9, 1, 7, 4, 1, 2, 2, 1, 3, 2, 7, 1, 7, 1, 1, 2, 1, 1, 4, 1, 46, 8, 2

Offset: 0

N. J. A. Sloane

			1.24573093961551732596668033... = 1 + 1/(4 + 1/(14 + 1/(2 + 1/(1 + ...)))). - _Harry J. Smith_, May 12 2009

H. P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..19999
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
J. Shallit & N. J. A. Sloane, Correspondence 1974-1975
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A005532 (decimal expansion).

ContinuedFraction[Surd[3,5],120] (* Harvey P. Dale, Dec 15 2018 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(3^(1/5)); for (n=1, 20000, write("b003117.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 12 2009

More terms from James Sellers, Sep 08 2000

Offset changed by Andrew Howroyd, Jul 05 2024

A003118 Continued fraction for fifth root of 4.

Original entry on oeis.org

1, 3, 7, 1, 2, 2, 1, 2, 4, 56, 1, 14, 2, 1, 1, 3, 5, 6, 2, 1, 1, 2, 1, 1, 8, 1, 2, 2, 1, 5, 1, 4, 1, 1, 3, 3, 1, 1, 3, 7, 4, 1, 10, 1, 2, 1, 8, 2, 4, 1, 1, 9, 2, 2, 2, 1, 2, 1, 1, 1, 92, 1, 26, 4, 31, 1, 2, 4, 1, 62, 8, 5, 1, 1, 1, 2, 1, 1, 63, 1, 2, 5, 4, 2, 1

Offset: 0

N. J. A. Sloane

			1.319507910772894259374001971... = 1 + 1/(3 + 1/(7 + 1/(1 + 1/(2 + ...)))). - _Harry J. Smith_, May 11 2009

H. P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..19999
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
J. Shallit & N. J. A. Sloane, Correspondence 1974-1975
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A005533 (decimal expansion).

ContinuedFraction[Surd[4,5],100] (* Harvey P. Dale, Sep 05 2025 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(4^(1/5)); for (n=1, 20000, write("b003118.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 11 2009

Offset changed by Andrew Howroyd, Jul 05 2024

cp's OEIS Frontend

A002951 Continued fraction for fifth root of 5.

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A128742 Number of compositions of n which avoid the pattern 112.

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A168396 Triangle, T(n,k) = number of compositions a(1),...,a(j) of n with a(1) = k, such that a(i+1) <= a(i) + 1 for 1 <= i < j.

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A224959 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) - p(j-1) <= 2.

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A002946 Continued fraction for cube root of 3.

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A002947 Continued fraction for cube root of 4.

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