A007531 -id:A007531 - cp's OEIS frontend

A269409 T(n,k)=Number of length-n 0..k arrays with no repeated value greater than or equal to the previous repeated value.

2, 3, 4, 4, 9, 6, 5, 16, 24, 9, 6, 25, 60, 63, 12, 7, 36, 120, 222, 159, 16, 8, 49, 210, 570, 804, 394, 20, 9, 64, 336, 1215, 2670, 2872, 957, 25, 10, 81, 504, 2289, 6960, 12380, 10132, 2292, 30, 11, 100, 720, 3948, 15477, 39560, 56890, 35383, 5419, 36, 12, 121

Offset: 1

R. H. Hardin, Feb 25 2016

Table starts
..2.....3......4.......5........6.........7.........8..........9.........10
..4.....9.....16......25.......36........49........64.........81........100
..6....24.....60.....120......210.......336.......504........720........990
..9....63....222.....570.....1215......2289......3948.......6372.......9765
.12...159....804....2670.....6960.....15477.....30744......56124......95940
.16...394...2872...12380....39560....104006....238224.....492312.....939360
.20...957..10132...56890...223320....695135...1837752....4302612....9168780
.25..2292..35383..259445..1253190...4623815..14121282...37478718...89241015
.30..5419.122480.1175355..6995660..30625210.108123624..325487010..866361210
.36.12678.420752.5293671.38870136.202067047.825227424.2819002698.8390905692

			Some solutions for n=6 k=4
..3. .2. .1. .0. .1. .3. .1. .0. .0. .1. .0. .2. .4. .3. .0. .4
..3. .0. .0. .3. .3. .4. .4. .2. .3. .4. .4. .4. .2. .4. .4. .4
..1. .3. .1. .0. .3. .4. .2. .4. .1. .1. .0. .2. .0. .2. .2. .2
..0. .4. .4. .3. .0. .0. .1. .0. .2. .4. .2. .4. .2. .4. .1. .4
..2. .0. .0. .2. .3. .0. .2. .4. .3. .2. .0. .0. .4. .1. .1. .2
..1. .0. .1. .4. .4. .4. .3. .4. .4. .3. .2. .0. .0. .3. .2. .4

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A002620(n+2).
Column 2 is A267960.
Column 3 is A267928.
Diagonal is A268205.
Row 1 is A000027(n+1).
Row 2 is A000290(n+1).
Row 3 is A007531(n+2).

Empirical for column k:
k=1: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4)
k=2: a(n) = 6*a(n-1) -9*a(n-2) -8*a(n-3) +24*a(n-4) -16*a(n-6)
k=3: [order 8]
k=4: [order 10]
k=5: [order 12]
k=6: [order 14]
k=7: [order 16]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 2*n
n=4: a(n) = n^4 + 4*n^3 + (7/2)*n^2 + (1/2)*n
n=5: a(n) = n^5 + 5*n^4 + (11/2)*n^3 + n^2 - (1/2)*n
n=6: a(n) = n^6 + 6*n^5 + 8*n^4 + (5/3)*n^3 - n^2 + (1/3)*n
n=7: a(n) = n^7 + 7*n^6 + 11*n^5 + (8/3)*n^4 - (11/6)*n^3 + (1/3)*n^2 - (1/6)*n

A055522 Largest area of a Pythagorean triangle with n as length of one of the three sides (in fact as a leg).

Original entry on oeis.org

6, 6, 30, 24, 84, 60, 180, 120, 330, 210, 546, 336, 840, 504, 1224, 720, 1710, 990, 2310, 1320, 3036, 1716, 3900, 2184, 4914, 2730, 6090, 3360, 7440, 4080, 8976, 4896, 10710, 5814, 12654, 6840, 14820, 7980, 17220, 9240, 19866, 10626, 22770, 12144, 25944

Offset: 3

Henry Bottomley, May 22 2000

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055523, A055524, A055525, A055526, A055527.

seq(piecewise(n mod 2 = 0,n*(n^2-4)/8,n*(n^2-1)/4),n=3..60); # C. Ronaldo

Table[n*(3*(n^2 - 2) - (n^2 + 2)*(-1)^n)/16, {n, 3, 50}] (* Wesley Ivan Hurt, Apr 27 2017 *)

a(n) = n*A055523(n)/2.
a(2k) = k*(k+1)*(k-1), a(2k+1) = k*(k+1)*(2k+1).
O.g.f.: 6*x^3*(x+1+x^2)/((1-x)^4*(1+x)^4). a(2k+1)=A055112(k). a(2k)=A007531(k+1). [R. J. Mathar, Aug 06 2008]
a(n) = n*(3*(n^2-2)-(n^2+2)*(-1)^n)/16. - Luce ETIENNE, Jul 17 2015

A093074 Greatest prime factor of n and its direct neighbors.

Original entry on oeis.org

2, 3, 3, 5, 5, 7, 7, 7, 5, 11, 11, 13, 13, 13, 7, 17, 17, 19, 19, 19, 11, 23, 23, 23, 13, 13, 13, 29, 29, 31, 31, 31, 17, 17, 17, 37, 37, 37, 19, 41, 41, 43, 43, 43, 23, 47, 47, 47, 7, 17, 17, 53, 53, 53, 11, 19, 29, 59, 59, 61, 61, 61, 31, 13, 13, 67, 67, 67, 23, 71, 71, 73

Offset: 1

Reinhard Zumkeller, Mar 18 2004

nonn

a(n) = A006530(n + A093075(n));
a(n) = max{A006530(n-1), A006530(n), A006530(n+1)}, n>1;
a(n) = A006530(A007531(n+1)), n>1;
for all primes p>2: a(p)=a(p-1)=p and if p is not the lesser member of a twin prime pair, then also a(p+1)=p;
(n,n+2) is a twin prime pair iff a(n-1)=a(n)=n and a(n+1)=a(n+2)=a(n+3)=n+2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A074399, A190136.

a093074 1 = 2
a093074 n = maximum $ map a006530 [n-1..n+1]
-- Reinhard Zumkeller, Jul 04 2012

a(n)=my(p=precprime(n+1));if(p>n-2,p,vecmax(apply(n->vecmax(factor(n)[,1]),[n-1,n,n+1]))) \\ Charles R Greathouse IV, Feb 19 2013

a(n) > 47 if n > 212381. - Charles R Greathouse IV, Feb 19 2013

A128961 a(n) = (n^3 - n)*3^n.

Original entry on oeis.org

0, 54, 648, 4860, 29160, 153090, 734832, 3306744, 14171760, 58458510, 233834040, 911952756, 3482001432, 13057505370, 48212327520, 175630621680, 632270238048, 2252462723046, 7949868434280, 27824539519980, 96653663595720, 333455139405234, 1143274763675088

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).

Cf. A000244, A007531, A036289, A128796.

Cf. A128960, A128962, A128963, A128964, A128965, A128967, A128969.

[(n^3-n)*3^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013

I:=[0,54,648,4860]; [n le 4 select I[n] else 12*Self(n-1)-54*Self(n-2)+108*Self(n-3)-81*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013

LinearRecurrence[{12, -54, 108, -81}, {0, 54, 648, 4860}, 30] (* or *) CoefficientList[Series[54 x/(1 - 3 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)

G.f.: 54*x^2/(1-3*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 12*a(n-1) - 54*a(n-2) + 108*a(n-3) - 81*a(n-4). - Vincenzo Librandi, Feb 12 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000244(n).
Sum_{n>=2} 1/a(n) = (2/3)*log(3/2) - 1/4.
Sum_{n>=2} (-1)^n/a(n) = (8/3)*log(4/3) - 3/4. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128962 a(n) = (n^3 - n)*4^n.

Original entry on oeis.org

0, 96, 1536, 15360, 122880, 860160, 5505024, 33030144, 188743680, 1038090240, 5536481280, 28789702656, 146565758976, 732828794880, 3607772528640, 17523466567680, 84112639524864, 399535037743104, 1880164883496960, 8774102789652480, 40637949762600960

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256).

Cf. A000302, A007531, A036289, A128796.

Cf. A128960, A128961, A128963, A128964, A128965, A128967, A128969.

[(n^3-n)*4^n: n in [1..20]]; // Vincenzo Librandi, Feb 09 2013

CoefficientList[Series[96 x / (1-4 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 09 2013 *)
Table[(n^3-n)4^n,{n,20}] (* or *) LinearRecurrence[{16,-96,256,-256},{0,96,1536,15360},20] (* Harvey P. Dale, Dec 31 2018 *)

G.f.: 96*x^2/(1-4*x)^4. - Vincenzo Librandi, Feb 09 2013
a(n) = 16*a(n-1) - 96*a(n-2) + 256*a(n-3) - 256*a(n-4). - Vincenzo Librandi, Feb 09 2013
a(n) = 96*A038846(n-2) for n>1. - Bruno Berselli, Feb 10 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000302(n).
Sum_{n>=2} 1/a(n) = (9/8)*log(4/3) - 5/16.
Sum_{n>=2} (-1)^n/a(n) = (25/8)*log(5/4) - 11/16. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128963 a(n) = (n^3 - n)*5^n.

Original entry on oeis.org

0, 150, 3000, 37500, 375000, 3281250, 26250000, 196875000, 1406250000, 9667968750, 64453125000, 418945312500, 2666015625000, 16662597656250, 102539062500000, 622558593750000, 3735351562500000, 22178649902343750, 130462646484375000, 761032104492187500

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625).

Cf. A000351, A007531, A036289, A081143, A128796.

Cf. A128960, A128961, A128962, A128964, A128965, A128967, A128969.

[(n^3-n)*5^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013

Table[(n^3-n)5^n,{n,20}] (* or *) LinearRecurrence[{20,-150,500,-625},{0,150,3000,37500},20] (* Harvey P. Dale, Jul 22 2012 *)
CoefficientList[Series[150 x/(1 - 5 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)

a(1)=0, a(2)=150, a(3)=3000, a(4)=37500, a(n)=20*a(n-1)-150*a(n-2)+ 500*a(n-3)- 625*a(n-4). - Harvey P. Dale, Jul 22 2012
G.f.: 150*x^2/(1 - 5*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 150*A081143(n+1). - Bruno Berselli, Feb 12 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000351(n).
Sum_{n>=2} 1/a(n) = (8/5)*log(5/4) - 7/20.
Sum_{n>=2} (-1)^n/a(n) = (18/5)*log(6/5) - 13/20. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128965 a(n) = (n^3 - n)*7^n.

Original entry on oeis.org

0, 294, 8232, 144060, 2016840, 24706290, 276710448, 2905459704, 29054597040, 279650496510, 2610071300760, 23751648836916, 211605598728888, 1851548988877770, 15951806673408480, 135590356723972080, 1138958996481365472, 9467596658251350486, 77968443067952298120

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (28,-294,1372,-2401).

Cf. A000420, A007531, A036289, A128796, A140107.

Cf. A128960, A128961, A128962, A128963, A128964, A128967, A128969.

[(n^3 - n)*7^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

LinearRecurrence[{28, -294, 1372, -2401}, {0, 294, 8232, 144060}, 30] (* Vincenzo Librandi, Feb 11 2013 *)
Table[(n^3-n)7^n,{n,20}] (* Harvey P. Dale, May 14 2020 *)

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 294x^2/(1-7x)^4.
a(n) = 294*A140107(n-2). (End)
a(n) = 28*a(n-1) - 294*a(n-2) + 1372*a(n-3) - 2401*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000420(n).
Sum_{n>=2} 1/a(n) = (18/7)*log(7/6) - 11/28.
Sum_{n>=2} (-1)^n/a(n) = (32/7)*log(8/7) - 17/28. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128967 a(n) = (n^3-n)*8^n.

Original entry on oeis.org

0, 384, 12288, 245760, 3932160, 55050240, 704643072, 8455716864, 96636764160, 1063004405760, 11338713661440, 117922622078976, 1200666697531392, 12006666975313920, 118219490218475520, 1148417904979476480, 11024811887802974208, 104735712934128254976

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (32,-384,2048,-4096).

Cf. A001018, A007531, A036289, A128796, A140802.

Cf. A128960, A128961, A128962, A128963, A128964, A128965, A128969.

[(n^3 - n)*8^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

LinearRecurrence[{32, -384, 2048, -4096}, {0, 384, 12288, 245760}, 30] (* Vincenzo Librandi, Feb 11 2013 *)

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 384x^2/(1-8x)^4.
a(n) = 384*A140802(n-2). (End)
a(n) = 32*a(n-1) - 384*a(n-2) + 2048*a(n-3) - 4096*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A001018(n).
Sum_{n>=2} 1/a(n) = (49/16)*log(8/7) - 13/32.
Sum_{n>=2} (-1)^n/a(n) = (81/16)*log(9/8) - 19/32. (End)

Corrected the offset. - Mohammad K. Azarian, Nov 20 2008

A128969 a(n) = (n^3 - n)*9^n.

Original entry on oeis.org

0, 486, 17496, 393660, 7085880, 111602610, 1607077584, 21695547384, 278942752080, 3451916556990, 41422998683880, 484649084601396, 5551434969070536, 62453643402043530, 691794203838020640, 7560322370515511280, 81651481601567521824, 872650209616752889494

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (36,-486,2916,-6561).

Cf. A001019, A007531, A036289, A038291, A128796.

Cf. A128960, A128961, A128962, A128963, A128964, A128965, A128967.

[(n^3-n)*9^n: n in [0..25]]; // Vincenzo Librandi, Feb 11 2013

I:=[0, 486, 17496, 393660]; [n le 4 select I[n] else 36*Self(n-1) - 486*Self(n-2) + 2916*Self(n-3) - 6561*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

CoefficientList[Series[486 x/(1 - 9 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2013 *)

From R. J. Mathar, Dec 19 2008 (Start)
G.f.: 486x^2/(1-9x)^4.
a(n) = 486*A038291(n+1,3). (End)
a(n) = 36*a(n-1) - 486*a(n-2) + 2916*a(n-3) - 6561*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A001019(n).
Sum_{n>=2} 1/a(n) = (32/9)*log(9/8) - 5/12.
Sum_{n>=2} (-1)^n/a(n) = (50/9)*log(10/9) - 7/12. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A215228 T(n,k) = number of length-n 0..k arrays connected end-around, with no sequence of L

Original entry on oeis.org

2, 3, 2, 4, 6, 0, 5, 12, 6, 0, 6, 20, 24, 12, 0, 7, 30, 60, 72, 0, 0, 8, 42, 120, 240, 120, 18, 0, 9, 56, 210, 600, 720, 408, 0, 0, 10, 72, 336, 1260, 2520, 2940, 840, 24, 0, 11, 90, 504, 2352, 6720, 12600, 10080, 2448, 0, 0, 12, 110, 720, 4032, 15120, 40110, 57960, 38640

Offset: 1

R. H. Hardin, Aug 06 2012

Table starts
  2  3      4       5         6          7           8          9         10
  2  6     12      20        30         42          56         72         90
  0  6     24      60       120        210         336        504        720
  0 12     72     240       600       1260        2352       4032       6480
  0  0    120     720      2520       6720       15120      30240      55440
  0 18    408    2940     12600      40110      105168     240408     496080
  0  0    840   10080     57960     228480      710640    1874880    4379760
  0 24   2448   38640    280560    1338120     4883424   14783328   38962080
  0  0   5760  140400   1330560    7761600    33384960  116212320  345945600
  0  0  15960  529440   6394680   45291120   228945360  915183360 3075040080
  0 66  39864 1956900  30548760  263674950  1568401296 7203324744
  0 72 108024 7335840 146516040 1537291560 10751253072
Empirical: row n is a polynomial of degree n.
Coefficients for rows 1-10, highest power first:
   1   1
   1   1   0
   1   0  -1   0
   1   0  -1   0   0
   1   0  -5   0   4   0
   1   0  -6   5   5  -5   0
   1   0  -7   0  14   0  -8   0
   1   0  -8   0  27 -12 -20  12   0
   1   0  -9   0  27   0 -31   0  12   0
   1   0 -10   0  35   9 -60 -25  34  16   0
Row n is divisible by n.
Column k is divisible by k+1.
From Robert Israel, Nov 23 2017: (Start)
Row n is a monic polynomial of degree n.
Proof: Let b(j,n,k) be the number of such arrays taking exactly j different values.
Then T(n,k) = Sum_{j <= n} b(j,n,k). But since the j values may be any combination of 0..k taken j at a time, b(j,n,k) = binomial(k+1,j)* b(j,n,j-1) which (if nonzero) is a polynomial in k of degree j.
In particular, b(n,n,n-1) = n!, so b(n,n,k) has degree n and leading coefficient 1. (End)

			Some solutions for n=5, k=4:
  3  0  1  1  1  0  4  4  0  1  3  2  2  3  1  0
  2  4  0  3  0  4  3  2  2  2  4  0  4  4  4  1
  0  2  2  2  2  3  0  3  1  4  0  4  3  1  0  0
  3  0  3  0  3  1  3  4  4  0  3  0  0  3  4  2
  1  3  2  4  0  2  1  0  1  4  2  1  4  0  2  3

R. H. Hardin, Table of n, a(n) for n = 1..165

Column 2 is A066297.
Row 2 is A002378.
Row 3 is A007531(n+1).
Row 4 is A047928(n+1).
Row 5 is A052787(n+2).

cp's OEIS Frontend

A269409 T(n,k)=Number of length-n 0..k arrays with no repeated value greater than or equal to the previous repeated value.

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A055522 Largest area of a Pythagorean triangle with n as length of one of the three sides (in fact as a leg).

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Maple

Mathematica

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A093074 Greatest prime factor of n and its direct neighbors.

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Haskell

PARI

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A128961 a(n) = (n^3 - n)*3^n.

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Magma

Magma

Mathematica

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A128962 a(n) = (n^3 - n)*4^n.

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Magma

Mathematica

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A128963 a(n) = (n^3 - n)*5^n.

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Magma

Mathematica

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A128965 a(n) = (n^3 - n)*7^n.

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Magma

Mathematica

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A128967 a(n) = (n^3-n)*8^n.

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