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.

A176642 Triangle T(n, k) = 8^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 64, 64, 1, 1, 512, 4096, 512, 1, 1, 4096, 262144, 262144, 4096, 1, 1, 32768, 16777216, 134217728, 16777216, 32768, 1, 1, 262144, 1073741824, 68719476736, 68719476736, 1073741824, 262144, 1, 1, 2097152, 68719476736, 35184372088832, 281474976710656, 35184372088832, 68719476736, 2097152, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 22 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,      1;
  1,      8,          1;
  1,     64,         64,           1;
  1,    512,       4096,         512,           1;
  1,   4096,     262144,      262144,        4096,          1;
  1,  32768,   16777216,   134217728,    16777216,      32768,      1;
  1, 262144, 1073741824, 68719476736, 68719476736, 1073741824, 262144, 1;

Links

Crossrefs

Cf. this sequence (q=2), A176643 (q=3), A176644 (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), this sequence (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).
Cf. A000567, A004247.

Programs

  • Magma
    [8^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 30 2021
  • Mathematica
    T[n_, k_, q_]:= (q*(3*q-2))^(k*(n-k)); Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten
With[{m=6}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)
  • Sage
    flatten([[8^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

Formula

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, q) = (q*(3*q - 2))^binomial(n+1,2) and q = 2.
T(n, k, q) = (q*(3*q-2))^(k*(n-k)) with q = 2.
T(n, k) = 8^A004247(n,k), where A004247 is interpreted as a triangle. [relation detected by sequencedb.net]. - R. J. Mathar, Jun 30 2021
T(n, k, m) = (m+2)^(k*(n-k)) with m = 6. - G. C. Greubel, Jun 30 2021

Extensions

Edited by R. J. Mathar and G. C. Greubel, Jun 30 2021

A176631 Triangle T(n, k) = 22^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 22, 1, 1, 484, 484, 1, 1, 10648, 234256, 10648, 1, 1, 234256, 113379904, 113379904, 234256, 1, 1, 5153632, 54875873536, 1207269217792, 54875873536, 5153632, 1, 1, 113379904, 26559922791424, 12855002631049216, 12855002631049216, 26559922791424, 113379904, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 22 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      22,           1;
  1,     484,         484,             1;
  1,   10648,      234256,         10648,           1;
  1,  234256,   113379904,     113379904,      234256,       1;
  1, 5153632, 54875873536, 1207269217792, 54875873536, 5153632, 1;

Links

Crossrefs

Cf. A000326.
Cf. A118190 (q=2), A176627 (q=3), this sequence (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), this sequence (m=20), A176641 (m=26), A176644 (m=38).

Programs

  • Magma
    [22^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 01 2021
  • Mathematica
    T[n_, k_, q_]= (Binomial[3*q,2]/3)^(k*(n-k)); Table[T[n,k,4], {n,0,12}, {k,0,n}]//Flatten
Table[22^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 01 2021 *)
  • Sage
    flatten([[22^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 01 2021

Formula

T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, k) = Product_{j=1..n} (q*(3*q - 1)/2)^j and q = 4.
T(n, k, q) = (binomial(3*q, 2)/3)^(k*(n-k)) with q = 4.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 20. - G. C. Greubel, Jul 01 2021

Extensions

Edited by G. C. Greubel, Jul 01 2021

A176643 Triangle T(n, k) = 21^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 21, 1, 1, 441, 441, 1, 1, 9261, 194481, 9261, 1, 1, 194481, 85766121, 85766121, 194481, 1, 1, 4084101, 37822859361, 794280046581, 37822859361, 4084101, 1, 1, 85766121, 16679880978201, 7355827511386641, 7355827511386641, 16679880978201, 85766121, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 22 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      21,           1;
  1,     441,         441,            1;
  1,    9261,      194481,         9261,           1;
  1,  194481,    85766121,     85766121,      194481,       1;
  1, 4084101, 37822859361, 794280046581, 37822859361, 4084101, 1;

Links

Crossrefs

Cf. A000567.
Cf. A176642 (q=2), this sequence (q=3), A176644 (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), this sequence (m=19), A176631 (m=20), A176641 (m=26).

Programs

  • Magma
    [(21)^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 01 2021
  • Mathematica
    T[n_, k_, q_]:= (q*(3*q-2))^(k*(n-k)); Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten
Table[21^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 01 2021 *)
  • Sage
    flatten([[(21)^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 01 2021

Formula

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, q) = (q*(3*q - 2))^binomial(n+1,2) and q = 3.
T(n, k, q) = (q*(3*q-2))^(k*(n-k)) with q = 3.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 19. - G. C. Greubel, Jul 01 2021

Extensions

Edited by G. C. Greubel, Jul 01 2021
Showing 1-3 of 3 results.

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

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