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.

A130295 Erroneous duplicate of A125026.

Original entry on oeis.org

1, 3, 2, 5, 7, 3, 7, 15, 13, 4, 9, 26, 34, 21, 5, 11, 40, 70, 65, 31, 6, 13, 57, 125, 155, 111, 43, 7, 15, 77, 203, 315, 301, 175, 57, 8, 17, 100, 308, 574, 686, 532, 260, 73, 9, 19, 126, 444, 966, 1386, 1344, 876369, 91, 10
Offset: 1

Views

Author

Gary W. Adamson, May 20 2007

Keywords

Comments

This sequence was initially defined as "A051340 * A007318". However, the matrix product A051340 * A007318 is not well defined, because all elements of A051340 are strictly positive integers, as are all elements of the lower left of A007318. Therefore the matrix A051340 must be truncated to its lower left (setting A[i,j]=0 if j>i), which actually equals A130296. Then the product yields this sequence, which is identical to A125026.
Row sums = A099035 (not A083706 as stated initially): (1, 5, 15, 39, 95, 223, 511, ...).

Examples

			First few rows of the triangle A125026:
   1;
   3,  2;
   5,  7,   3;
   7, 15,  13,   4;
   9, 26,  34,  21,   5;
  11, 40,  70,  65,  31,  6;
  13, 57, 125, 155, 111, 43, 7;
  ...

Crossrefs

Cf. A051340, A099035.

Formula

(A051340) * A007318 as infinite lower triangular matrices. [Here (A051340) is that matrix with the upper right triangle set to zero, which is actually A130296. - M. F. Hasler, Aug 15 2015]

Extensions

Restored and edited by M. F. Hasler, Aug 15 2015

A128224 Duplicate of A125026.

Original entry on oeis.org

1, 3, 2, 5, 7, 3, 7, 15, 13, 4, 9, 26, 34, 21, 5, 11, 40, 70, 65, 31, 6, 13, 57, 125, 155, 111, 43, 7, 15, 77, 203, 315, 301, 175, 57, 8, 17, 100, 308, 574, 686, 532, 260, 73, 9, 19, 126, 444, 966, 1386, 876, 369, 91, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 19 2007

Keywords

Comments

Row sums = A099035: (1, 5, 15, 39, 95, 223, 511, 1151, ...).

Examples

			First few rows of the triangle:
   1;
   3,  2;
   5,  7,   3;
   7, 15,  13,   4;
   9, 26,  34,  21,   5;
  11, 40,  70,  65,  31,  6;
  13, 57, 125, 155, 111, 43, 7;
  ...

References

Crossrefs

Cf. A007318, A127899, A099035.

A051340 A simple 2-dimensional array, read by antidiagonals: T[i,j] = 1 for j>0, T[i,0] = i+1; i,j = 0,1,2,3,...

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Warning: contributions from Kimberling refer to an alternate version indexed by 1 instead of 0. Other contributors (Adamson in A125026/A130301/A130295) refer to this considering the upper right triangle set to zero, T[i,j]=0 for j>i. - M. F. Hasler, Aug 15 2015
From Clark Kimberling, Feb 05 2011: (Start)
A member of the accumulation chain:
... < A051340 < A141419 < A185874 < A185875 < A185876 < ...
(See A144112 for the definition of accumulation array.)
In the m-th accumulation array of A051340,
row_1 = C(m,1) and column_1 = C(1,m+1), for m>=0. (End)

Examples

			Northwest corner:
  1...1...1...1...1...1...1
  2...1...1...1...1...1...1
  3...1...1...1...1...1...1
  4...1...1...1...1...1...1
  5...1...1...1...1...1...1
  6...1...1...1...1...1...1
The Mathematica code shows that the weight array of this array (i.e., the array of which this array is the accumulation array), has northwest corner
  1....0...0...0...0...0...0
  1...-1...0...0...0...0...0
  1...-1...0...0...0...0...0
  1...-1...0...0...0...0...0
  1...-1...0...0...0...0...0. - _Clark Kimberling_, Feb 05 2011

Links

Crossrefs

Cf. A144112, A141419, A185874, A185875, A185876.

Programs

  • Magma
    [k eq n select n+1 else 1: k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 18 2023
  • Maple
    A051340 := proc(n, k) if k=0 then n+1; else 1; end if; end proc: # R. J. Mathar, Jul 16 2015
  • Mathematica
    (* This program generates A051340, then its accumulation array A141419, then its weight array described under Example. *)
f[n_,0]:=0; f[0,k_]:=0;  (* needed for the weight array *)
f[n_,1]:=n; f[n_,k_]:=1/;k>1;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A051340 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A141419 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* weight array *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* Clark Kimberling, Feb 05 2011 *)
f[n_] := Join[ Table[1, {n - 1}], {n}]; Array[ f, 14] // Flatten (* Robert G. Wilson v, Mar 04 2012 *)
Table[PadLeft[{n},n,1],{n,15}]//Flatten (* Harvey P. Dale, Jun 17 2025 *)
  • Python
    from math import comb, isqrt
def A051340(n):
    a = (m:=isqrt(k:=n+2<<1))+(k>m*(m+1))
    return 1 if n-comb(a,2)+1 else a-1 # Chai Wah Wu, Jun 21 2025
  • SageMath
    def A051340(n,k): return n+1 if (k==n) else 1
flatten([[A051340(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Mar 18 2023

Formula

For n>0, a(n(n+3)/2)=n+1, and if k is not of the form n*(n+3)/2, then a(k)=1. - Benoit Cloitre, Oct 31 2002, corrected by M. F. Hasler, Aug 15 2015
T(n,0) = n+1 and T(n,k) = 1 if k >= 0, for n >= 0. - Clark Kimberling, Feb 05 2011

Extensions

Edited by M. F. Hasler, Aug 15 2015
Showing 1-3 of 3 results.

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

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