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 1-9 of 9 results.

A162611 Triangle read by rows in which row n lists n+1 terms, starting with n, such that the difference between successive terms is equal to n^2 - 1.

Original entry on oeis.org

0, 1, 1, 2, 5, 8, 3, 11, 19, 27, 4, 19, 34, 49, 64, 5, 29, 53, 77, 101, 125, 6, 41, 76, 111, 146, 181, 216, 7, 55, 103, 151, 199, 247, 295, 343, 8, 71, 134, 197, 260, 323, 386, 449, 512, 9, 89, 169, 249, 329, 409, 489, 569, 649, 729, 10, 109, 208, 307, 406, 505, 604
Offset: 0

Views

Author

Omar E. Pol, Jul 09 2009

Keywords

Comments

Note that the last term of the n-th row is the n-th cube A000578(n).
See also A159797, A162614 and A162622.

Examples

			Triangle begins:
0;
1, 1;
2, 5, 8;
3,11,19,27;
4,19,34,49,64;
5,29,53,77,101,125;
6,41,76,111,146,181,216;

Crossrefs

Cf. A000578, A159797, A162612, A162613, A162614, A162622.

Programs

  • Mathematica
    Flatten[Join[{0,1,1},Table[Range[n,(n+1)(n^2-1),n^2-1],{n,2,10}]]] (* Harvey P. Dale, Jun 18 2014 *)

Extensions

Edited by Omar E. Pol, Jul 18 2009

A162614 Triangle read by rows in which row n lists n+1 terms, starting with n, such that the difference between successive terms is equal to n^3 - 1.

Original entry on oeis.org

0, 1, 1, 2, 9, 16, 3, 29, 55, 81, 4, 67, 130, 193, 256, 5, 129, 253, 377, 501, 625, 6, 221, 436, 651, 866, 1081, 1296, 7, 349, 691, 1033, 1375, 1717, 2059, 2401, 8, 519, 1030, 1541, 2052, 2563, 3074, 3585, 4096, 9, 737, 1465, 2193, 2921, 3649, 4377, 5105, 5833
Offset: 0

Views

Author

Omar E. Pol, Jul 15 2009

Keywords

Comments

Note that the last term of the n-th row is the fourth power of n, A000583(n).
See also the triangles of A162615 and A162616.

Examples

			Triangle begins:
  0;
  1,   1;
  2,   9,  16;
  3,  29,  55,  81;
  4,  67, 130, 193, 256;
  5, 129, 253, 377, 501,  625;
  6, 221, 436, 651, 866, 1081, 1296;
  ...

Crossrefs

Cf. A000583, A068601, A159797, A162609, A162610, A162611, A162612, A162613, A162615, A162616, A162622, A162623, A162624.

Programs

  • Python
    def A162614(n,k):
    return n+k*(n**3-1)
print([A162614(n,k) for n in range(20) for k in range(n+1)])
# R. J. Mathar, Oct 20 2009

Formula

Sum_{k=0..n} T(n,k) = n*(n^2-n+1)*(n+1)^2/2 (row sums). - R. J. Mathar, Jul 20 2009
T(n,k) = n + k*(n^3-1). - R. J. Mathar, Oct 20 2009

Extensions

More terms from R. J. Mathar, Oct 20 2009

A162609 Triangle read by rows in which row n lists n terms, starting with 1, with gaps = n-2 between successive terms.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 5, 7, 1, 4, 7, 10, 13, 1, 5, 9, 13, 17, 21, 1, 6, 11, 16, 21, 26, 31, 1, 7, 13, 19, 25, 31, 37, 43, 1, 8, 15, 22, 29, 36, 43, 50, 57, 1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81
Offset: 1

Views

Author

Omar E. Pol, Jul 09 2009

Keywords

Comments

Equals A081493 when first column is removed. - Georg Fischer, Jul 25 2023

Examples

			Triangle begins:
  1;
  1,  1;
  1,  2,  3;
  1,  3,  5,  7;
  1,  4,  7, 10, 13;
  1,  5,  9, 13, 17, 21;
  1,  6, 11, 16, 21, 26, 31;

Crossrefs

Cf. A002061, A159797, A159798, A162610, A161611, A162612, A162613.
Cf. A060354 (row sums), A081493 (without first column).

Programs

  • Mathematica
    Table[NestList[#+(n-2)&,1,n-1],{n,20}]//Flatten (* Harvey P. Dale, Oct 23 2017 *)

Formula

T(n,n) = A002061(n-1).
T(n,k) = A076110(n-1,k) = 1+(n-2)*(k-1). - R. J. Mathar, Mar 30 2023

A162622 Triangle read by rows in which row n lists n+1 terms, starting with n, such that the difference between successive terms is equal to n^4 - 1.

Original entry on oeis.org

0, 1, 1, 2, 17, 32, 3, 83, 163, 243, 4, 259, 514, 769, 1024, 5, 629, 1253, 1877, 2501, 3125, 6, 1301, 2596, 3891, 5186, 6481, 7776, 7, 2407, 4807, 7207, 9607, 12007, 14407, 16807, 8, 4103, 8198, 12293, 16388, 20483, 24578, 28673, 32768, 9, 6569, 13129
Offset: 0

Views

Author

Omar E. Pol, Jul 15 2009

Keywords

Comments

Note that the last term of the n-th row is the 5th power of n, A000584(n).
See also the triangles of A162623 and A162624.

Examples

			Triangle begins:
  0;
  1,    1;
  2,   17,    32;
  3,   83,   163,   243;
  4,  259,   514,   769,  1024;
  5,  629,  1253,  1877,  2501,  3125;
  6, 1301,  2596,  3891,  5186,  6481,  7776;
  7, 2407,  4807,  7207,  9607, 12007, 14407, 16807;
  8, 4103,  8198, 12293, 16388, 20483, 24578, 28673, 32768;
  9, 6569, 13129, 19689, 26249, 32809, 39369, 45929, 52489, 59049; etc.

Links

Crossrefs

Cf. A000583, A000584, A123865, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162615, A162616, A162623, A162624.

Programs

  • Magma
    /* Triangle: */ [[n+k*(n^4-1): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Dec 14 2012
  • Maple
    A162622 := proc(n,k) n+k*(n^4-1) ; end proc: seq(seq( A162622(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Feb 11 2010
  • Mathematica
    Flatten[Table[NestList[#+n^4-1&,n,n],{n,0,9}]] (* Harvey P. Dale, Jun 23 2013 *)

Formula

Sum_{k=0..n} T(n,k) = n*(n+1)*(1+n^4)/2 (row sums). [R. J. Mathar, Jul 20 2009]

Extensions

7th and later rows from R. J. Mathar, Feb 11 2010

A162612 Triangle read by rows in which row n lists n terms, starting with n^2+n-1, with gaps = n^2-1 between successive terms.

Original entry on oeis.org

1, 5, 8, 11, 19, 27, 19, 34, 49, 64, 29, 53, 77, 101, 125, 41, 76, 111, 146, 181, 216, 55, 103, 151, 199, 247, 295, 343, 71, 134, 197, 260, 323, 386, 449, 512, 89, 169, 249, 329, 409, 489, 569, 649, 729, 109, 208, 307, 406, 505, 604, 703, 802, 901, 1000, 131, 251
Offset: 1

Views

Author

Omar E. Pol, Jul 09 2009

Keywords

Comments

Note that the last term of the n-th row is the n-th cube A000578(n).

Examples

			Triangle begins:
   1;
   5,   8;
  11,  19,  27;
  19,  34,  49,  64;
  29,  53,  77, 101, 125;
  41,  76, 111, 146, 181, 216;

Links

Crossrefs

Cf. A000578, A028387, A159797, A159798, A162610, A162611, A162613.

Programs

Formula

Sum_{k=1..n} T(n,k)= n*(n^3 + n^2 + n - 1)/2 (row sums). - R. J. Mathar, Jul 20 2009

Extensions

More terms from Franklin T. Adams-Watters, Aug 06 2009

A162615 Triangle read by rows in which row n lists n terms, starting with n, such that the difference between successive terms is equal to n^3 - 1 = A068601(n).

Original entry on oeis.org

1, 2, 9, 3, 29, 55, 4, 67, 130, 193, 5, 129, 253, 377, 501, 6, 221, 436, 651, 866, 1081, 7, 349, 691, 1033, 1375, 1717, 2059, 8, 519, 1030, 1541, 2052, 2563, 3074, 3585, 9, 737, 1465, 2193, 2921, 3649, 4377, 5105, 5833, 10, 1009, 2008, 3007, 4006, 5005, 6004
Offset: 1

Views

Author

Omar E. Pol, Jul 12 2009

Keywords

Comments

See also the triangles of A162614 and A162616.

Examples

			Triangle begins:
  1;
  2,   9;
  3,  29,  55;
  4,  67, 130, 193;
  5, 129, 253, 377, 501;
  6, 221, 436, 651, 866, 1081;
  ...

Links

Crossrefs

Cf. A000583, A068601, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162616.

Programs

  • Maple
    A162615 := proc(n,k) n+(k-1)*(n^3-1) ; end proc: seq(seq(A162615(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Feb 05 2010
  • Mathematica
    Flatten[Table[c=n^3-1;NestList[#+c&,n,n-1],{n,10}]] (* Harvey P. Dale, Nov 13 2011 *)

Formula

Row sums: n*(n^4 - n^3 + n + 1)/2. - R. J. Mathar, Jul 20 2009

Extensions

Terms beyond the 6th row from R. J. Mathar and Max Alekseyev, Feb 05 2010

A162616 Triangle read by rows in which row n lists n terms, starting with n^3 + n - 1, such that the difference between successive terms is equal to n^3 - 1 = A068601(n).

Original entry on oeis.org

1, 9, 16, 29, 55, 81, 67, 130, 193, 256, 129, 253, 377, 501, 625, 221, 436, 651, 866, 1081, 1296, 349, 691, 1033, 1375, 1717, 2059, 2401, 519, 1030, 1541, 2052, 2563, 3074, 3585, 4096, 737, 1465, 2193, 2921, 3649, 4377, 5105, 5833, 6561, 1009, 2008
Offset: 1

Views

Author

Omar E. Pol, Jul 12 2009

Keywords

Comments

Note that the last term of the n-th row is the fourth power of n, A000583(n).
See also the triangles of A162614 and A162615.

Examples

			Triangle begins:
    1;
    9,  16;
   29,  55,  81;
   67, 130, 193, 256;
  129, 253, 377, 501,  625;
  221, 436, 651, 866, 1081, 1296;
  ...

Links

Crossrefs

Cf. A000583, A068601, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162615.

Programs

  • Maple
    A162616 := proc(n,k) n^3+n-1+(k-1)*(n^3-1) ; end proc: seq(seq(A162616(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Feb 05 2010
  • Mathematica
    Table[NestList[#+n^3-1&,n^3+n-1,n-1],{n,10}]//Flatten (* Harvey P. Dale, Dec 17 2021 *)

Formula

Row sums: n*(n^2 + n - 1)*(n^2+1)/2. - R. J. Mathar, Jul 20 2009

Extensions

More terms from R. J. Mathar, Feb 05 2010

A162624 Triangle read by rows in which row n lists n terms, starting with n^4 + n - 1, such that the difference between successive terms is equal to n^4 - 1 = A123865(n).

Original entry on oeis.org

1, 17, 32, 83, 163, 243, 259, 514, 769, 1024, 629, 1253, 1877, 2501, 3125, 1301, 2596, 3891, 5186, 6481, 7776, 2407, 4807, 7207, 9607, 12007, 14407, 16807, 4103, 8198, 12293, 16388, 20483, 24578, 28673, 32768, 6569, 13129, 19689, 26249, 32809
Offset: 1

Views

Author

Omar E. Pol, Jul 12 2009

Keywords

Comments

Note that the last term of the n-th row is the 5th power of n, A000584(n).
See also the triangles of A162622 and A162623.

Examples

			Triangle begins:
     1;
    17,   32;
    83,  163,  243;
   259,  514,  769, 1024;
   629, 1253, 1877, 2501, 3125;
  1301, 2596, 3891, 5186, 6481, 7776;
  ...

Links

Crossrefs

Cf. A000584, A123865, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162615, A162616, A162622, A162623.

Programs

Formula

Row sums: n*(n^5 + n^4 + n - 1)/2. - R. J. Mathar, Jul 20 2009

A162623 Triangle read by rows in which row n lists n terms, starting with n, such that the difference between successive terms is equal to n^4 - 1 = A123865(n).

Original entry on oeis.org

1, 2, 17, 3, 83, 163, 4, 259, 514, 769, 5, 629, 1253, 1877, 2501, 6, 1301, 2596, 3891, 5186, 6481, 7, 2407, 4807, 7207, 9607, 12007, 14407, 8, 4103, 8198, 12293, 16388, 20483, 24578, 28673, 9, 6569, 13129, 19689, 26249, 32809, 39369, 45929, 52489, 10
Offset: 1

Views

Author

Omar E. Pol, Jul 12 2009

Keywords

Comments

See also the triangles of A162622 and A162624.

Examples

			Triangle begins:
  1;
  2,   17;
  3,   83,  163;
  4,  259,  514,  769;
  5,  629, 1253, 1877, 2501;
  6, 1301, 2596, 3891, 5186, 6481;

Crossrefs

Cf. A000583, A000584, A123865, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162615, A162616, A162622, A162624.

Programs

  • Maple
    A162623 := proc(n,k) n+k*(n^4-1) ; end: seq(seq(A162623(n,k),k=0..n-1),n=1..15) ; # R. J. Mathar, Sep 27 2009
  • Mathematica
    dst[n_]:=Module[{c=n^4-1},Range[n,n*c,c]]; Flatten[Join[{1},Table[dst[n],{n,2,10}]]] (* Harvey P. Dale, Jul 29 2014 *)

Formula

Row sums: n*(n^5 - n^4 + n + 1)/2. - R. J. Mathar, Jul 20 2009

Extensions

More terms from R. J. Mathar, Sep 27 2009
Showing 1-9 of 9 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.