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.

A061803 Sum of n-th row of triangle of 4th powers: 1; 1 16 1; 1 16 81 16 1; 1 16 81 256 81 16 1; ... (cf. A133824).

Original entry on oeis.org

1, 18, 115, 452, 1333, 3254, 6951, 13448, 24105, 40666, 65307, 100684, 149981, 216958, 305999, 422160, 571217, 759714, 995011, 1285332, 1639813, 2068550, 2582647, 3194264, 3916665, 4764266, 5752683, 6898780, 8220717, 9737998
Offset: 1

Views

Author

Amarnath Murthy, May 28 2001

Keywords

Examples

			a(3) = 115 = 1 + 16 + 81 + 16 + 1

Links

Crossrefs

Cf. A133824.

Programs

  • Mathematica
    Table[Total[2Range[n-1]^4]+n^4,{n,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,18,115,452,1333,3254},30] (* Harvey P. Dale, Aug 23 2016 *)
  • PARI
    a(n) = { n*(6*n^4 + 10*n^2 - 1)/15 } \\ Harry J. Smith, Jul 28 2009

Formula

a(n) = n*(6*n^4 + 10*n^2 - 1)/15. - Dean Hickerson, Jun 06 2001
G.f.: x*(1+x)^2*(1+10*x+x^2)/(1-x)^6. - Colin Barker, Apr 20 2012
E.g.f.: exp(x)*x*(15 + 120*x + 160*x^2 + 60*x^3 + 6*x^4)/15. - Stefano Spezia, Dec 08 2024

Extensions

More terms from Larry Reeves (larryr(AT)acm.org) and Jason Earls, May 28 2001

A124258 Triangle whose rows are sequences of increasing and decreasing squares: 1; 1,4,1; 1,4,9,4,1; ...

Original entry on oeis.org

1, 1, 4, 1, 1, 4, 9, 4, 1, 1, 4, 9, 16, 9, 4, 1, 1, 4, 9, 16, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 49, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 49, 64, 49, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 49, 64, 81, 64, 49, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16
Offset: 1

Views

Author

Jonathan Vos Post, Dec 16 2006

Keywords

Comments

The triangle A003983 with individual entries squared and each 2nd row skipped.
Analogous to A004737. - Peter Bala, Sep 25 2007
T(n,k) = min(n,k)^2. The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

Examples

			Triangle starts
  1;
  1, 4, 1;
  1, 4, 9, 4, 1:
  1, 4, 9, 16, 9, 4, 1:
From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1...1...
  1...4...4...4...4...4...
  1...4...9...9...9...9...
  1...4...9..16..16..16...
  1...4...9..16..25..25...
  1...4...9..16..25..36...
  ...
The start of the sequence as triangle array read by rows:
  1;
  1, 4, 1;
  1, 4, 9,  4,  1;
  1, 4, 9, 16,  9,  4,  1;
  1, 4, 9, 16, 25, 16,  9,  4, 1;
  1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 1;
  ...
Row number k contains 2*k-1 numbers 1,4,...,(k-1)^2,k^2,(k-1)^2,...,4,1. (End)

Links

Crossrefs

Cf. A005900 (row sums), A004737, A133819, A133823, A133824, A133825, A003983.

Programs

  • Maple
    A003983 := proc(n,k) min(n,k) ; end: A124258 := proc(n,k) A003983(n,k)^2 ; end: for d from 1 to 20 by 2 do for c from 1 to d do printf("%d, ",A124258(d+1-c,c)) ; od: od: # R. J. Mathar, Sep 21 2007
# second Maple program:
T:= n-> i^2$i=1..n, (n-i)^2$i=1..n-1:
seq(T(n), n=1..10);  # Alois P. Heinz, Feb 15 2022
  • Mathematica
    Flatten[Table[Join[Range[n]^2,Range[n-1,1,-1]^2],{n,10}]] (* Harvey P. Dale, Jun 14 2015 *)

Formula

O.g.f.: (1+qx)^2/((1-x)(1-qx)^2(1-q^2x)) = 1 + x(1 + 4q + q^2) + x^2(1 + 4q + 9q^2 + 4q^3 + q^4) + ... . - Peter Bala, Sep 25 2007
From Boris Putievskiy, Jan 13 2013: (Start)
a(n) = (A004737(n))^2.
a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^2. (End)

Extensions

More terms from R. J. Mathar, Sep 21 2007
Edited by N. J. A. Sloane, Jun 30 at the suggestion of R. J. Mathar

A133823 Triangle whose rows are sequences of increasing and decreasing cubes:1; 1,8,1; 1,8,27,8,1; ... .

Original entry on oeis.org

1, 1, 8, 1, 1, 8, 27, 8, 1, 1, 8, 27, 64, 27, 8, 1, 1, 8, 27, 64, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 512, 343, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 512, 729
Offset: 0

Views

Author

Peter Bala, Sep 25 2007

Keywords

Comments

Reading the triangle by rows produces the sequence 1,1,8,1,1,8,27,8,1,..., analogous to A004737.
T(n,k) = min(n,k)^3. The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

Examples

			Triangle starts
  1;
  1, 8, 1;
  1, 8, 27, 8, 1;
  1, 8, 27, 64, 27, 8, 1;
From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1...1...
  1...8...8...8...8...8...
  1...8..27..27..27..27...
  1...8..27..64..64..64...
  1...8..27..64.125.125...
  1...8..27..64.125.216...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  1,8,1;
  1,8,27,8,1;
  1,8,27,64,27,8,1;
  1,8,27,64,125,64,27,8,1;
  1,8,27,64,125,216,125,64,27,8,1;
  . . .
Row number k contains 2*k-1 numbers 1,8,...,(k-1)^3,k^3,(k-1)^3,...,8,1. (End)

Links

Crossrefs

Cf. A004737, A037270 (row sums), A133820, A124258, A133824, A003983.

Programs

  • Mathematica
    Table[Join[Range[n]^3,Range[n-1,1,-1]^3],{n,10}]//Flatten (* Harvey P. Dale, May 29 2019 *)

Formula

O.g.f.: (1+qx)(1+4qx+q^2x^2)/((1-x)(1-qx)^3(1-q^2x)) = 1 + x(1 + 8q + q^2) + x^2(1 + 8q + 27q^2 + 8q^3 + q^4) + ... .
From Boris Putievskiy, Jan 13 2013: (Start)
a(n) = (A004737(n))^3.
a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^3. (End)

A133821 Triangle whose rows are sequences of increasing fourth powers: 1; 1,16; 1,16,81; ... .

Original entry on oeis.org

1, 1, 16, 1, 16, 81, 1, 16, 81, 256, 1, 16, 81, 256, 625, 1, 16, 81, 256, 625, 1296, 1, 16, 81, 256, 625, 1296, 2401, 1, 16, 81, 256, 625, 1296, 2401, 4096, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000
Offset: 1

Views

Author

Peter Bala, Sep 25 2007

Keywords

Comments

Reading the triangle by rows produces the sequence 1,1,16,1,16,81,1,16,81,256,..., analogous to A002260.

Examples

			Triangle starts
1;
1, 16;
1, 16; 81;
1, 16, 81, 256;
1, 16, 81, 256, 625;

Links

Crossrefs

Cf. A000538 (row sums), A002260, A133819, A133820, A133824.

Programs

  • Haskell
    a133821 n k = a133821_tabl !! (n-1) !! (k-1)
a133821_row n = a133821_tabl !! (n-1)
a133821_tabl = map (`take` (tail a000583_list)) [1..]
-- Reinhard Zumkeller, Nov 11 2012
  • Mathematica
    Module[{nn=10,fp},fp=Range[(nn(nn+1))/2]^4;Table[TakeList[fp,{n}],{n,nn}]]//Flatten (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Mar 29 2020 *)

Formula

O.g.f.: (1+11qx+11q^2x^2+q^3x^3)/((1-x)(1-qx)^5) = 1 + x(1 + 16q) + x^2(1 + 16q + 81q^2) + ... . Cf. 4th row of A008292.

Extensions

Offset changed by Reinhard Zumkeller, Nov 11 2012
Showing 1-4 of 4 results.

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