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 1-3 of 3 results.

A117142 Number of partitions of n in which any two parts differ by at most 2.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 14, 15, 20, 21, 27, 28, 35, 36, 44, 45, 54, 55, 65, 66, 77, 78, 90, 91, 104, 105, 119, 120, 135, 136, 152, 153, 170, 171, 189, 190, 209, 210, 230, 231, 252, 253, 275, 276, 299, 300, 324, 325, 350, 351, 377, 378, 405, 406, 434, 435, 464, 465
Offset: 1

Views

Author

Emeric Deutsch, Feb 27 2006

Keywords

Comments

Equals row sums of triangle A177991. - Gary W. Adamson, May 16 2010
Positive numbers that are either triangular (A000217) or triangular minus 1 (A000096). - Jon E. Schoenfield, Jun 12 2010

Examples

			a(6) = 9 because we have
  1: [6],
  2: [4, 2],
  3: [3, 3],
  4: [3, 2, 1],
  5: [3, 1, 1, 1],
  6: [2, 2, 2],
  7: [2, 2, 1, 1],
  8: [2, 1, 1, 1, 1],
  9: [1, 1, 1, 1, 1, 1]
([5,1] and [4,1,1] do not qualify).

Links

Crossrefs

Cf. A000096, A000217, A117143, A152271, A177991, A152919.
Column k=2 of A194621. - Alois P. Heinz, Oct 17 2012

Programs

  • GAP
    List([1..60],n->(2*n^2+10*n+3+(-1)^n*(2*n-3))/16); # Muniru A Asiru, Dec 21 2018
  • Magma
    [(2*n*(n+5) +3 +(-1)^n*(2*n-3))/16: n in [1..60]]; // G. C. Greubel, Jul 18 2023
  • Maple
    g:=sum(x^k/(1-x^k)/(1-x^(k+1))/(1-x^(k+2)),k=1..75): gser:=series(g,x=0,70): seq(coeff(gser,x^n),n=1..65); with(combinat): for n from 1 to 7 do P:=partition(n): A:={}: for j from 1 to nops(P) do if P[j][nops(P[j])]-P[j][1]<3 then A:=A union {P[j]} else A:=A fi od: print(A); od: # this program yields the partitions
  • Mathematica
    Table[Count[IntegerPartitions[n], ?(Max[#] - Min[#] <= 2 &)], {n, 30}] (* _Birkas Gyorgy, Feb 20 2011 *)
Table[(2*n^2 +10*n +3 +(-1)^n*(2*n-3))/16, {n,30}] (* Birkas Gyorgy, Feb 20 2011 *)
Table[Sum[If[EvenQ[k], 1, (k+1)/2], {k,0,n}], {n,0,60}] (* Jon Maiga, Dec 21 2018 *)
  • PARI
    Vec(x*(x^2-x-1)/((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Mar 05 2015
  • SageMath
    [(2*n*(n+5) +3 +(-1)^n*(2*n-3))/16 for n in range(1,61)] # G. C. Greubel, Jul 18 2023

Formula

G.f.: Sum_{k>=1} x^k/((1 - x^k)*(1 - x^(k + 1))*(1 - x^(k + 2))). More generally, the g.f. of the number of partitions in which any two parts differ by at most b is Sum_{k>=1} (x^k/(Product_{j=k..k+b} 1 - x^j)).
a(n) = (2*n^2 + 10*n + 3 + (-1)^n * (2*n - 3))/16. - Birkas Gyorgy, Feb 20 2011
G.f.: (1 + x)/(1 - x)/(Q(0) - x^2 - x^3), where Q(k) = 1 + x^2 + x^3 + k*x*(1 + x^2) - x^2*(1 + x*(k + 2))*(1 + k*x)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jan 05 2014
G.f.: x*(1 + x - x^2)/((1 - x)^3*(1 + x)^2). - Colin Barker, Mar 05 2015
a(n) = Sum_{k=0..n-1} A152271(k). - Jon Maiga, Dec 21 2018
E.g.f.: (1/16)*( (3 + 2*x)*exp(-x) + (3 + 12*x + 2*x^2)*exp(x) ). - G. C. Greubel, Jul 18 2023
a(n) = A152919(n+1)/2. - Ridouane Oudra, Oct 29 2024

A177990 Triangle read by rows, variant of A070909, a cellular automaton.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, May 16 2010

Keywords

Comments

A070909 * A177990 = A177991
A007318 * A177990 = A177992
A177990 * A007318 = A177993

Examples

			First few rows of the triangle =
1;
0, 1;
0, 1, 1;
0, 1, 0, 1;
0, 1, 0, 1, 1;
0, 1, 0, 1, 0, 1;
0, 1, 0, 1, 0, 1, 1;
0, 1, 0, 1, 0, 1, 0, 1;
0, 1, 0, 1, 0, 1, 0, 1, 1;
0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1;
0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
...

Crossrefs

Cf. A070909, A177991, A177992, A177993.

Programs

  • Mathematica
    rows = 10; ca = CellularAutomaton[28, {{1}, 0}, rows]; Table[ca[[k, 2 ;; k]], {k, 2, rows+1}] // Flatten (* Jean-François Alcover, Jul 12 2017 *)

Formula

Infinite lower triangular matrix, variant of A070909: Columns alternate between (1,0,0,0,...) and (1,1,1,...); whereas A070909 leftmost column = (1,1,1,...).

A177994 Triangle read by rows, A177990 * A070909.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, May 16 2010

Keywords

Comments

Row sums = A101881: (1, 2, 4, 5, 8, 9, 13, 14,...).
Double Riordan array ( 1/((1 - x)*(1 - x^2)); x*(1 - x^2), x/(1 - x^2) ) as defined in Davenport et al. The set of double Riordan arrays of the form ( g(x); x*f_1(x), x*f_2(x) ), where f_1(x)*f_2(x) = 1, forms a group under matrix multiplication. Here g, f_1 and f_2 denote power series with constant term equal to 1. This is the array (( 1/((1 - x)*(1 - x^2)), 1/(1 - x) )) in the notation of the Bala link. - Peter Bala, Aug 26 2021

Examples

			First few rows of the triangle =
1;
1, 1;
2, 1, 1;
2, 1, 1, 1;
3, 1, 2, 1, 1;
3, 1, 2, 1, 1, 1;
4, 1, 3, 1, 2, 1, 1;
4, 1, 3, 1, 2, 1, 1, 1;
5, 1, 4, 1, 3, 1, 2, 1, 1;
5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
7, 1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
7, 1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
...

Links

Crossrefs

Cf. A177991 = A070909 * A177990.
Cf. A101881.

Programs

  • Haskell
    a177994 n k = a177994_tabl !! n !! k
a177994_row n = a177994_tabl !! n
a177994_tabl = [1] : [1,1] : map f a177994_tabl
               where f xs@(x:_) = (x + 1) : 1 : xs
-- Reinhard Zumkeller, Feb 20 2015

Formula

As infinite lower triangular matrices, A177990 * A070909
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.