A069756 -id:A069756 - cp's OEIS frontend

A052762 Products of 4 consecutive integers: a(n) = n(n-1)(n-2)*(n-3).

0, 0, 0, 0, 24, 120, 360, 840, 1680, 3024, 5040, 7920, 11880, 17160, 24024, 32760, 43680, 57120, 73440, 93024, 116280, 143640, 175560, 212520, 255024, 303600, 358800, 421200, 491400, 570024, 657720, 755160, 863040, 982080, 1113024

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Also, starting with n=4, the square of area of cyclic quadrilateral with sides n, n-1, n-2, n-3. - Zak Seidov, Jun 20 2003
Number of n-colorings of the complete graph on 4 vertices, which is also the tetrahedral graph. - Eric M. Schmidt, Oct 31 2012
Cf. A130534 for relations to colored forests and disposition of flags on flagpoles. - Tom Copeland, Apr 05 2014
Number of 4-permutations of the set {1, 2, ..., n}. - Joerg Arndt, Apr 05 2014

Eric M. Schmidt, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 719
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014.
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv:1508.07894 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Cyclic Quadrilateral
Wikipedia, Pochhammer symbol.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001094, A002378, A007531, A052787.

[n*(n-1)*(n-2)*(n-3): n in [0..30]]; // G. C. Greubel, Nov 19 2017

spec := [S,{B=Set(Z),S=Prod(Z,Z,Z,Z,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(numbperm (n,4), n=0..34); # Zerinvary Lajos, Apr 26 2007
G(x):=x^4*exp(x): f[0]:=G(x): for n from 1 to 34 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..34); # Zerinvary Lajos, Apr 05 2009

Table[n*(n+1)*(n+2)*(n+3), {n,-3,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)
Times@@@Partition[Range[-3,60], 4, 1] (* Harvey P. Dale, May 09 2012 *)
LinearRecurrence[ {5,-10,10,-5,1}, {0,0,0,0,24}, 60] (* Harvey P. Dale, May 09 2012 *)

A052762(n):=n*(n-1)*(n-2)*(n-3)$
makelist(A052762(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n)=24*binomial(n,4) \\ Charles R Greathouse IV, Nov 20 2011

a(n) = n*(n-1)*(n-2)*(n-3) = n!/(n-4)! (for n >= 4).
a(n) = A001094(n) - n.
E.g.f.: x^4*exp(x).
Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (-1-n)*a(n) + (n-3)*a(n+1)}.
a(n) + 1 = A062938(n-4) for n > 4. - Amarnath Murthy, Dec 13 2003
a(n) = numbperm(n, 4). - Zerinvary Lajos, Apr 26 2007
O.g.f.: -24*x^4/(-1+x)^5. - R. J. Mathar, Nov 23 2007
For n > 4: a(n) = A173333(n, n-4). - Reinhard Zumkeller, Feb 19 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=24. - Harvey P. Dale, May 09 2012
a(n) = a(n-1) + 4*A007531(n). - J. M. Bergot, May 30 2012
a(n) = (n)4 = Pochhammer(n,4), using the "falling factorial" convention; other authors write Pochhammer(x,k) for what is denoted x^(k) in the Wikipedia article, then a(n) = (n-3)^(4). - _M. F. Hasler, Oct 20 2013
a(n) - 1 = A069756(n-2) for n >= 4. - Jean-Christophe Hervé, Nov 01 2015
a(n) = 24 * A000332(n). - Bruce J. Nicholson, Apr 03 2017
From R. J. Mathar, Jun 30 2021: (Start)
Sum_{n>=4} 24*(-1)^n/a(n) = A242023.
Sum_{n>=4} 1/a(n) = 1/18. (End)

More terms from Henry Bottomley, Mar 20 2000

Formula corrected by Philippe Deléham, Dec 12 2003

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

Original entry on oeis.org

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

A264031 Minimum of the sum r + s of the coefficients of a linear combination of consecutive squares rk^2 + s(k+1)^2 equals to n, with r, s and k >=0.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 2, 1, 4, 5, 3, 2, 5, 6, 1, 3, 2, 7, 5, 4, 3, 8, 6, 1, 4, 3, 7, 6, 5, 4, 2, 7, 3, 5, 1, 8, 7, 6, 5, 2, 8, 4, 6, 5, 9, 8, 3, 1, 2, 9, 5, 7, 6, 10, 9, 3, 7, 5, 10, 2, 8, 7, 1, 10, 3, 8, 6, 11, 7, 9, 2, 4, 11, 3, 9, 7, 12, 8, 5, 1

Offset: 1

Jean-Christophe Hervé, Nov 01 2015

nonn

Every number n >= (k+2)*(k+1)*k*(k-1) - 1 = A069756(k) is of the form r*k^2 + s*(k+1)^2 with r, s and k positive integers. For any n >= 1, a(n) gives the minimum value of r + s for n = r*k^2 + s*(k+1)^2.

			7 = 2^2 + 3*1^2, the sum of the coefficients of the linear combination is 1+3 = 4; The only other linear combination of consecutive squares giving 7 is 7*1^2 + 0, thus a(7) = 4, the minimum sum of the coefficients.

Cf. A001105, A069756, A230812.

a(k^2) = 1, a(A001105(k)) = 2 for k > 0 and a(A230812(k)) = 2; for any other values, a(n) >= 3.

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A052762 Products of 4 consecutive integers: a(n) = n(n-1)(n-2)*(n-3).

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

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A264031 Minimum of the sum r + s of the coefficients of a linear combination of consecutive squares rk^2 + s(k+1)^2 equals to n, with r, s and k >=0.

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cp's OEIS Frontend

A052762 Products of 4 consecutive integers: a(n) = n*(n-1)*(n-2)*(n-3).

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Magma

Maple

Mathematica

Maxima

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

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Author

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Crossrefs

Programs

Mathematica

Formula

Extensions

A264031 Minimum of the sum r + s of the coefficients of a linear combination of consecutive squares r*k^2 + s*(k+1)^2 equals to n, with r, s and k >=0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A052762 Products of 4 consecutive integers: a(n) = n(n-1)(n-2)*(n-3).

A264031 Minimum of the sum r + s of the coefficients of a linear combination of consecutive squares rk^2 + s(k+1)^2 equals to n, with r, s and k >=0.