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.

A069764 Frobenius number of the numerical semigroup generated by consecutive octahedral numbers.

Original entry on oeis.org

89, 773, 3611, 12179, 33349, 78889, 167383, 326471, 595409, 1027949, 1695539, 2690843, 4131581, 6164689, 8970799, 12769039, 17822153, 24441941, 32995019, 43908899, 57678389, 74872313, 96140551, 122221399, 153949249, 192262589, 238212323, 292970411, 357838829
Offset: 2

Views

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 18 2002

Keywords

Comments

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since consecutive octahedral numbers are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generated semigroup has the formula ab-a-b.

Examples

			a(2)=89 because 89 is not a nonnegative linear combination of 6 and 19 (the second and third octahedral numbers), but all integers greater than 89 are.

Links

Crossrefs

Cf. A005900, A037165, A059769, A069755-A069763.

Programs

  • Mathematica
    FrobeniusNumber/@Partition[Rest[Table[(n(2n^2+1))/3,{n,30}]],2,1] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{89,773,3611,12179,33349,78889,167383},30] (* Harvey P. Dale, Nov 19 2015 *)

Formula

a(n) = ((1/3)*n*(2*n^2+1)-1)*((1/3)*(n+1)*(2*(n+1)^2+1)-1)-1.
G.f.: x^2*(89+150*x+69*x^2+20*x^3-13*x^4+6*x^5-x^6)/(1-x)^7. - Colin Barker, Feb 12 2012
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+ 21*a(n-5)- 7*a(n-6)+a(n-7). - Harvey P. Dale, Nov 19 2015

Extensions

More terms from Carl Najafi, Sep 10 2011

A069756 Frobenius number of the numerical semigroup generated by consecutive squares.

Original entry on oeis.org

23, 119, 359, 839, 1679, 3023, 5039, 7919, 11879, 17159, 24023, 32759, 43679, 57119, 73439, 93023, 116279, 143639, 175559, 212519, 255023, 303599, 358799, 421199, 491399, 570023, 657719, 755159, 863039, 982079, 1113023, 1256639, 1413719, 1585079, 1771559
Offset: 2

Views

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 05 2002

Keywords

Comments

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1, ..., a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since consecutive squares are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generated semigroup has the formula ab-a-b.
Given the set {n, n+1, n+2, n+3} and starting at n=0, the sum of all possible products of the terms in all possible subsets = a(n+2). Example for n=5, 5+6+7+8=26; 5(6+7+8)+6*(7+8)+7*8=277; 5*(6*7+6*8+7*8)+6*7*8=1066; 5*6*7*8=1680 and the sum of these 15 possible subsets is 3023 = a(5+2) = a(7). The sum is a(n+2) = n^4 + 10*n^3 + 35*n^2 + 50*n + 23. - J. M. Bergot, Apr 17 2013

Examples

			a(2)=23 because 23 is not a nonnegative linear combination of 4 and 9, but all integers greater than 23 are.

Links

Crossrefs

Cf. A000290, A037165, A059769, A069755, A069757-A069764.

Programs

  • Maple
    seq(n^4+2*n^3-n^2-2*n-1, n=2..50); # Robert Israel, Nov 01 2015
  • Mathematica
    Table[(n^2-1)((n+1)^2-1)-1, {n,2,30}] (* T. D. Noe, Nov 27 2006 *)
FrobeniusNumber/@Partition[Range[2,40]^2,2,1] (* Harvey P. Dale, Jul 25 2012 *)
  • PARI
    x='x+O('x^50); Vec(x^2*(23+4*x-6*x^2+4*x^3-x^4)/(1-x)^5) \\ Altug Alkan, Nov 01 2015

Formula

a(n) = n^2*(n+1)^2-n^2-(n+1)^2 = n^4+2*n^3-n^2-2*n-1.
a(n) = Numerator of ((n + 2)! - (n - 2)!)/n!, n >=2. - Artur Jasinski, Jan 09 2007
G.f.: x^2*(23+4*x-6*x^2+4*x^3-x^4)/(1-x)^5. [Colin Barker, Feb 14 2012]
a(n) = (n-1)*n*(n+1)*(n+2) - 1 = A052762(n+2) - 1. - Jean-Christophe Hervé, Nov 01 2015

Extensions

Corrected by T. D. Noe, Nov 27 2006

A069757 Frobenius number of the numerical semigroup generated by three consecutive pentagonal numbers.

Original entry on oeis.org

43, 133, 287, 1699, 921, 1569, 3006, 3197, 4129, 12915, 6445, 8621, 14087, 13549, 16753, 43144, 20783, 25793, 38854, 35769, 43321, 101747, 48147, 57764, 82815, 74393, 89017, 198120, 93689, 108983, 151478, 133957, 159025, 341659, 162180
Offset: 2

Views

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 05 2002

Keywords

Comments

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the greatest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since three consecutive pentagonal numbers are relatively prime, they generate a numerical semigroup with a Frobenius number.

Examples

			a(2)=43 because 43 is not a nonnegative linear combination of 5, 12 and 22, but all integers greater than 43 are.

Links

Crossrefs

Cf. A000326, A037165, A059769, A069755-A069764.

Programs

  • Mathematica
    FrobeniusNumber/@Partition[PolygonalNumber[5,Range[2,40]],3,1] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 16 2018 *)

A069759 Frobenius number of the numerical semigroup generated by consecutive hex numbers.

Original entry on oeis.org

107, 647, 2159, 5399, 11339, 21167, 36287, 58319, 89099, 130679, 185327, 255527, 343979, 453599, 587519, 749087, 941867, 1169639, 1436399, 1746359, 2103947, 2513807, 2980799, 3509999, 4106699, 4776407
Offset: 1

Views

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 08 2002

Keywords

Comments

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since consecutive hex numbers are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generated semigroup has the formula ab-a-b.

Examples

			a(1)=107 because 107 is not a nonnegative linear combination of 7 and 19, but all integers greater than 107 are.

Links

Crossrefs

Cf. A003215, A037165, A059769, A069755-A069764.

Programs

  • Mathematica
    FrobeniusNumber/@Partition[Table[3n^2+3n+1,{n,30}],2,1] (* Harvey P. Dale, Dec 25 2018 *)

Formula

a(n) = 9*n^4+36*n^3+45*n^2+18*n-1; with offset 2, a(n) = 9*n^4-9*n^2-1.
G.f.: x*(107+112*x-6*x^2+4*x^3-x^4)/(1-x)^5. - Colin Barker, Feb 14 2012

A069761 Frobenius number of the numerical semigroup generated by four consecutive tetrahedral numbers.

Original entry on oeis.org

41, 249, 253, 853, 1243, 1571, 2619, 5059, 5357, 9437, 11801, 13609, 18327, 27607, 28919, 41951, 49169, 54473, 67253, 90573, 94051, 124099, 140347, 152027, 178989, 226141, 233369, 291089, 321839, 343639, 392631, 475999, 488993, 587633, 639653, 676181, 756779
Offset: 2

Views

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 09 2002

Keywords

Comments

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since four consecutive tetrahedral numbers are relatively prime, they generate a numerical semigroup with a Frobenius number.

Examples

			a(2) = 41 because 41 is not a nonnegative linear combination of 4, 10, 20 and 35, but all integers greater than 43 are.

Links

Crossrefs

Cf. A000292, A037165, A059769, A069755-A069764.

Programs

  • Mathematica
    FrobeniusNumber/@Partition[Binomial[Range[2,50]+2,3],4,1] (* Harvey P. Dale, Jan 22 2012 *)

Formula

Conjecture: a(n)= +a(n-1) +4*a(n-6) -4*a(n-7) -6*a(n-12) +6*a(n-13) +4*a(n-18) -4*a(n-19) -a(n-24) +a(n-25). - R. J. Mathar, Aug 15 2025
Conjectured g.f.: x^2*(-4*x^2 -600*x^3 -390*x^4 -1680*x^9 -282*x^8 -496*x^11 -804*x^10 -208*x -312*x^15 -144*x^14 -768*x^13 -772*x^12-41 -32*x^18 -40*x^17 -102*x^16 -2*x^20 -8*x^19 -1608*x^7 +x^24 -884*x^6 -328*x^5) / ( (1+x)^4 *(x^2-x+1)^4 *(1+x+x^2)^4 *(x-1)^5 ). - R. J. Mathar, Aug 15 2025

Extensions

Sequence terms corrected and extended by Harvey P. Dale, Jan 22 2012
Offset corrected and example corrected by Harvey P. Dale, Jan 24 2012

A069762 Frobenius number of the numerical semigroup generated by three consecutive pyramidal numbers.

Original entry on oeis.org

51, 191, 609, 1324, 2813, 4711, 8576, 13894, 23319, 34165, 51661, 71126, 100529, 136239, 187543, 241586, 321251, 404839, 516704, 645358, 813141, 982651, 1221299, 1463734, 1767473, 2106271, 2524101, 2940909, 3500209, 4061663, 4736456, 5474526, 6352219, 7228469
Offset: 2

Views

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 18 2002

Keywords

Comments

Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since three consecutive pyramidal numbers are relatively prime, they generate a numerical semigroup with a Frobenius number.

Examples

			a(2)=51 because 51 is not a nonnegative linear combination of 5, 14 and 30, but all integers greater than 51 are.

Links

  • R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Crossrefs

Cf. A000330, A037165, A059769, A069755-A069764.

Extensions

More terms from and offset corrected by Sean A. Irvine, May 19 2024

A069763 Frobenius number of the numerical semigroup generated by consecutive cubes.

Original entry on oeis.org

181, 1637, 7811, 26659, 73529, 174761, 372007, 727271, 1328669, 2296909, 3792491, 6023627, 9254881, 13816529, 20114639, 28641871, 39988997, 54857141, 74070739, 98591219, 129531401, 168170617, 215970551, 274591799, 345911149
Offset: 2

Views

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 18 2002

Keywords

Comments

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since consecutive cubes are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generated semigroup has the formula ab-a-b.

Examples

			a(2)=181 because 181 is not a nonnegative linear combination of 8 and 27, but all integers greater than 181 are.

Links

  • R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Crossrefs

Cf. A000578, A037165, A059769, A069755-A069764.

Formula

a(n) = n^3*(n+1)^3-n^3-(n+1)^3 = n^6+3*n^5+3*n^4-n^3-3*n^2-3*n-1.
G.f.: x^2*(181+370*x+153*x^2+24*x^3-13*x^4+6*x^5-x^6)/(1-x)^7. [Colin Barker, Feb 14 2012]

A069758 Frobenius number of the numerical semigroup generated by three consecutive hexagonal numbers.

Original entry on oeis.org

65, 377, 395, 797, 1589, 6029, 3347, 4571, 6035, 10997, 10979, 12212, 19409, 47246, 24023, 29003, 35357, 52112, 50603, 50411, 73049, 158207, 78155, 90203, 102005, 144443, 138467, 131474, 183077
Offset: 2

Views

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 08 2002

Keywords

Comments

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since three consecutive hexagonal numbers are relatively prime, they generate a numerical semigroup with a Frobenius number.

Examples

			a(2)=65 because 65 is not a nonnegative linear combination of 6, 15 and 28, but all integers greater than 65 are.

Links

  • R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Crossrefs

Cf. A000384, A037165, A059769, A069755-A069764.

Programs

  • Mathematica
    FrobeniusNumber/@Partition[Table[n(2n-1),{n,2,35}],3,1] (* Harvey P. Dale, Jul 25 2011 *)

A069760 Frobenius number of the numerical semigroup generated by consecutive centered square numbers.

Original entry on oeis.org

47, 287, 959, 2399, 5039, 9407, 16127, 25919, 39599, 58079, 82367, 113567, 152879, 201599, 261119, 332927, 418607, 519839, 638399, 776159, 935087, 1117247, 1324799, 1559999, 1825199, 2122847
Offset: 1

Views

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 09 2002

Keywords

Comments

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since consecutive centered squares are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generator semigroup is ab-a-b.

Examples

			a(1)=47 because 47 is not a nonnegative linear combination of 5 and 13, but all integers greater than 47 are.

Links

Crossrefs

Cf. A001844, A037165, A059769, A069755-A069764.

Programs

  • Mathematica
    Table[4n^4+16n^3+20n^2+8n-1,{n,30}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{47,287,959,2399,5039},30] (* Harvey P. Dale, Apr 25 2011 *)

Formula

a(n) = 4*n^4+16*n^3+20*n^2+8*n-1.
a(n) = 5*a(n-1)-10*a(n-2) +10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Apr 25 2011
G.f.: x*(47+52*x-6*x^2+4*x^3-x^4)/(1-x)^5. - Colin Barker, Feb 14 2012
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