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 1-10 of 16 results. Next

A198457 Consider triples (a, b, c) where a <= b < c and (a^2+b^2-c^2)/(c-a-b) = 2, ordered by a and then b; sequence gives a, b and c values in that order.

Original entry on oeis.org

3, 6, 7, 4, 4, 6, 5, 16, 17, 6, 10, 12, 7, 8, 11, 7, 30, 31, 8, 18, 20, 9, 14, 17, 9, 48, 49, 10, 12, 16, 10, 28, 30, 11, 70, 71, 12, 18, 22, 12, 40, 42, 13, 16, 21, 13, 30, 33, 13, 96, 97, 14, 25, 29, 14, 54, 56, 15, 22, 27, 15, 40, 43, 15, 126, 127, 16, 20, 26
Offset: 1

Views

Author

Charlie Marion, Nov 09 2011

Keywords

Comments

See A198453.
Because either all sides or only one side of a Pythagorean (-+2)-triangle ABC is even their sum is always even. Thus csc(C) = -(a+b+c+k)/k is an integer. So ((a+2)^2 + (b+2)^2 - (c+2)^2)|(2*(a+2)*(b+2)) resp. (a^2 + b^2 - c^2)|(2*a*b). - Ralf Steiner, Sep 18 2019

Examples

			3*5 +  6*8  =  7*9;
4*6 +  4*6  =  6*8;
5*7 + 16*17 = 17*18;
6*8 + 10*12 = 12*14;
7*9 +  8*10 = 11*13;
7*9 + 30*32 = 31*33.

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198456, A198458-A198468.

Extensions

More terms from David A. Corneth, Sep 22 2019

A198456 Consider triples a<=b

Original entry on oeis.org

3, 6, 10, 8, 15, 11, 21, 28, 13, 36, 16, 23, 45, 28, 55, 18, 23, 66, 21, 27, 78, 46, 91, 20, 23, 36, 53, 105, 26, 41, 120, 136, 28, 52, 77, 153, 31, 58, 86, 171, 40, 49, 190, 33, 44, 54, 71, 210, 36, 41, 78, 116
Offset: 1

Views

Author

Charlie Marion, Oct 26 2011

Keywords

Comments

See A198453.
The definition amounts to saying that T_a+T_b=T_c where T_i denotes a triangular number (A000217). - N. J. A. Sloane, Apr 01 2020

Examples

			2*3 + 2*3 = 3*4
3*4 + 5*6 = 6*7
4*5 + 9*10 = 10*11
5*6 + 6*7 = 8*9
5*6 + 14*15 = 15*16
6*7 + 9*10 = 11*12

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A000217, A156682, A198453-A198455, A198457-A198469, A333530, A333531.

A198455 Consider triples a<=b

Original entry on oeis.org

2, 5, 9, 6, 14, 9, 20, 27, 10, 35, 13, 21, 44, 26, 54, 14, 20, 65, 17, 24, 77, 44, 90, 14, 18, 33, 51, 104, 21, 38, 119, 135, 22, 49, 75, 152, 25, 55, 84, 170, 35, 45, 189, 26, 39, 50, 68, 209, 29, 35, 75, 114, 230, 125
Offset: 1

Views

Author

Charlie Marion, Oct 26 2011

Keywords

Comments

See A198453.
The definition amounts to saying that T_a+T_b=T_c where T_i denotes a triangular number (A000217). - N. J. A. Sloane, Apr 01 2020

Examples

			2*3 + 2*3 = 3*4
3*4 + 5*6 = 6*7
4*5 + 9*10 = 10*11
5*6 + 6*7 = 8*9
5*6 + 14*15 = 15*16
6*7 + 9*10 = 11*12

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A000217, A156681, A198453, A198454, A198456-A198469, A333530, A333531.

A198454 Consider triples a<=b

Original entry on oeis.org

2, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 23, 23, 23, 24
Offset: 1

Views

Author

Charlie Marion, Oct 26 2011

Keywords

Comments

See A198453.

Examples

			2*3 + 2*3 = 3*4
3*4 + 5*6 = 6*7
4*5 + 9*10 = 10*11
5*6 + 6*7 = 8*9
5*6 + 14*15 = 15*16
6*7 + 9*10 = 11*12

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A009004, A198453-A198469.

A198458 Consider triples a<=b

Original entry on oeis.org

3, 4, 5, 6, 7, 7, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 28, 28, 28, 29, 30, 30, 30, 30, 30, 30, 31, 31, 31
Offset: 1

Views

Author

Charlie Marion, Nov 15 2011

Keywords

Comments

See A198453 and A198457.

Examples

			3*5 + 6*8 = 7*9
4*6 + 4*6 = 6*8
5*7 + 16*17 = 17*18
6*8 + 10*12 12*14
7*9 + 8*10 = 11*13
7*9 + 30*32 = 31*33

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198457, A198459-A198469.

A198459 Consider triples a<=b

Original entry on oeis.org

6, 4, 16, 10, 8, 30, 18, 14, 48, 12, 28, 70, 18, 40, 16, 30, 96, 25, 54, 22, 40, 126, 20, 33, 70, 160, 26, 42, 88, 24, 64, 198, 52, 108, 30, 78, 240, 28, 40, 63, 130, 54, 286, 34, 48, 75, 154, 32, 64, 110, 336, 88, 180, 38, 128, 390, 28, 36, 66, 102, 208, 448, 33, 42
Offset: 1

Views

Author

Charlie Marion, Nov 15 2011

Keywords

Comments

See A198453 and A198457.

Examples

			3*5 + 6*8 = 7*9
4*6 + 4*6 = 6*8
5*7 + 16*17 = 17*18
6*8 + 10*12 12*14
7*9 + 8*10 = 11*13
7*9 + 30*32 = 31*33

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198458, A198460-A198469.

A198460 Consider triples a<=b

Original entry on oeis.org

7, 6, 17, 12, 11, 31, 20, 17, 49, 16, 30, 71, 22, 42, 21, 33, 97, 29, 56, 27, 43, 127, 26, 37, 72, 161, 32, 46, 90, 31, 67, 199, 56, 110, 37, 81, 241, 36, 46, 67, 132, 59, 287, 42, 54, 79, 156, 41, 69, 113, 337, 92, 182, 47, 131, 391, 40, 46, 72, 106, 210, 449, 45, 52
Offset: 1

Views

Author

Charlie Marion, Nov 15 2011

Keywords

Comments

See A198453 and A198457.

Examples

			3*5 + 6*8 = 7*9
4*6 + 4*6 = 6*8
5*7 + 16*17 = 17*18
6*8 + 10*12 12*14
7*9 + 8*10 = 11*13
7*9 + 30*32 = 31*33

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198459, A198461-A198469.

A198461 Consider triples a<=b

Original entry on oeis.org

2, 3, 4, 3, 7, 8, 4, 12, 13, 5, 18, 19, 6, 6, 9, 6, 11, 13, 6, 25, 26, 7, 15, 17, 7, 33, 34, 8, 42, 43, 9, 10, 14, 9, 15, 18, 9, 52, 53, 10, 30, 32, 10, 63, 64, 11, 36, 38, 11, 75, 76, 12, 14, 19, 12, 19, 23, 12, 27, 30, 12, 88, 89, 13, 102, 103, 14, 57, 59, 14, 117, 118
Offset: 1

Views

Author

Charlie Marion, Nov 26 2011

Keywords

Comments

See A198453.

Examples

			2*5 + 3*6 = 4*7
3*6 + 7*10 = 8*11
4*7 + 12*15 = 13*16
5*8 + 18*21 = 19*22
6*9 + 6*9 = 9*12
6*9 + 11*14 = 13*16

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198460, A198462-A198469.

A198462 Consider triples a<=b

Original entry on oeis.org

2, 3, 4, 5, 6, 6, 6, 7, 7, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 14, 14, 15, 15, 15, 15, 15, 16, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 27, 27, 28
Offset: 1

Views

Author

Charlie Marion, Nov 26 2011

Keywords

Comments

See A198453.

Examples

			2*5 + 3*6 = 4*7
3*6 + 7*10 = 8*11
4*7 + 12*15 = 13*16
5*8 + 18*21 = 19*22
6*9 + 6*9 = 9*12
6*9 + 11*14 = 13*16

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198461, A198463-A198469.

A198463 Consider triples a<=b

Original entry on oeis.org

3, 7, 12, 18, 6, 11, 25, 15, 33, 42, 10, 15, 52, 30, 63, 36, 75, 14, 19, 27, 88, 102, 75, 117, 18, 23, 42, 65, 133, 150, 30, 39, 168, 22, 27, 60, 92, 187, 102, 207, 42, 54, 228, 22, 26, 31, 81, 250, 51, 135, 273, 147, 297, 30, 35, 105, 322, 45, 66, 84, 348
Offset: 1

Views

Author

Charlie Marion, Nov 26 2011

Keywords

Comments

See A198453.

Examples

			2*5 + 3*6 = 4*7
3*6 + 7*10 = 8*11
4*7 + 12*15 = 13*16
5*8 + 18*21 = 19*22
6*9 + 6*9 = 9*12
6*9 + 11*14 = 13*16

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198462, A198464-A198469.
Showing 1-10 of 16 results. Next

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

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