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 1-6 of 6 results.

A079211 Duplicate of A026835.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 2, 1, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 6, 3, 2, 1, 1, 1, 1, 1, 8, 5, 3, 2, 1, 1, 1, 1, 1, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 12, 7, 4, 3, 2, 1, 1, 1, 1, 1, 1, 15, 8, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 18, 10, 6, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 22, 12, 7, 4
Offset: 1

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Author

Keywords

A085246 Satisfies a(1)=1, a(A026835(n+1)) = a(n)+1, with a(m)=0 for all m not found in A026835, where A026835(n+2)=A026835(n+1)+a(n)+1.

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 1, 4, 0, 0, 0, 1, 1, 2, 0, 5, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 3, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 0, 2, 0, 2, 0, 3, 0, 0, 1, 3, 0, 0, 1, 4, 0, 0, 0, 1, 1, 2, 0, 7, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 0, 1, 3, 0, 0, 1, 3, 0, 0, 1, 4, 0, 0, 0, 1, 1, 2, 0, 4, 0
Offset: 1

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Author

Paul D. Hanna, Aug 22 2003

Keywords

Comments

Removing the leading '1' and all zeros results in the original sequence with every term incremented by 1.

Examples

			Initialize all terms to zero. Set a(1)=1, go one term forward,
set a(2)=a(1)+1=2, go 2 terms forward,
set a(4)=a(2)+1=3, go 3 terms forward,
set a(7)=a(3)+1=1, go 1 term forward,
set a(8)=a(4)+1=4, go 4 terms forward,
set a(12)=a(5)+1=1, etc.
The indices 1,2,4,7,8,12,... form A085262.

Crossrefs

Cf. A085262.

Formula

Records occur at 2^n: a(2^n)=n+1.

A219922 Smallest m such that n is contained in m-th row of A026835.

Original entry on oeis.org

1, 3, 5, 6, 7, 8, 11, 9, 15, 10, 19, 11, 23, 29, 12, 27, 16, 13, 37, 20, 17, 14, 24, 21, 18, 28, 15, 22, 19, 29, 26, 16, 36, 33, 20, 27, 37, 17, 31, 44, 21, 38, 35, 25, 32, 18, 29, 22, 36, 46, 26, 56, 40, 19, 30, 23, 44, 34, 27, 51, 64, 31, 58, 20, 97, 24
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 01 2012

Keywords

Comments

A026835(a(n),k) = n for some k.

Links

Programs

  • Haskell
    import Data.List (findIndex)
import Data.Maybe (fromJust)
a219922 n = (fromJust $ findIndex (n `elem`) a026835_tabl) + 1

A026807 Triangular array T read by rows: T(n,k) = number of partitions of n in which every part is >=k, for k=1,2,...,n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 7, 2, 1, 1, 1, 11, 4, 2, 1, 1, 1, 15, 4, 2, 1, 1, 1, 1, 22, 7, 3, 2, 1, 1, 1, 1, 30, 8, 4, 2, 1, 1, 1, 1, 1, 42, 12, 5, 3, 2, 1, 1, 1, 1, 1, 56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1, 77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1, 101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 135, 34, 13
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

T(n,g) is also the number of not necessarily connected 2-regular graphs with girth at least g: the part i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Feb 05 2012

Examples

			Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))) = y*x+(2*y+y^2)*x^2+(3*y+y^2+y^3)*x^3+(5*y+2*y^2+y^3+y^4)*x^4+(7*y+2*y^2+y^3+y^4+y^5)*x^5+...
Triangle starts:  - _Jason Kimberley_, Feb 05 2012
1;
2, 1;
3, 1, 1;
5, 2, 1, 1;
7, 2, 1, 1, 1;
11, 4, 2, 1, 1, 1;
15, 4, 2, 1, 1, 1, 1;
22, 7, 3, 2, 1, 1, 1, 1;
30, 8, 4, 2, 1, 1, 1, 1, 1;
42, 12, 5, 3, 2, 1, 1, 1, 1, 1;
56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1;
77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1;
101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1;
From _Tilman Piesk_, Feb 20 2016: (Start)
n = 12, k = 4, t = A000217(k-1) = 6
vp = A000041(n..n-t) = A000041(12..6) = (77, 56, 42, 30, 22, 15, 11)
vc = A231599(k-1, 0..t) = A231599(3, 0..6) = (1,-1,-1, 0, 1, 1,-1)
T(12, 4) = vp * transpose(vc) = 77-56-42+22+15-11 = 5
(End)

Links

Crossrefs

Row sums give A046746.
Cf. A026835.
Cf. A026794.
Cf. A231599.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: this sequence (triangle); columns of this sequence: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9). For g >= 3, girth at least g implies no loops or parallel edges. - Jason Kimberley, Feb 05 2012
Not necessarily connected 2-regular simple graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 05 2012
Cf. A002260.

Programs

  • Haskell
    import Data.List (tails)
a026807 n k = a026807_tabl !! (n-1) !! (k-1)
a026807_row n = a026807_tabl !! (n-1)
a026807_tabl = map
   (\row -> map (p $ last row) $ init $ tails row) a002260_tabl
   where p 0  _ = 1
         p _ [] = 0
         p m ks'@(k:ks) = if m < k then 0 else p (m - k) ks' + p m ks
-- Reinhard Zumkeller, Dec 01 2012
  • Maple
    T:= proc(n, k) option remember;
      `if`(k<1 or k>n, 0, `if`(n=k, 1, T(n, k+1) +T(n-k, k)))
    end:
seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Mar 28 2012
  • Mathematica
    T[n_, k_] := T[n, k] = If[ k<1 || k>n, 0, If[n == k, 1, T[n, k+1] + T[n-k, k]]]; Table [Table[ T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)
  • Python
    from see_there import a231599_row  # A231599
from sympy.ntheory import npartitions  # A000041
def a026807(n, k):
    if k > n:
        return 0
    elif k > n/2:
        return 1
    else:
        vc = a231599_row(k-1)
        t = len(vc)
        vp_range = range(n-t, n+1)
        vp_range = vp_range[::-1]  # reverse
        r = 0
        for i in range(0, t):
            r += vc[i] * npartitions(vp_range[i])
        return r
# Tilman Piesk, Feb 21 2016

Formula

T(n,1)=A000041(n), T(n,2)=A002865(n) for n>1, T(n,3)=A008483(n) for n>2, T(n,4)=A008484(n) for n>3.
G.f.: Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))). - Vladeta Jovovic, Jun 22 2003
T(n, k) = T(n, k+1) + T(n-k, k), T(n, k) = 1 if n/2 < k <= n. - Franklin T. Adams-Watters, Jan 24 2005; Tilman Piesk, Feb 20 2016
T(n, k) = A000041(n..n-t) * transpose(A231599(k-1, 0..t)) with t = A000217(k-1). - Tilman Piesk, Feb 20 2016
Equals A026794 * A000012 as infinite lower triangular matrices. - Gary W. Adamson, Jan 31 2008

A035294 Number of ways to partition 2n into distinct positive integers.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 15, 22, 32, 46, 64, 89, 122, 165, 222, 296, 390, 512, 668, 864, 1113, 1426, 1816, 2304, 2910, 3658, 4582, 5718, 7108, 8808, 10880, 13394, 16444, 20132, 24576, 29927, 36352, 44046, 53250, 64234, 77312, 92864, 111322, 133184, 159046
Offset: 0

Views

Author

N. J. A. Sloane, Bill Gosper

Keywords

Comments

Also, number of partitions of 2n into odd numbers. - Vladeta Jovovic, Aug 17 2004
This sequence was originally defined as the expansion of sum ( q^n / product( 1-q^k, k=1..2*n), n=0..inf ). The present definition is due to Reinhard Zumkeller. Michael Somos points out that the equivalence of the two definitions follows from Andrews, page 19.
Also, number of partitions of 2n with max descent 1 and last part 1. - Wouter Meeussen, Mar 31 2013

Examples

			a(4)=6 [8=7+1=6+2=5+3=5+2+1=4+3+1=2*4].
G.f. = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 10*x^5 + 15*x^6 + 22*x^7 + 46*x^9 + ...
G.f. = q + q^49 + 2*q^97 + 4*q^145 + 6*q^193 + 10*q^241 + 15*q^289 + ...

References

  • G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.

Links

Crossrefs

Cf. A000009, A000041, A058686, A262987, A282893.
Cf. A078408, A078406, A078407.
Cf. A079122, A079126, A079124, A079125, A067953.
Cf. A005408.

Programs

  • Haskell
    import Data.MemoCombinators (memo2, integral)
a035294 n = a035294_list !! n
a035294_list = f 1 where
   f x = (p' 1 (x - 1)) : f (x + 2)
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m = if m < k then 0 else p' k (m - k) + p' (k + 2) m
-- Reinhard Zumkeller, Nov 27 2015
  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(2*n, 2*n-1):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 11 2015
  • Mathematica
    Table[Count[IntegerPartitions[2 n], q_ /; Union[q] == Sort[q]], {n, 16}];
Table[Count[IntegerPartitions[2 n], q_ /; Count[q, _?EvenQ] == 0], {n, 16}];
Table[Count[IntegerPartitions[2 n], q_ /; Last[q] == 1 && Max[q - PadRight[Rest[q], Length[q]]] <= 1 ], {n, 16}];
(* Wouter Meeussen, Mar 31 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] /QPochhammer[ x], {x, 0, 2 n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ x^8] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, May 06 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^(8*k+1)) * (1+x^(8*k+2))^2 * (1+x^(8*k+3))^2 * (1+x^(8*k+4))^3 * (1+x^(8*k+5))^2 * (1+x^(8*k+6))^2 * (1+x^(8*k+7)) * (1+x^(8*k+8))^3, {k,0,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 06 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-2] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[2n, 2n-1]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)
  • PARI
    {a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A), n))};/* Michael Somos, Nov 01 2005 */

Formula

a(n) = A000009(2*n). - Michael Somos, Mar 03 2003
Expansion of Sum_{n >= 0} q^n / Product_{k = 1..2*n} (1 - q^k).
a(n) = T(2*n, 0), T as defined in A026835.
G.f.: Product_{i >= 0} ((1 + x^(8*i + 1)) * (1 + x^(8*i + 2))^2 * (1 + x^(8*i + 3))^2 * (1 + x^(8*i + 4))^3 * (1 + x^(8*i + 5))^2 * (1 + x^(8*i + 6))^2 * (1 + x^(8*i + 7)) * (1 + x^(8*i + 8))^3). - Vladeta Jovovic, Oct 10 2004
G.f.: (Sum_{k>=0} x^A074378(k)) / (Product_{k>0} (1 - x^k)) = f( x^3, x^5) / f(-x, -x^2) where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 01 2005
Euler transform of period 16 sequence [1, 1, 2, 1, 2, 0, 1, 0, 1, 0, 2, 1, 2, 1, 1, 0, ...]. - Michael Somos, Dec 17 2002
a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(11/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2015
a(n) = A000041(n) + A282893(n). - Michael Somos, Feb 24 2017
Convolution with A000041 is A058696. - Michael Somos, Feb 24 2017
Convolution with A097451 is A262987. - Michael Somos, Feb 24 2017
G.f.: 1/(1 - x)*Sum_{n>=0} x^floor((3*n+1)/2)/Product_{k = 1..n} (1 - x^k). - Peter Bala, Feb 04 2021
G.f.: Product_{n >= 1} (1 - q^(8*n))*(1 + q^(8*n-3))*(1 + q^(8*n-5))/(1 - q^n). - Peter Bala, Dec 30 2024

A087156 Nonnegative numbers excluding 1.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
Offset: 1

Views

Author

N. J. A. Sloane, Oct 11 2008

Keywords

Comments

The old entry with this sequence number was a duplicate of A026835.
A063524(a(n)) = 0. - Reinhard Zumkeller, Oct 11 2008
Inverse binomial transform of A006589. - Philippe Deléham, Nov 25 2008
a(n) = maximum value of j, where 1 <= j <= n-1, such that floor(j^2 / n) > 0 for each n.

Links

Crossrefs

Cf. A000027, A166373.

Programs

Formula

G.f.: x^2*(2-x)/(1-x)^2 . E.g.f.: x*(exp(x)-1). - Philippe Deléham, Nov 25 2008
a(n) = A163300(n)/2. - Juri-Stepan Gerasimov, Aug 14 2009
a(n) = n mod sigma_k(n), where sigma_k is the k divisor sigma function. -Enrique Pérez Herrero, Nov 11 2009
a(n+1) = floor((n+sqrt(n^2+8n))/2). - Philippe Deléham, Oct 03 2011
a(n) = n mod n^2. - Andrew Secunda, Aug 21 2015

Extensions

Comment and cross-reference added by Christopher Hunt Gribble, Oct 14 2009, Oct 17 2009
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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