A059769 -id:A059769 - cp's OEIS frontend

A069755 Frobenius number of the numerical semigroup generated by 3 consecutive triangular numbers.

17, 29, 89, 125, 251, 323, 539, 659, 989, 1169, 1637, 1889, 2519, 2855, 3671, 4103, 5129, 5669, 6929, 7589, 9107, 9899, 11699, 12635, 14741, 15833, 18269, 19529, 22319, 23759, 26927, 28559, 32129, 33965, 37961, 40013, 44459, 46739, 51659

Offset: 2

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

The Frobenius number of the 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. Any three successive triangular numbers are relatively prime, so they generate a numerical semigroup with a Frobenius number.

			a(2)=17 because 17 is not a nonnegative linear combination of 3, 6 and 10 but all numbers greater than 17 are.

Harvey P. Dale, Table of n, a(n) for n = 2..1000
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
Aureliano M. Robles-Pérez, José Carlos Rosales, The Frobenius number for sequences of triangular and tetrahedral numbers, arXiv:1706.04378 [math.NT], 2017.

Cf. A000217, A037165, A059769, A069756-A069762.

tri=Range[40]Range[2,41]/2; Table[t=CoefficientList[Series[1/(1-x^tri[[n]])/(1-x^tri[[n+1]])/(1-x^tri[[n+2]]), {x,0,n(n+1)(n+2)}], x]; Last[Position[t,0]-1][[1]], {n,2,33}] (* T. D. Noe, Nov 27 2006 *)
Rest[FrobeniusNumber/@Partition[Accumulate[Range[50]],3,1]] (* Harvey P. Dale, Oct 04 2011 *)

Conjectures from Colin Barker, Nov 22 2012: (Start)
a(n) = (-14 + 6*(-1)^n + (3+9*(-1)^n)*n + 3*(5+(-1)^n)*n^2 + 6*n^3)/8.
G.f.: x^2*(17 + 12*x + 9*x^2 - 3*x^4 + x^6) / ((1 - x)^4*(1 + x)^3). (End)
Conjectures from Colin Barker, Mar 21 2017: (Start)
a(n) = (6*n^3 + 18*n^2 + 12*n - 8)/8 for n even.
a(n) = (6*n^3 + 12*n^2 - 6*n - 20)/8 for n odd. (End)

Corrected by T. D. Noe, Nov 27 2006

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

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

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.

			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.

Harvey P. Dale, Table of n, a(n) for n = 2..1000
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

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

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 *)

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

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

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

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

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

T. D. Noe, Table of n, a(n) for n = 2..1000
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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

seq(n^4+2*n^3-n^2-2*n-1, n=2..50); # Robert Israel, Nov 01 2015

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 *)

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

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

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

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

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.

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

Harvey P. Dale, Table of n, a(n) for n = 2..1000
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

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

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

A305412 a(n) = F(n)*F(n+1) + F(n+2), where F = A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 3, 5, 11, 23, 53, 125, 307, 769, 1959, 5039, 13049, 33929, 88451, 230957, 603667, 1578823, 4130829, 10810469, 28295411, 74067401, 193893263, 507590495, 1328842801, 3478880593, 9107706243, 23844088085, 62424315227, 163428464759, 427860443429, 1120151837069

Offset: 0

Vincenzo Librandi, Jun 05 2018

Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Cf. A001654, A269803.
Cf. A059769: F(n)*F(n+1) - F(n+2), with offset 3.
Equals A000045 + A286983.
First differences are listed in A059727 (after 0).

List([0..35], n -> Fibonacci(n)*Fibonacci(n+1)+Fibonacci(n+2)); # Muniru A Asiru, Jun 06 2018

[Fibonacci(n)*Fibonacci(n+1)+Fibonacci(n+2): n in [0..30]];

Table[Fibonacci[n] Fibonacci[n+1] + Fibonacci[n+2], {n, 0, 30}]

G.f.: (1 - 5*x^2 - 2*x^3 + x^4)/((x + 1)*(1 - 3*x + x^2)*(1 - x - x^2)).
a(n) = 3*a(n-1) + a(n-2) - 5*a(n-3) - a(n-4) + a(n-5).
5*a(n) = (-1)^(n+1) +5*F(n+2) + A002878(n). - R. J. Mathar, Nov 14 2019

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

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

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.

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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

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

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

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

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.

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

Harvey P. Dale, Table of n, a(n) for n = 2..100
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

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

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

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

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

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

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.

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

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

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

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

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

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.

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

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

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

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]

A105392 Frobenius number of the subsemigroup of the natural numbers generated by successive pairs of Lucas numbers.

Original entry on oeis.org

0, 5, 17, 59, 169, 475, 1287, 3449, 9149, 24155, 63557, 166919, 437839, 1147645, 3006777, 7875419, 20623889, 54003395, 141397847, 370208849, 969258949, 2537616955, 6643671117, 17393524559, 45537109919, 119218140725

Offset: 1

Jonathan Vos Post, May 01 2005

			a(3) = 17 because the 3rd and 4th Lucas numbers are 4 and 7, so
a(3) = (4-1)*(7-1)-1 = 17. Or, a(3)=17 because 17 is the largest positive
integer that is not a nonnegative linear combination of 4 and 7.

R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
Eric Weisstein's World of Mathematics, Lucas numbers.

Cf. A000204, A059769.

A000204 := proc(n) option remember; if n = 1 then 1; elif n = 2 then 3; else procname(n-1)+procname(n-2) ; end if; end proc:
A105392 := proc(n) A000204(2*n+1)-A000204(n+2)+(-1)^n ; end proc:
seq(A105392(n),n=0..20) ; # R. J. Mathar, Nov 16 2010

a(n) = (L(n)-1)*(L(n+1)-1)-1 where L(n) = A000204(n).
a(n) = A002878(n)-A000204(n+2)+(-1)^n, for n>1. [Ralf Stephan, Nov 15 2010, index shifted by R. J. Mathar, Nov 16 2010]
G.f.: x^2*(5+2*x+3*x^2-x^4)/(1+x)/(1-3*x+x^2)/(1-x-x^2). [Colin Barker, Feb 17 2012]

cp's OEIS Frontend

A069755 Frobenius number of the numerical semigroup generated by 3 consecutive triangular numbers.

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

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

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

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A305412 a(n) = F(n)*F(n+1) + F(n+2), where F = A000045 (Fibonacci numbers).

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

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

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