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 1-4 of 4 results.

A245093 Triangle read by rows in which row n lists the first n terms of A000203.

Original entry on oeis.org

1, 1, 3, 1, 3, 4, 1, 3, 4, 7, 1, 3, 4, 7, 6, 1, 3, 4, 7, 6, 12, 1, 3, 4, 7, 6, 12, 8, 1, 3, 4, 7, 6, 12, 8, 15, 1, 3, 4, 7, 6, 12, 8, 15, 13, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28
Offset: 1

Views

Author

Omar E. Pol, Jul 15 2014

Keywords

Comments

Reluctant sequence of A000203.
Row sums give A024916.
Has a symmetric representation - for more information see A237270.

Examples

			Triangle begins:
1;
1, 3;
1, 3, 4;
1, 3, 4, 7;
1, 3, 4, 7, 6;
1, 3, 4, 7, 6, 12;
1, 3, 4, 7, 6, 12, 8;
1, 3, 4, 7, 6, 12, 8, 15;
1, 3, 4, 7, 6, 12, 8, 15, 13;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28;

Links

Crossrefs

Cf. A000203, A024916, A027293, A104765, A140207, A221529, A237270, A237593, A245099, A265652.

Programs

  • Haskell
    import Data.List (inits)
a245093 n k = a245093_tabl !! (n-1) !! (k-1)
a245093_row n = a245093_tabl !! (n-1)
a245093_tabl = tail $ inits $ a000203_list
-- Reinhard Zumkeller, Dec 12 2015

Formula

T(n,k) = A000203(k), 1<=k<=n.

A104764 Triangle T(n,k) = Lucas(n-k+1) read by rows, 1<=k<=n.

Original entry on oeis.org

1, 3, 1, 4, 3, 1, 7, 4, 3, 1, 11, 7, 4, 3, 1, 18, 11, 7, 4, 3, 1, 29, 18, 11, 7, 4, 3, 1, 47, 29, 18, 11, 7, 4, 3, 1, 76, 47, 29, 18, 11, 7, 4, 3, 1, 123, 76, 47, 29, 18, 11, 7, 4, 3, 1, 199, 123, 76, 47, 29, 18, 11, 7, 4, 3, 1, 322, 199, 123
Offset: 1

Views

Author

Gary W. Adamson, Mar 24 2005

Keywords

Examples

			First few rows of the triangle are:
1;
3, 1;
4, 3, 1;
7, 4, 3, 1;
11, 7, 4, 3, 1;
18, 11, 7, 4, 3, 1;
...

Crossrefs

Cf. A104765, A000204.

Programs

Formula

T(n,k) = A000204(n-k+1).

A104796 Triangle read by rows: T(n,k) = (n+1-k)*Fibonacci(n+2-k), for n>=1, 1<=k<=n.

Original entry on oeis.org

1, 4, 1, 9, 4, 1, 20, 9, 4, 1, 40, 20, 9, 4, 1, 78, 40, 20, 9, 4, 1, 147, 78, 40, 20, 9, 4, 1, 272, 147, 78, 40, 20, 9, 4, 1, 495, 272, 147, 78, 40, 20, 9, 4, 1, 890, 495, 272, 147, 78, 40, 20, 9, 4, 1, 1584, 890, 495, 272, 147, 78, 40, 20, 9, 4, 1, 2796, 1584, 890, 495, 272
Offset: 1

Views

Author

Gary W. Adamson, Mar 26 2005

Keywords

Comments

The first column is A023607 (without the leading zero).

Examples

			Rows 1,2,3,4,5,6 and columns 1,2,3,4,5,6 of the triangle are:
1;
4, 1;
9, 4, 1;
20, 9, 4, 1;
40, 20, 9, 4, 1;
78, 40, 20, 9, 4, 1;
...
Row 3 for example is 3*F(4), 2*F(3), 1*F(2) = 3*3, 2*2, 1*1 = 9, 4, 1.
Row 4 is 4*F(5), 3*F(4), 2*F(3), 1*F(2) = 4*5, 3*3, 2*2, 1*1 = 20, 9, 4, 1.
Reading the rows backwards gives an initial segment of the terms of A023607 (but without the initial zero).

Links

Crossrefs

Row sums are in A094584.
Cf. A023607, A104762, A104765, A000045.

Programs

  • Mathematica
    Table[(n+1-k)Fibonacci[n+2-k],{n,20},{k,n}]//Flatten (* Harvey P. Dale, Sep 24 2020 *)
Module[{nn=20,c},c=LinearRecurrence[{2,1,-2,-1},{1,4,9,20},nn];Table[ Reverse[ Take[c,n]],{n,nn}]]//Flatten (* Harvey P. Dale, Sep 25 2020 *)

Extensions

Edited by Ralf Stephan, Apr 05 2009
Entry revised by N. J. A. Sloane, Sep 23 2020

A104793 Triangle T(n,k) = A023537(n-k), n >= 1, 0 <= k < n, read by rows.

Original entry on oeis.org

1, 5, 1, 13, 5, 1, 28, 13, 5, 1, 54, 28, 13, 5, 1, 98, 54, 28, 13, 5, 1, 171, 98, 54, 28, 13, 5, 1, 291, 171, 98, 54, 28, 13, 5, 1, 487, 291, 171, 98, 54, 28, 13, 5, 1, 806, 487, 291, 171, 98, 54, 28, 13, 5, 1, 1324, 806, 487, 291, 171, 98, 54, 28, 13, 5, 1
Offset: 1

Views

Author

Gary W. Adamson, Mar 26 2005

Keywords

Comments

Repeatedly writing the sequence A023537 backwards.

Examples

			First few rows of the triangle are:
   1;
   5,  1;
  13,  5,  1;
  28, 13,  5,  1;
  54, 28, 13,  5, 1;
  98, 54, 28, 13, 5, 1; ...

Links

Crossrefs

Row sums are in A027963.
Cf. A104765.

Programs

  • GAP
    Flat(List([1..12], n-> List([0..n-1], k-> Lucas(1, -1, n-k+4)[2] -3*n+3*k-7 ))); # G. C. Greubel, Jun 01 2019
  • Magma
    [[Lucas(n-k+4) -(3*n-3*k+7): k in [0..n-1]]: n in [1..12]]; // G. C. Greubel, Jun 01 2019
  • Mathematica
    Table[LucasL[n-k+4] -3*n+3*k-7, {n,1,12}, {k,0,n-1}]//Flatten (* G. C. Greubel, Jun 01 2019 *)
  • PARI
    {T(n,k) = fibonacci(n-k+5) + fibonacci(n-k+3) -3*n +3*k - 7}; \\ G. C. Greubel, Jun 01 2019
  • Sage
    [[lucas_number2(n-k+4, 1, -1) -3*n+3*k-7 for k in (0..n-1)] for n in (1..12)] # G. C. Greubel, Jun 01 2019

Formula

From Ralf Stephan, Apr 05 2009: (Start)
T(n,k) = Lucas(n-k+4) - (3*n - 3*k + 7).
T(n,k) = A023537(A004736(n, k+1)). (End)

Extensions

Edited by Ralf Stephan, Apr 05 2009
Showing 1-4 of 4 results.

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

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