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.

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A246649 Integers of the form (2^(k+1) - 1)/(2*k + 1).

Original entry on oeis.org

1, 1, 17, 1334440654591915542993625911497130241, 948042080603099421350928003060030968743284199473954197137709371401
Offset: 1

Views

Author

Clark Kimberling, Sep 01 2014

Keywords

Comments

The next term has 192 digits. - Harvey P. Dale, Feb 05 2019

Examples

			The sum of the numbers row 7 of the triangular array at A027926 is 2^8 - 1 = 255, and the number of numbers in row 7 is 15, and 255/15 = 17; thus 7 is in this sequence, and 17 is in A246649.

Crossrefs

Cf. A246637, A246648.

Programs

  • Mathematica
    z = 140000; u = Select[Range[0, z], IntegerQ[(2^(# + 1) - 1)/(2 # + 1)] &]   (* A246648 *)
v = Table[(2^(u[[k]] + 1) - 1)/(2 u[[k]] + 1), {k, 1, 6}] (* A246649 *)
Select[Table[(2^(n+1)-1)/(2n+1),{n,0,250}],IntegerQ] (* Harvey P. Dale, Feb 05 2019 *)

A055800 Triangle T read by rows: T(i,0)=1 for i >= 0; T(i,i)=0 for i >= 1; T(i,j) = Sum_{k=1..floor(i/2)} T(i-2k,j-2k+1) for 1 <= j <= i-1, where T(m,n) := 0 if m < 0 or n < 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 2, 2, 1, 0, 1, 1, 1, 2, 2, 1, 0, 0, 1, 1, 1, 2, 3, 4, 3, 1, 0, 1, 1, 1, 2, 3, 4, 3, 1, 0, 0, 1, 1, 1, 2, 3, 5, 7, 7, 4, 1, 0, 1, 1, 1, 2, 3, 5, 7, 7, 4, 1, 0, 0, 1, 1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1, 0
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Comments

T(i+j,j) is the number of strings (s(1),...,s(m)) of nonnegative integers s(k) such that m <= i+1, s(m)=j and s(k)-s(k-1) is an odd positive integer for k=2,3,...,m.
T(i+j,j) is the number of compositions of j consisting of at most i parts, all positive odd integers.

Examples

			Triangle begins:
  1;
  1,0;
  1,1,0;
  1,1,0,0;
  1,1,1,1,0;
  ...
T(10,5) counts the strings 012345, 0125, 0145, 0345, 05.
T(10,5) counts the compositions 11111, 113, 131, 311, 5.

Links

Crossrefs

Row sums are powers of 2: A016116.
T(2n, n)=A000045(n) for n >= 1 (Fibonacci numbers).
Cf. A027926.

Programs

  • GAP
    T:= function(n,k)
    if n<0 or k<0 then return 0;
    elif k=0 then return 1;
    elif k=n then return 0;
    else return Sum([1..Int(n/2)], j-> T(n-2*j, k-2*j+1));
    fi; end;
Flat(List([0..15], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jan 23 2020
  • Magma
    function T(n,k)
  if n lt 0 or k lt 0 then return 0;
  elif k eq 0 then return 1;
  elif k eq n then return 0;
  else return (&+[T(n-2*j, k-2*j+1): j in [1..Floor(n/2)]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 23 2020
  • Maple
    T:= proc(n,k) option remember;
    if n<0 or k<0 then 0;
    elif k=0 then 1;
    elif k=n then 0;
    else add(T(n-2*j, k-2*j+1), j=1..floor(n/2)) ;
    end if; end proc:
seq(seq(T(n,k), k=0..n), n=0..15); # G. C. Greubel, Jan 24 2020
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0 || k<0, 0, If[k==0, 1, If[k==n, 0, Sum[T[n-2*j, k- 2*j+1], {j, Floor[n/2]}]]]]; Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 24 2020 *)
  • PARI
    T(n,k) = if(n<0 || k<0, 0, if(k==0, 1, if(k==n, 0, sum(j=1, n\2, T(n-2*j, k-2*j+1) ))));
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jan 23 2020
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0 or k<0): return 0
    elif (k==0): return 1
    elif (k==n): return 0
    else: return sum(T(n-2*j, k-2*j+1) for j in (1..floor(n/2)))
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Jan 23 2020

Formula

G.f. for k-th diagonal: (1-x^2-x*(x/(1-x^2))^k)/(1-x-x^2). - Vladeta Jovovic, Mar 10 2005

Extensions

a(88)-a(90) from Michel Marcus, Jan 21 2019

A096601 Triangle (read by rows) where the number of row entries increases by steps of 2 and the entries are stacked in a rectangular fashion. The end entries are set to 1. Rest of entries are set to the sum of the entry directly above in the previous row plus the entry (if present) in the previous row that is either to the left (if the current row number is even) or to the right (if odd).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 4, 3, 2, 1, 1, 2, 4, 7, 7, 5, 3, 1, 1, 1, 1, 3, 6, 11, 14, 12, 8, 4, 2, 1, 1, 2, 4, 9, 17, 25, 26, 20, 12, 6, 3, 1, 1, 1, 1, 3, 6, 13, 26, 42, 51, 46, 32, 18, 9, 4, 2, 1, 1, 2, 4, 9, 19, 39, 68, 93, 97, 78, 50, 27, 13, 6, 3, 1, 1
Offset: 1

Views

Author

Gerald McGarvey, Aug 14 2004

Keywords

Comments

The row sums are 2^n-1 (A000225). Shares many entries with A027926., e.g. the 'central column' (and greatest numbers in the rows) is A027988 (Greatest number in row n of array T given by A027926.)

Examples

			........................1........................
.....................1..1..1.....................
..................1..2..2..1..1..................
...............1..1..3..4..3..2..1...............
............1..2..4..7..7..5..3..1..1............
.........1..1..3..6.11.14.12..8..4..2..1.........
......1..2..4..9.17.25.26.20.12..6..3..1..1......
...1..1..3..6.13.26.42.51.46.32.18..9..4..2..1...
1..2..4..9.19.39.68.93.97.78.50.27.13..6..3..1..1

Crossrefs

Cf. A000225, A027926, A027988.
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