A045991 -id:A045991 - cp's OEIS frontend

A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 2, 0, 18, 6, 3, 0, 192, 48, 12, 4, 0, 2500, 500, 100, 20, 5, 0, 38880, 6480, 1080, 180, 30, 6, 0, 705894, 100842, 14406, 2058, 294, 42, 7, 0, 14680064, 1835008, 229376, 28672, 3584, 448, 56, 8, 0, 344373768, 38263752, 4251528, 472392, 52488, 5832, 648, 72, 9

Offset: 0

Alois P. Heinz, Aug 19 2013

			T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,1) = 2: [1,2], [2,1].
T(2,2) = 2: [1,1], [2,2].
T(3,1) = 18: [1,1,2], [1,1,3], [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3], [3,3,1], [3,3,2].
T(3,2) = 6: [1,2,2], [1,3,3], [2,1,1], [2,3,3], [3,1,1], [3,2,2].
T(3,3) = 3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
  1;
  0,        1;
  0,        2,       2;
  0,       18,       6,      3;
  0,      192,      48,     12,     4;
  0,     2500,     500,    100,    20,    5;
  0,    38880,    6480,   1080,   180,   30,   6;
  0,   705894,  100842,  14406,  2058,  294,  42,  7;
  0, 14680064, 1835008, 229376, 28672, 3584, 448, 56,  8;

Alois P. Heinz, Rows n = 0..140, flattened

Row sums give: A000312.
Columns k=0-4 give: A000007, A066274(n) = 2*A081131(n) for n>1, A053506(n) for n>2, A055865(n-1) = A085389(n-1) for n>3, A085390(n-1) for n>4.
Main diagonal gives: A028310.
Lower diagonals include (offsets may differ): A002378, A045991, A085537, A085538, A085539.
Cf. A228154, A228617.

T:= (n, k)-> `if`(n=0 and k=0, 1, `if`(k<1 or k>n, 0,
             `if`(k=n, n, (n-1)*n^(n-k)))):
seq(seq(T(n,k), k=0..n), n=0..12);

f[0,0]=1;
f[n_,k_]:=Which[1<=k<=n-1,n^(n-k)*(n-1),k<1,0,k==n,n,k>n,0];
Table[Table[f[n,k],{k,0,n}],{n,0,10}]//Grid (* Geoffrey Critzer, May 19 2014 *)

T(0,0) = 1, else T(n,k) = 0 for k<1 or k>n, else T(n,n) = n, else T(n,k) = (n-1)*n^(n-k).
Sum_{k=0..n}   T(n,k) = A000312(n).
Sum_{k=0..n} k*T(n,k) = A031972(n).

A334705 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three points from an n X k grid of points which are the vertices of a triangle of nonzero area.

Original entry on oeis.org

0, 0, 4, 0, 18, 76, 0, 48, 200, 516, 0, 100, 412, 1056, 2148, 0, 180, 738, 1884, 3820, 6768, 0, 294, 1200, 3052, 6176, 10922, 17600, 0, 448, 1824, 4628, 9352, 16516, 26588, 40120, 0, 648, 2632, 6668, 13456, 23740, 38192, 57588, 82608, 0, 900, 3650, 9232, 18612, 32812, 52758, 79508, 114000, 157252

Offset: 1

N. J. A. Sloane, Jun 13 2020

It follows from the definitions that T(n,k) + A334704(n,k) = A334703(n,k) for 1 <= k <= n.

			Triangle begins:
0,
0, 4,
0, 18, 76,
0, 48, 200, 516,
0, 100, 412, 1056, 2148,
0, 180, 738, 1884, 3820, 6768,
0, 294, 1200, 3052, 6176, 10922, 17600,
0, 448, 1824, 4628, 9352, 16516, 26588, 40120,
0, 648, 2632, 6668, 13456, 23740, 38192, 57588, 82608,
0, 900, 3650, 9232, 18612, 32812, 52758, 79508, 114000, 157252,
0, 1210, 4900, 12380, 24940, 43934, 70608, 106364, 152456, 210234, 280988,
...
This is the lower half of a symmetric array. The full symmetric array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 4, 18, 48, 100, 180, 294, 448, 648, 900, 1210, 1584, ...
0, 18, 76, 200, 412, 738, 1200, 1824, 2632, 3650, 4900, 6408, ...
0, 48, 200, 516, 1056, 1884, 3052, 4628, 6668, 9232, 12380, 16176, ...
0, 100, 412, 1056, 2148, 3820, 6176, 9352, 13456, 18612, 24940, 32568, ...
0, 180, 738, 1884, 3820, 6768, 10922, 16516, 23740, 32812, 43934, 57336, ...
0, 294, 1200, 3052, 6176, 10922, 17600, 26588, 38192, 52758, 70608, 92112, ...
0, 448, 1824, 4628, 9352, 16516, 26588, 40120, 57588, 79508, 106364, 138708, ...
0, 648, 2632, 6668, 13456, 23740, 38192, 57588, 82608, 114000, 152456, 198760, ...
0, 900, 3650, 9232, 18612, 32812, 52758, 79508, 114000, 157252, 210234, 274016 , ...
0, 1210, 4900, 12380, 24940, 43934, 70608, 106364, 152456, 210234, 280988, 366152, ...
...

This is a companion to the triangles A334703 and A334704.

Rows (or columns) 2,3,4,5 of the full array are A045991, A262402, A296367, A334707. The main diagonal is A045996.

Rows 6 onwards from Tom Duff (see the Duff link in A334704). - N. J. A. Sloane, Jun 19 2020

A133070 a(n) = n^5 - n^3 - n^2.

Original entry on oeis.org

0, -1, 20, 207, 944, 2975, 7524, 16415, 32192, 58239, 98900, 159599, 246960, 368927, 534884, 755775, 1044224, 1414655, 1883412, 2468879, 3191600, 4074399, 5142500, 6423647, 7948224, 9749375, 11863124, 14328495, 17187632, 20485919, 24272100, 28598399, 33520640, 39098367, 45394964

Offset: 0

Omar E. Pol, Nov 01 2007

Exponents are prime numbers in decreasing order.

			a(7)=16415 because 7^5=16807, 7^3=343, 7^2=49 and we can write 16807-343-49=16415.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Cf. A000290, A000578, A000584, A011379, A045991, A100019.

Cf. A133071, A133072, A133073.

[n^5-n^3-n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

Table[n^5-n^3-n^2,{n,0,40}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{0,-1,20,207,944,2975},41] (* Harvey P. Dale, Jul 23 2011 *)

for(n=0,50, print1(n^5 - n^3 - n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

a(n) = n^5 - n^3 - n^2.
G.f.: x*(-1 +26*x + 72*x^2 + 22*x^3 + x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007
a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6), with a(0)=0, a(1)=-1, a(2)=20, a(3)=207, a(4)=944, a(5)=2975. - Harvey P. Dale, Jul 23 2011

More terms from Vincenzo Librandi, Dec 15 2010

A133071 a(n) = n^5 - n^3 + n^2.

Original entry on oeis.org

0, 1, 28, 225, 976, 3025, 7596, 16513, 32320, 58401, 99100, 159841, 247248, 369265, 535276, 756225, 1044736, 1415233, 1884060, 2469601, 3192400, 4075281, 5143468, 6424705, 7949376, 9750625, 11864476, 14329953, 17189200, 20487601, 24273900, 28600321, 33522688, 39100545, 45397276

Offset: 0

Omar E. Pol, Nov 01 2007

nonn

Exponents are prime numbers in decreasing order.

			a(7)=16513 because 7^5=16807, 7^3=343, 7^2=49 and we can write 16807-343+49=16513.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000290, A000578, A000584, A011379, A045991, A100019.

Cf. A133070, A133072, A133073.

[n^5-n^3+n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

Table[n^5 - n^3 + n^2, {n,0,50}] (* G. C. Greubel, Oct 20 2017 *)

for(n=0,50, print1(n^5 - n^3 + n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

a(n) = n^5 - n^3 + n^2.

G.f.: x*(1 + 22*x + 72*x^2 + 26*x^3 - x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007

More terms from Vincenzo Librandi, Dec 15 2010

A133072 a(n) = n^5 + n^3 - n^2.

Original entry on oeis.org

0, 1, 36, 261, 1072, 3225, 7956, 17101, 33216, 59697, 100900, 162261, 250416, 373321, 540372, 762525, 1052416, 1424481, 1895076, 2482597, 3207600, 4092921, 5163796, 6447981, 7975872, 9780625, 11898276, 14367861, 17231536, 20534697, 24326100, 28657981, 33586176, 39170241, 45473572

Offset: 0

Omar E. Pol, Nov 01 2007

nonn

Exponents are the prime numbers in decreasing order.

			a(7)=17101 because 7^5=16807, 7^3=343, 7^2=49 and we can write 16807+343-49=17101.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000290, A000578, A000584, A011379, A045991, A100019.

Cf. A133070, A133071, A133073.

[n^5+n^3-n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

Table[n^5 + n^3 - n^2, {n, 0, 50}] (* G. C. Greubel, Oct 20 2017 *)

for(n=0,50, print1(n^5 + n^3 - n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

a(n) = n^5 + n^3 - n^2.

G.f.: x*(1 + 30*x + 60*x^2 + 26*x^3 + 3*x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007

More terms from Vincenzo Librandi, Dec 15 2010

A133073 a(n) = n^5 + n^3 + n^2.

Original entry on oeis.org

0, 3, 44, 279, 1104, 3275, 8028, 17199, 33344, 59859, 101100, 162503, 250704, 373659, 540764, 762975, 1052928, 1425059, 1895724, 2483319, 3208400, 4093803, 5164764, 6449039, 7977024, 9781875, 11899628, 14369319, 17233104, 20536379, 24327900, 28659903, 33588224, 39172419, 45475884

Offset: 0

Omar E. Pol, Nov 01 2007

nonn

Exponents are prime numbers in decreasing order.

			a(7) = 17199 because 7^5 = 16807, 7^3 = 343, 7^2 = 49 and we can write 16807 + 343 + 49 = 17199.

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A000290, A000578, A000584, A011379, A045991, A100019.

Cf. A133070, A133071, A133072.

[n^5+n^3+n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

Total[#^{5,3,2}]&/@Range[0,40]  (* Harvey P. Dale, Jan 18 2011 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,3,44,279,1104,3275},35] (* James C. McMahon, Mar 10 2025 *)

for(n=0,50, print1(n^5 + n^3 + n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

G.f.: x*(3 + 26*x + 60*x^2 + 30*x^3 + x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007

a(n) = n^2*(n^3 + n + 1). - Wesley Ivan Hurt, Mar 02 2023

More terms from Vincenzo Librandi, Dec 15 2010

A152416 Decimal expansion of 2 - Pi^2/6.

Original entry on oeis.org

3, 5, 5, 0, 6, 5, 9, 3, 3, 1, 5, 1, 7, 7, 3, 5, 6, 3, 5, 2, 7, 5, 8, 4, 8, 3, 3, 3, 5, 3, 9, 7, 4, 8, 1, 0, 7, 8, 1, 0, 5, 0, 0, 9, 8, 7, 9, 3, 2, 0, 1, 5, 6, 2, 2, 6, 4, 4, 4, 1, 7, 7, 0, 6, 2, 9, 9, 9, 2, 5, 2, 9, 5, 9, 6, 7, 9, 9, 1, 2, 6, 1, 6, 6, 3, 7, 1, 0, 9, 9, 3, 8, 0, 2, 4, 1, 2, 9, 4, 6, 9, 5, 9, 9, 5

Offset: 0

R. J. Mathar, Dec 03 2008

Essentially the 9's complement of the digits of A013661, starting with the second. Consider the constants N(s) = Sum_{n >= 2} 1/(n^s*(n-1)) = s - Sum_{k=2..s} Zeta(k), where Zeta is Riemann's zeta function. N(1)=1 and this constant here is N(2).

The proportion of triangles formed by random lines in a plane (see Theorem 6 in Miles link). - Michel Marcus, Sep 04 2015

			Equals 0.355065933151773563527584833353974810781050098793201562264441770...

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 (2009) Section 4.1.
Mathematical Reflections, Solution to Problem U268, Issue 3, 2013, p. 17.
R. E. Miles, Random polygons determined by random lines in a plane, PNAS 1964 52 (4) 901-907.
Index entries for transcendental numbers

Cf. A013661, A045991.

Maple
```
evalf(2-Pi^2/6);
```

First@ RealDigits[N[2 - Pi^2/6, 120]] (* Michael De Vlieger, Sep 04 2015 *)

2 - Pi^2/6 \\ Michel Marcus, Jan 06 2017

Equals 2 - A013661.
Equals lim_{n->oo} (1/n^2)*Sum_{k=2..n^2-1} (fractional_part(n/sqrt(k))). See Mathematical Reflections link. - Michel Marcus, Jan 06 2017
From Amiram Eldar, Aug 09 2020: (Start)
Equals Sum_{k>=1} 1/(k*(k+1)^2) = Sum_{k>=2} 1/A045991(k).
Equals Integral_{x=0..1} log(x)*log(1-x) dx. (End)
Equals Sum_{i, j >= 1} 1/((i + 1)^2*binomial(i+j+1, i)). - Peter Bala, Aug 05 2025

A153257 a(n) = n^3 - (n+1)^2.

Original entry on oeis.org

-1, -3, -1, 11, 39, 89, 167, 279, 431, 629, 879, 1187, 1559, 2001, 2519, 3119, 3807, 4589, 5471, 6459, 7559, 8777, 10119, 11591, 13199, 14949, 16847, 18899, 21111, 23489, 26039, 28767, 31679, 34781, 38079, 41579, 45287, 49209, 53351, 57719, 62319, 67157, 72239

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 22 2008

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A045991, A069778, A081437, A140719.

Table[n^3-(n+1)^2,{n,0,40}] (* Harvey P. Dale, Oct 05 2022 *)

my(x='x+O('x^43)); Vec((x^3+5*x^2+x-1)/(x-1)^4) \\ Elmo R. Oliveira, Aug 27 2025

From Elmo R. Oliveira, Aug 27 2025: (Start)
G.f.: (-1 + x + 5*x^2 + x^3)/(1 - x)^4.
E.g.f.: (-1 + x)*(1 + 3*x + x^2)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

More terms from Elmo R. Oliveira, Aug 27 2025

A047929 a(n) = n^2(n-1)(n-2).

Original entry on oeis.org

0, 18, 96, 300, 720, 1470, 2688, 4536, 7200, 10890, 15840, 22308, 30576, 40950, 53760, 69360, 88128, 110466, 136800, 167580, 203280, 244398, 291456, 345000, 405600, 473850, 550368, 635796, 730800, 836070, 952320, 1080288, 1220736

Offset: 2

N. J. A. Sloane

There are 5 ways to put parentheses in the expression a - b - c - d: (a - (b - c)) - d, ((a - b) - c) - d, (a - b) - (c - d), a - (b - (c - d)), a - ((b - c) - d). This sequence describes how many sets of natural numbers [a,b,c,d] can be produced with the numbers {0,1,2,3,...,n} such that all the distinct expressions take different values. A045991 describes the similar process for a - b - c. For example, no sets can be produced with only 0's or only 0's and 1's; with {0,1,2,3}, 18 such sets can be produced. - Asher Auel, Jan 26 2000

For n >= 3, a(n)/6 is the number of permutations of n symbols that 3-commute with an n-cycle (see A233440 for definition). - Luis Manuel Rivera Martínez, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for sequences related to parenthesizing
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A004320, A045991, A233440.

[n^2*(n-1)*(n-2): n in [2..40]]; // Vincenzo Librandi, May 02 2011

Drop[CoefficientList[Series[6 x^3*(3 + x)/(1 - x)^5, {x, 0, 34}], x], 2] (* Michael De Vlieger, May 21 2021 *)

a(n)=n^4 - 3*n^3 + 2*n^2 \\ Charles R Greathouse IV, May 02 2011

a(n) = A004320(n-2)*6.
G.f.: 6*x^3*(3 + x)/(1 - x)^5. - Stefano Spezia, May 20 2021
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, May 22 2021
From Amiram Eldar, May 25 2021: (Start)
Sum_{n>=3} 1/a(n) = (Pi^2 - 9)/12.
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/24 + 2*log(2) - 7/4. (End)

A133280 Triangle formed by: 1 even, 2 odd, 3 even, 4 odd, ... starting with zero.

Original entry on oeis.org

0, 1, 3, 4, 6, 8, 9, 11, 13, 15, 16, 18, 20, 22, 24, 25, 27, 29, 31, 33, 35, 36, 38, 40, 42, 44, 46, 48, 49, 51, 53, 55, 57, 59, 61, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120

Offset: 0

Omar E. Pol, Aug 27 2008

This sequence is related to the Connell sequence (A001614).
First member of every row is a square (A000290).
A127366(T(n,k)) mod 2 = 0 or equal parity of T(n,k) and A000196(T(n,k)); complement of A195437. - Reinhard Zumkeller, Oct 12 2011
Written as a square array the main diagonal gives A002943. - Omar E. Pol, Aug 13 2013
Last member of every row is one less than a square (A005563). - Harvey P. Dale, Oct 02 2013

			Written as a triangle the sequence begins:
    0;
    1,   3;
    4,   6,   8;
    9,  11,  13,  15;
   16,  18,  20,  22,  24;
   25,  27,  29,  31,  33,  35;
   36,  38,  40,  42,  44,  46,  48;
   49,  51,  53,  55,  57,  59,  61,  63;
   64,  66,  68,  70,  72,  74,  76,  78,  80;
   81,  83,  85,  87,  89,  91,  93,  95,  97,  99;
  100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120;

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened

Column 1 is A000290. Right border gives A005563.
Cf. A001614.
Cf. A045991 (row sums). - R. J. Mathar, Jul 20 2009

a133280 n k = a133280_tabl !! n !! k
a133280_tabl = f 0 1 [0..] where
   f m j xs = (filter ((== m) . (`mod` 2)) ys) : f (1 - m) (j + 2) xs'
     where (ys,xs') = splitAt j xs
b133280 = bFile' "A133280" (concat $ take 101 a133280_tabl) 0
-- Reinhard Zumkeller, Oct 12 2011

Flatten[Table[Range[(n-1)^2,n^2-1,2],{n,20}]] (* Harvey P. Dale, Oct 02 2013 *)

T(n,k) = n^2 + 2*k;
for(n=0,10,for(k=0,n,print1(T(n,k),", "))); \\ Joerg Arndt, Aug 13 2013

from math import isqrt
def A133280(n): return (m:=(n<<1)+1)-((isqrt(m+1<<2)+1)>>1) # Chai Wah Wu, Aug 01 2022

a(n) = A005408(n) - A002024(n+1). - Ivan N. Ianakiev, Aug 13 2013

T(n,k) = n^2 + 2*k. - Joerg Arndt, Aug 13 2013

cp's OEIS Frontend

A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A334705 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three points from an n X k grid of points which are the vertices of a triangle of nonzero area.

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A133070 a(n) = n^5 - n^3 - n^2.

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A133071 a(n) = n^5 - n^3 + n^2.

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A133072 a(n) = n^5 + n^3 - n^2.

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Magma

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A133073 a(n) = n^5 + n^3 + n^2.

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Magma

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A152416 Decimal expansion of 2 - Pi^2/6.

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A047929 a(n) = n^2(n-1)(n-2).