A052762 -id:A052762 - cp's OEIS frontend

A062938 a(n) = n(n+1)(n+2)(n+3)+1 = (n^2 + 3n + 1)^2.

1, 25, 121, 361, 841, 1681, 3025, 5041, 7921, 11881, 17161, 24025, 32761, 43681, 57121, 73441, 93025, 116281, 143641, 175561, 212521, 255025, 303601, 358801, 421201, 491401, 570025, 657721, 755161, 863041, 982081, 1113025, 1256641

Offset: 0

Amarnath Murthy, Jul 05 2001

a(n) = product of first four terms of an arithmetic progression + n^4, where the first term is 1 and the common difference is n. E.g. a(1) = 1*2*3*4 +1^4 =25, a(4) = 1*5*9*13 + 4^4= 841 etc. - Amarnath Murthy, Sep 19 2003
Is it possible for one of the squares to be the sum of two or more lesser squares each used only once? - J. M. Bergot, Feb 17 2011
Yes, in fact a(1)-a(11) are examples. - Charles R Greathouse IV, Jun 28 2011
This sequence demonstrates that the product of any 4 consecutive integers plus 1 is a square.  The square roots are in A028387. - Harvey P. Dale, Oct 19 2011
The sum of three consecutive terms of the sequence is divisible by 3. The quotient is a square number: [a(n)+a(n+1)+a(n+2)]/3=(n^2+5*n+7)^2. - Carmine Suriano, Jan 23 2012
All terms end with 1 or 5. - Uri Geva, Jan 06 2024

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 19.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 85.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A028387, A052762, A082043.

[(n^2+3*n+1)^2: n in [0..50]]; // G. C. Greubel, Dec 24 2022

Table[(n^2+3*n+1)^2, {n,0,50}]
Times@@#+1&/@Partition[Range[0,50],4,1]  (* Harvey P. Dale, Apr 02 2011 *)

j=[]; for(n=0,70,j=concat(j,(n^2+3*n+1)^2)); j

{ for (n=0, 1000, write("b062938.txt", n, " ", (n^2 + 3*n + 1)^2) ) } \\ Harry J. Smith, Aug 14 2009

[(n^2+3*n+1)^2 for n in range(51)] # G. C. Greubel, Dec 24 2022

a(n+1) = numerator( ((n+2)! + (n-2)!)/n! ), for n>=2. - Artur Jasinski, Jan 09 2007; corrected by Michel Marcus, Dec 25 2022
a(n) = A028387(n)^2. - Jaroslav Krizek, Oct 31 2010
a(n) = n*(n+1)*(n+2)*(n+3)+1^4 = 1*(1+n)*(1+2*n)*(1+3*n)+n^4 =(n^2+3*n+1)^2; in general, n*(n+k)*(n+2*k)*(n+3*k)+k^4 = k*(k+n)*(k+2*n)*(k+3*n)+n^4 = (n^2+3*k*n+k^2)^2. - Charlie Marion, Jan 13 2011
G.f.: (1+20*x+6*x^2-4*x^3+x^4)/(1-x)^5. - Colin Barker, Jun 30 2012
a(n) = A052762(n+3) + 1. - Bruce J. Nicholson, Apr 22 2017
Sum_{n>=0} 1/a(n) = (Pi^2/5)*(1+t^2) - 2*sqrt(5)*Pi*t/25 - 1, where t = tan(Pi*sqrt(5)/2). - Amiram Eldar, Apr 03 2022
E.g.f.: (1 +24*x +36*x^2 +12*x^3 +x^4)*exp(x). - G. C. Greubel, Dec 24 2022

More terms from Jason Earls, Harvey P. Dale and Dean Hickerson, Jul 06 2001

A053625 Product of 6 consecutive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 5040, 20160, 60480, 151200, 332640, 665280, 1235520, 2162160, 3603600, 5765760, 8910720, 13366080, 19535040, 27907200, 39070080, 53721360, 72681840, 96909120, 127512000, 165765600, 213127200, 271252800, 342014400, 427518000, 530122320

Offset: 0

Henry Bottomley, Mar 20 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A002378, A007531, A045619, A052762, A052787.

Cf. A000579, A173333.

F:=Factorial;; Concatenation([0,0,0,0,0,0], List([6..30], n-> F(n)/F(n-5) )); # G. C. Greubel, Aug 27 2019

I:=[0,0,0,0,0,0,720]; [n le 7 select I[n] else 7*Self(n-1) -21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6) +Self(n-7): n in [1..30]]; // Vincenzo Librandi, Apr 28 2012

seq(combinat[numbperm](n, 6), n=0..31); # Zerinvary Lajos, Apr 26 2007

CoefficientList[Series[720*x^6/(1-x)^7,{x,0,30}],x] (* Vincenzo Librandi, Apr 28 2012 *)
Times@@@Partition[Range[-5,30],6,1] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,0,0,720},30] (* Harvey P. Dale, Nov 13 2015 *)
Pochhammer[Range[30]-6, 6] (* G. C. Greubel, Aug 27 2019 *)

a(n)=factorback([n-5..n]) \\ Charles R Greathouse IV, Oct 07 2015

[rising_factorial(n-5,6) for n in (0..30)] # G. C. Greubel, Aug 27 2019

a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5) = n!/(n-6)! = A052787(n)*(n-6) = a(n-1)*n/(n-6).
E.g.f.: x^6*exp(x).
a(n) = 720 * A000579(n). - Zerinvary Lajos, Apr 26 2007
For n > 5: a(n) = A173333(n, n-6). - Reinhard Zumkeller, Feb 19 2010
G.f.: 720*x^6/(1-x)^7. - Colin Barker, Mar 27 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). - Vincenzo Librandi, Apr 28 2012
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=6} 1/a(n) = 1/600.
Sum_{n>=6} (-1)^n/a(n) = 4*log(2)/15 - 661/3600. (End)

A122046 Partial sums of floor(n^2/8).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 10, 16, 24, 34, 46, 61, 79, 100, 124, 152, 184, 220, 260, 305, 355, 410, 470, 536, 608, 686, 770, 861, 959, 1064, 1176, 1296, 1424, 1560, 1704, 1857, 2019, 2190, 2370, 2560, 2760, 2970, 3190, 3421, 3663, 3916, 4180, 4456, 4744, 5044, 5356, 5681, 6019, 6370

Offset: 0

Roger L. Bagula, Sep 13 2006

Degree of the polynomial P(n+1,x), defined by P(n,x) = [x^(n-1)*P(n-1,x)*P(n-4,x)+P(n-2,x)*P(n-3,x)]/P(n-5,x) with P(1,x)=P(0,x)=P(-1,x)=P(-2,x)=P(-3,x)=1.
Define the sequence b(n) = 1, 4, 10, 20, 36, 60,... for n>=0 with g.f. 1/((1+x)*(1+x^2)*(1-x)^5). Then a(n+3) = b(n)-b(n-1) and b(n)+b(n+1)+b(n+2)+b(n+3) = A052762(n+7)/24. - J. M. Bergot, Aug 21 2013
Maximum Wiener index of all maximal 4-degenerate graphs with n-1 vertices.  (A maximal 4-degenerate graph can be constructed from a 4-clique by iteratively adding a new 4-leaf (vertex of degree 4) adjacent to four existing vertices.)  The extremal graphs are 4th powers of paths, so the bound also applies to 4-trees. - Allan Bickle, Sep 15 2022

			a(6) = 10 = 0 + 0 + 0 + 1 + 2 + 3 + 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
A. N. W. Hone, Comments on A122046
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004; arXiv:0807.2538 [nlin.SI], 2008. [Set a(n)=d(n+3) on p. 8]
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=4]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1,-3,3,-1).

Cf. A014125, A052762, A057077, A090294, A122047, A144678, A190718.
Partial sums of A001972.
Maximum Wiener index of all maximal k-degenerate graphs for k=1..6: A000292, A002623, A014125, A122046 (this sequence), A122047, A175724.

[Round((2*n^3+3*n^2-8*n)/48): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011

A122046 := proc(n) round((2*n^3+3*n^2-8*n)/48) ; end proc: # Mircea Merca

p[n_] := p[n] = Cancel[Simplify[ (x^(n - 1)p[n - 1]p[n - 4] + p[n - 2]*p[n - 3])/p[n - 5]]]; p[ -5] = 1;p[ -4] = 1;p[ -3] = 1;p[ -2] = 1;p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]
Accumulate[Floor[Range[0,60]^2/8]] (* or *) LinearRecurrence[{3,-3,1,1,-3,3,-1},{0,0,0,1,3,6,10},60] (* Harvey P. Dale, Dec 23 2019 *)

a(n)=(2*n^3+3*n^2-8*n+3)\48 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = Sum_{k=0..n} floor(k^2/8).
a(n) = round((2*n^3 + 3*n^2 - 8*n)/48) = round((4*n^3 + 6*n^2 - 16*n - 9)/96) = floor((2*n^3 + 3*n^2 - 8*n + 3)/48) = ceiling((2*n^3 + 3*n^2 - 8*n - 12)/48). - Mircea Merca
a(n) = a(n-8) + (n-4)^2 + n, n > 8. - Mircea Merca
From Andrew Hone, Jul 15 2008: (Start)
a(n+1) = cos((2*n+1)*Pi/4)/(4*sqrt(2)) + (2*n+3)*(2*n^2 + 6*n - 5)/96 + (-1)^n/32.
a(n+1) = A057077(n+1)/8 + A090294(n-1)/32 + (-1)^n/32.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4) - 3*a(n-5) + 3*a(n-6) - a(n-7). (End)
O.g.f.: x^3 / ( (1+x)*(x^2+1)*(x-1)^4 ). - R. J. Mathar, Jul 15 2008
From Johannes W. Meijer, May 20 2011: (Start)
a(n+3) = A144678(n) + A144678(n-1) + A144678(n-2) + A144678(n-3);
a(n+3) = Sum_{k=0..6} min(6-k+1,k+1)* A190718(n+k-6). (End)
a(n) = (4*n^3 + 6*n^2 - 16*n - 9 - 3*(-1)^n + 12*(-1)^((2*n - 1 + (-1)^n)/4))/96. - Luce ETIENNE, Mar 21 2014
E.g.f.: ((2*x^3 + 9*x^2 - 3*x - 6)*cosh(x) + 6*(cos(x) + sin(x)) + (2*x^3 + 9*x^2 - 3*x - 3)*sinh(x))/48. - Stefano Spezia, Apr 05 2023

Edited by N. J. A. Sloane, Sep 17 2006, Jul 11 2008, Jul 12 2008

More formulas and better name from Mircea Merca, Nov 19 2010

A268283 Number of distinct directed Hamiltonian cycles of the Platonic graphs (in the order of tetrahedral, cubical, octahedral, dodecahedral, and icosahedral graph).

Original entry on oeis.org

6, 12, 32, 60, 2560

Offset: 1

Melvin Peralta, Jan 29 2016

a(n)/2 is the number of distinct undirected Hamiltonian cycles of the Platonic graph corresponding to a(n).

Eric Weisstein's World of Mathematics, Tetrahedral Graph
Eric Weisstein's World of Mathematics, Cubical Graph
Eric Weisstein's World of Mathematics, Octahedral Graph
Eric Weisstein's World of Mathematics, Dodecahedral Graph
Eric Weisstein's World of Mathematics, Icosahedral Graph

Cf. A052762 (tetrahedral graph), A140986 (cubical graph), A115400 (octahedral graph), A218513 (dodecahedral graph), A218514 (icosahedral graph).

A218514 Number of n-colorings of the icosahedral graph.

Original entry on oeis.org

0, 0, 0, 0, 240, 80400, 4012560, 76848240, 825447840, 6005512800, 33014872800, 146953113120, 554770648080, 1835249610480, 5448481998960, 14778817981200, 37135461679680, 87386816771520, 194264943433920, 410876964198720, 831638579799600, 1618744884780240

Offset: 0

Eric M. Schmidt, Oct 31 2012

N. Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993. See p. 69.

Eric M. Schmidt, Table of n, a(n) for n = 0..1000
Eric W. Weisstein, Icosahedral Graph
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A052762, A140986, A115400, A218513, A296916, A296917..

A218514(n):=n*(n-1)*(n-2)*(n-3)*(n^8 -24*n^7 +260*n^6 -1670*n^5 +6999*n^4 -19698*n^3 +36408*n^2 -40240*n +20170)$
makelist(A218514(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */

def A218514(n) : return n*(n-1)*(n-2)*(n-3)*(n^8 -24*n^7 +260*n^6 -1670*n^5 +6999*n^4 -19698*n^3 +36408*n^2 -40240*n +20170);

a(n) = n(n-1)(n-2)(n-3)(n^8 -24n^7 +260n^6 -1670n^5 +6999n^4 -19698n^3 +36408n^2 -40240n +20170).
Hence a(n) = n^12 - 30*n^11 + 415*n^10 - 3500*n^9 + 20023*n^8 - 81622*n^7 + 241605*n^6 - 517360*n^5 + 780286*n^4 - 782108*n^3 + 463310*n^2 - 121020*n (cf. A296917) - N. J. A. Sloane, Dec 23 2017
G.f.: -240*x^4*(12547*x^8 +131518*x^7 +481078*x^6 +743494*x^5 +485740*x^4 +128698*x^3 +12442*x^2 +322*x +1)/(x-1)^13. [Colin Barker, Nov 06 2012]

A239035 Product of 8 consecutive integers. a(n) = RisingFactorial(n, 8).

Original entry on oeis.org

0, 40320, 362880, 1814400, 6652800, 19958400, 51891840, 121080960, 259459200, 518918400, 980179200, 1764322560, 3047466240, 5079110400, 8204716800, 12893126400, 19769460480, 29654190720, 43609104000, 62990928000, 89513424000, 125318793600, 173059286400

Offset: 0

Michel Marcus, Mar 09 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vincent Thill, Nombres consécutifs et carrés, author's blog, Aug 03 2013 (consecutive numbers and squares).
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. Product of n consecutive integers: A002378 (n=2), A007531 (n=3), A052762 (n=4), A052787 (n=5), A053625 (n=6), A159083 (n=7).

[(n^4+14*n^3+63*n^2+98*n+28)^2-(28+8*n)^2: n in [0..30]]; // Vincenzo Librandi, Mar 11 2014

CoefficientList[Series[40320 x/(1 - x)^9, {x, 0, 30}], x] (* Vincenzo Librandi, Mar 11 2014 *)

PARI
```
a(n) = prod(k=0, 7, n+k);
```

concat([0], Vec(40320*x/(1-x)^9  + O(x^100))) \\ Colin Barker, Mar 09 2014

print([rising_factorial(n, 8) for n in range(23)]) # Peter Luschny, Mar 22 2022

a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7).
a(n) = (n^4+14*n^3+63*n^2+98*n+28)^2 - (28+8*n)^2. (see Thill link).
G.f.: 40320*x / (1-x)^9. - Colin Barker, Mar 09 2014
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/35280.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(2)/315 - 1163/66150. (End)

A159083 Products of 7 consecutive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 5040, 40320, 181440, 604800, 1663200, 3991680, 8648640, 17297280, 32432400, 57657600, 98017920, 160392960, 253955520, 390700800, 586051200, 859541760, 1235591280, 1744364160, 2422728000, 3315312000, 4475671200, 5967561600, 7866331200

Offset: 0

Zerinvary Lajos, Apr 05 2009

Michael De Vlieger and Harvey P. Dale, Table of n, a(n) for n = 0..10000 (first 1000 terms by Harvey P. Dale.)
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A052762, A052787, A053625, A173333.

Equals A008279(n,7) (for n>=7).

I:=[0,0,0,0,0,0,0,5040]; [n le 8 select I[n] else 8*Self(n-1) - 28*Self(n-2) +56*Self(n-3) -70*Self(n-4) +56*Self(n-5) -28*Self(n-6) +8*Self(n-7) -Self(n-8): n in [1..30]]; // G. C. Greubel, Jun 28 2018

G(x):=x^7*exp(x): f[0]:=G(x): for n from 1 to 36 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..33);

Table[Times@@(n+Range[0,6]),{n,-6,25}] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,0,0,0,0,0,0,5040},30] (* Harvey P. Dale, Apr 07 2018 *)

my(x='x+O('x^30)); concat([0,0,0,0,0,0,0], Vec(5040*x^7/(1-x)^8)) \\ G. C. Greubel, Jun 28 2018

E.g.f.: x^7*exp(x).
For n>=8: a(n) = A173333(n,n-7). - Reinhard Zumkeller, Feb 19 2010
G.f.: 5040*x^7/(1-x)^8. - Colin Barker, Mar 27 2012
From Amiram Eldar, Mar 08 2022: (Start)
a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6) = n!/(n-7)!.
Sum_{n>=7} 1/a(n) = 1/4320.
Sum_{n>=7} (-1)^(n+1)/a(n) = 4*log(2)/45 - 1327/21600. (End)

A162995 A scaled version of triangle A162990.

Original entry on oeis.org

1, 3, 1, 12, 4, 1, 60, 20, 5, 1, 360, 120, 30, 6, 1, 2520, 840, 210, 42, 7, 1, 20160, 6720, 1680, 336, 56, 8, 1, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1

Offset: 1

Johannes W. Meijer, Jul 27 2009

We get this scaled version of triangle A162990 by dividing the coefficients in the left hand columns by their 'top-values' and then taking the square root.
T(n,k) = A173333(n+1,k+1), 1 <= k <= n. - Reinhard Zumkeller, Feb 19 2010
T(n,k) = A094587(n+1,k+1), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012

			The first few rows of the triangle are:
[1]
[3, 1]
[12, 4, 1]
[60, 20, 5, 1]

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A094587.
A056542(n) equals the row sums for n>=1.
A001710, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431 are related to the left hand columns.
A000012, A009056, A002378, A007531, A052762, A052787, A053625 and A159083 are related to the right hand columns.

a162995 n k = a162995_tabl !! (n-1) !! (k-1)
a162995_row n = a162995_tabl !! (n-1)
a162995_tabl = map fst $ iterate f ([1], 3)
   where f (row, i) = (map (* i) row ++ [1], i + 1)
-- Reinhard Zumkeller, Jul 04 2012

a := proc(n, m): (n+1)!/(m+1)! end: seq(seq(a(n, m), m=1..n), n=1..9); # Johannes W. Meijer, revised Nov 23 2012

Table[(n+1)!/(m+1)!, {n, 10}, {m, n}] (* Paolo Xausa, Mar 31 2024 *)

a(n,m) = (n+1)!/(m+1)! for n = 1, 2, 3, ..., and m = 1, 2, ..., n.

A218513 Number of n-colorings of the dodecahedral graph.

Original entry on oeis.org

0, 0, 0, 7200, 168506880, 112603286160, 15108392957760, 775405390866960, 20886647215714560, 353998543659193440, 4231366997071432320, 38508081275604409920, 281586666065022616320, 1720887594454493527920, 9053942417801770507200, 41955877772610102690480

Offset: 0

Eric M. Schmidt, Oct 31 2012

N. Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993. See pp. 69-70.

Eric M. Schmidt, Table of n, a(n) for n = 0..1000
Eric W. Weisstein, Dodecahedral Graph
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Cf. A052762, A140986, A115400, A218514, A296918, A296919.

A218513(n):=n*(n-1)*(n-2)*(n^17 -27*n^16 +352*n^15 -2950*n^14 +17839*n^13 -82777*n^12 +305866*n^11 -921448*n^10 +2297495*n^9 -4783425*n^8 +8347700*n^7 -12195590*n^6 +14808795*n^5 -14713381*n^4 +11613602*n^3 -6892084*n^2 +2751604*n -555984)$
makelist(A218513(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */

def A218513(n) : return n*(n-1)*(n-2)*(n^17 -27*n^16 +352*n^15 -2950*n^14 +17839*n^13 -82777*n^12 +305866*n^11 -921448*n^10 +2297495*n^9 -4783425*n^8 +8347700*n^7 -12195590*n^6 +14808795*n^5 -14713381*n^4 +11613602*n^3 -6892084*n^2 +2751604*n -555984);

a(n) = n(n-1)(n-2)(n^17 - 27n^16 + 352n^15 - 2950n^14 + 17839n^13 - 82777n^12 + 305866n^11 - 921448n^10 + 2297495n^9 - 4783425n^8 + 8347700n^7 - 12195590n^6 + 14808795n^5 - 14713381n^4 + 11613602n^3 - 6892084n^2 + 2751604n - 555984).
See A296919 for the coefficients of the expanded form of a(n). - N. J. A. Sloane, Dec 23 2017
G.f.: -240*x^3*(2007273*x^17 +678113783*x^16 +62897280675*x^15 +2149103163405*x^14 +32571452423195*x^13 +246267894384141*x^12 +1000326687571911*x^11 +2283861589692665*x^10 +3002531231655465*x^9 +2288662487004975*x^8 +1001857651156729*x^7 +244960098705399*x^6 +31779760521705*x^5 +2006465657455*x^4 +53246253405*x^3 +454442307*x^2 +701482*x +30)/(x-1)^21. - Colin Barker, Nov 06 2012

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

cp's OEIS Frontend

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