A118057 -id:A118057 - cp's OEIS frontend

A118061 9800n^2-5740n-4059.

1, 23661, 66921, 129781, 212241, 314301, 435961, 577221, 738081, 918541, 1118601, 1338261, 1577521, 1836381, 2114841, 2412901, 2730561, 3067821, 3424681, 3801141, 4197201, 4612861, 5048121, 5502981, 5977441, 6471501

Offset: 1

Charlie Marion, Apr 26 2006

In general, all sequences of equations which contain every positive integer in order exactly once (a pairwise equal summed, ordered partition of the positive integers) may be defined as follows: For all k, let x(k)=A001652(k) and z(k)=A001653(k). Then if we define a(n) to be (x(k)+z(k))n^2-(z(k)-1)n-x(k), the following equation is true: a(n)+(a(n)+1)+...+(a(n)+(x(k)+z(k))n+(2x(k)+z(k)-1)/2)=(a(n)+ (x(k)+z(k))n+(2x(k)+z(k)+1)/2)+...+(a(n)+2(x(k)+z(k))n+x(k)); a(n)+2(x(k)+z(k))n+x(k))=a(n+1)-1; e.g., in this sequence, x(5)=A001652(5)=4059 and z(5)=A001653(5)=5741; cf. A000290,A118057-A118060.

			a(3)= 9800*3^2-5740*3-4059=66921, a(4)=9800*4^2-5740*4-4059=129781 and 66921+66922+...+103250=103251+...+129780

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

[9800*n^2-5740*n-4059: n in [1..40]]; // Vincenzo Librandi, Jul 09 2012

CoefficientList[Series[(1+23658*x-4059*x^2)/(1-x)^3,{x,0,40}],x] (* Vincenzo Librandi, Jul 09 2012 *)

a(n)=9800*n^2-5740*n-4059 \\ Charles R Greathouse IV, May 10 2016

a(n)+(a(n)+1)+...+(a(n)+9800n+6929)=(a(n)+9800n+6930)+...+(a(n)+19600n+4059); a(n)+19600n+4059=a(n+1)-1; a(n+1)-1=a(n)+19600n+4059.
a(n)+(a(n)+1)+...+(a(n)+576n+203)=35(140n-41)(140n+29)(140n+99); e.g., 66921+66922+...+103250=3091156215=35*379*449*519.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(1+23658*x-4059*x^2)/(1-x)^3. - Colin Barker, Jul 01 2012

Corrected by T. D. Noe, Nov 13 2006

A118058 a(n) = 49n^2 - 28n - 20.

Original entry on oeis.org

1, 120, 337, 652, 1065, 1576, 2185, 2892, 3697, 4600, 5601, 6700, 7897, 9192, 10585, 12076, 13665, 15352, 17137, 19020, 21001, 23080, 25257, 27532, 29905, 32376, 34945, 37612, 40377, 43240, 46201, 49260, 52417, 55672, 59025, 62476, 66025

Offset: 1

Charlie Marion, Apr 26 2006

In general, all sequences of equations which contain every positive integer in order exactly once (a pairwise equal summed, ordered partition of the positive integers) may be defined as follows: For all k, let x(k)=A001652(k) and z(k)=A001653(k). Then if we define a(n) to be (x(k)+z(k))n^2-(z(k)-1)n-x(k), the following equation is true: a(n)+(a(n)+1)+...+(a(n)+(x(k)+z(k))n+(2x(k)+z(k)-1)/2)=(a(n)+ (x(k)+z(k))n+(2x(k)+z(k)+1)/2)+...+(a(n)+2(x(k)+z(k))n+x(k)); a(n)+2(x(k)+z(k))n+x(k))=a(n+1)-1; e.g., in this sequence, x(2)=A001652(2) and z(2)=A001653(2)=29; cf. A000290,A118057,A118059-A118061.

			a(3)=49*3^2-28*3-20=337, a(4)=49*4^2-28*4-20=652 and 337+338+...+518=519+...+651.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

[49*n^2-28*n-20: n in [1..50]]; // Vincenzo Librandi, Jul 08 2012

Table[49*n^2 - 28*n - 20, {n, 10}] (* Vincenzo Librandi, Jul 08 2012 *)

a(n)=49*n^2-28*n-20 \\ Charles R Greathouse IV, Jun 17 2017

a(n)+(a(n)+1)+...+(a(n)+98n+34)=7(7n-2)(7n+5)(14n+3)/2; e.g., 337+338+...+518=77805=7*19*26*45/2.

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(1+117*x-20*x^2)/(1-x)^3. - Colin Barker, Jun 30 2012

Corrected by Franklin T. Adams-Watters and T. D. Noe, Oct 25 2006

A118059 288n^2 - 168n - 119.

Original entry on oeis.org

1, 697, 1969, 3817, 6241, 9241, 12817, 16969, 21697, 27001, 32881, 39337, 46369, 53977, 62161, 70921, 80257, 90169, 100657, 111721, 123361, 135577, 148369, 161737, 175681, 190201, 205297, 220969, 237217, 254041, 271441, 289417, 307969

Offset: 1

Charlie Marion, Apr 26 2006

In general, all sequences of equations which contain every positive integer in order exactly once (a pairwise equal summed, ordered partition of the positive integers) may be defined as follows: For all k, let x(k)=A001652(k) and z(k)=A001653(k). Then if we define a(n) to be (x(k)+z(k))n^2-(z(k)-1)n-x(k), the following equation is true: a(n)+(a(n)+1)+...+(a(n)+(x(k)+z(k))n+(2x(k)+z(k)-1)/2)=(a(n)+(x(k)+z(k))n+(2x(k)+z(k)+1)/2)+...+(a(n)+2(x(k)+z(k))n+x(k)); a(n)+2(x(k)+z(k))n+x(k))=a(n+1)-1; e.g., in this sequence, x(3)=A001652(3)=119 and z(3)=A001653(3)=169; cf. A000290, A118057-A118058, A118060-A118061.

			a(3)=288*3^2-168*3-119=337, a(4)=288*4^2-168*4-119=3817 and 1969+1970+...+3036=3037+...+3816

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

[288*n^2 - 168*n - 119: n in [1..50]]; // Vincenzo Librandi, Jul 08 2012

Table[288*n^2 - 168*n - 119, {n, 100}] (* Vincenzo Librandi, Jul 08 2012 *)

a(n)=288*n^2-168*n-119 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(1+694*x-119*x^2)/(1-x)^3. - Colin Barker, Jul 01 2012
a(n)+(a(n)+1)+...+(a(n)+288n+203)=(a(n)+288n+204)+...+a(n+1)-1; a(n+1)-1=a(n)+576n+119.
a(n)+(a(n)+1)+...+(a(n)+288n+203)=6(24n-7)(24n+5)(24n+17); e.g., 1969+1970+...+3036=2672670=6*65*77*89.

Corrected by T. D. Noe, Nov 13 2006

A118060 a(n) = 1681n^2 - 984n - 696.

Original entry on oeis.org

1, 4060, 11481, 22264, 36409, 53916, 74785, 99016, 126609, 157564, 191881, 229560, 270601, 315004, 362769, 413896, 468385, 526236, 587449, 652024, 719961, 791260, 865921, 943944, 1025329, 1110076, 1198185, 1289656, 1384489, 1482684, 1584241

Offset: 1

Charlie Marion, Apr 26 2006

In general, all sequences of equations which contain every positive integer in order exactly once (a pairwise equal summed, ordered partition of the positive integers) may be defined as follows: For all k, let x(k)=A001652(k) and z(k)=A001653(k). Then if we define a(n) to be (x(k)+z(k))n^2-(z(k)-1)n-x(k), the following equation is true: a(n)+(a(n)+1)+...+(a(n)+(x(k)+z(k))n+(2x(k)+z(k)-1)/2)=(a(n)+ (x(k)+z(k))n+(2x(k)+z(k)+1)/2)+...+(a(n)+2(x(k)+z(k))n+x(k)); a(n)+2(x(k)+z(k))n+x(k))=a(n+1)-1; e.g., in this sequence, x(4)=A001652(4)=696 and z(4)=A001653(4)=985; cf. A000290, A118057-A118059, A118061.

			a(3)=1681*3^2-984*3-696=11481, a(4)=1681*4^2-984*4-696=22264 and 11481+11482+...+17712=17713+...+22263

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

[1681*n^2 - 984*n - 696: n in [1..40]]; // Vincenzo Librandi, Jul 09 2012

CoefficientList[Series[(1+4057*x-696*x^2)/(1-x)^3,{x,0,40}],x] (* Vincenzo Librandi, Jul 09 2012 *)
LinearRecurrence[{3,-3,1},{1,4060,11481},40] (* Harvey P. Dale, Oct 28 2016 *)

a(n)=1681*n^2-984*n-696 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(1+4057*x-696*x^2)/(1-x)^3. - Colin Barker, Jul 01 2012
a(n)+(a(n)+1)+...+(a(n)+1681n+1188) = (a(n)+1681n+1189)+ ... +a(n+1)-1; a(n+1)-1 = a(n)+3362n+696.
a(n)+(a(n)+1)+...+(a(n)+1681n+1188)=41(41n-12)(41n+29)(82n+17)/2; e.g., 11481+11482+...+17712=90965388=41*111*152*263/2.

Corrected by T. D. Noe, Nov 13 2006

A338369 Triangle read by rows: T(n,k) = (Sum_{i=0..n-k}(1+ki)^3)/(Sum_{i=0..n-k} (1+ki)) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 7, 1, 1, 10, 17, 13, 1, 1, 15, 31, 34, 21, 1, 1, 21, 49, 64, 57, 31, 1, 1, 28, 71, 103, 109, 86, 43, 1, 1, 36, 97, 151, 177, 166, 121, 57, 1, 1, 45, 127, 208, 261, 271, 235, 162, 73, 1, 1, 55, 161, 274, 361, 401, 385, 316, 209, 91, 1, 1, 66, 199, 349, 477, 556, 571, 519, 409, 262, 111, 1

Offset: 0

Werner Schulte, Nov 26 2020

Seen as a square array: (1) A(n,k) = T(n+k,k) = (k^2*n^2+k*(k+2)*n+2)/2 for n,k >= 0; (2) A(n,k) = A(n-1,k) + k*(1 + k*n) for k >= 0 and n > 0; (3) A(n,k) = A(n,k-1) + k*n*(n+1) - n*(n-1)/2 for n >= 0 and k > 0; (4) G.f. of row n >= 0: (2 + (n^2+3*n-4)*x + (n^2-n+2)*x^2) / (2*(1-x)^3).

			The triangle T(n,k) for 0 <= k <= n starts:
n \k :  0   1    2    3    4    5    6    7    8    9   10   11   12
====================================================================
   0 :  1
   1 :  1   1
   2 :  1   3    1
   3 :  1   6    7    1
   4 :  1  10   17   13    1
   5 :  1  15   31   34   21    1
   6 :  1  21   49   64   57   31    1
   7 :  1  28   71  103  109   86   43    1
   8 :  1  36   97  151  177  166  121   57    1
   9 :  1  45  127  208  261  271  235  162   73    1
  10 :  1  55  161  274  361  401  385  316  209   91    1
  11 :  1  66  199  349  477  556  571  519  409  262  111    1
  12 :  1  78  241  433  609  736  793  771  673  514  321  133    1
etc.

Cf. A000012 (column 0, main diagonal), A000217 (column 1), A056220 (column 2), A081271 (column 3), A118057 (column 4), A002061 (1st subdiagonal), A056109 (2nd subdiagonal), A085473 (3rd subdiagonal), A272039 (4th subdiagonal).

T[n_, k_] := Sum[(1 + k*i)^3, {i, 0, n - k}]/Sum[1 + k*i, {i, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 26 2020 *)

for(n=0,12,for(k=0,n,print1((k^2*(n-k)^2+k*(k+2)*(n-k)+2)/2,", "));print(" "))

T(n,k) = (k^2*(n-k)^2 + k*(k+2)*(n-k) + 2)/2 for 0 <= k <= n.
T(n,0) = T(n,n) = 1 for n >= 0.
T(n,k) = T(n-1,k-1) + k*(n-k)*(n-k+1) - (n-k)*(n-k-1)/2 for 0 < k <= n.
T(n,k) = T(n-1,k) + k * (1+k*(n-k)) for 0 <= k < n.
G.f. of column k >= 0: (1 + (k^2+k-2)*t + (1-k)*t^2) * t^k / (1-t)^3.
E.g.f.: exp(x+y)*(2 + (x^2 + 2*x - 2)*y + (x^2 - 4*x + 2)*y^2 - (2*x - 5)*y^3 + y^4)/2. - Stefano Spezia, Nov 27 2020

cp's OEIS Frontend

A118061 9800*n^2-5740*n-4059.

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A118058 a(n) = 49n^2 - 28n - 20.

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A118059 288*n^2 - 168*n - 119.

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A118060 a(n) = 1681*n^2 - 984*n - 696.

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A338369 Triangle read by rows: T(n,k) = (Sum_{i=0..n-k}(1+k*i)^3)/(Sum_{i=0..n-k} (1+k*i)) for 0 <= k <= n.

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A118061 9800n^2-5740n-4059.

A118059 288n^2 - 168n - 119.

A118060 a(n) = 1681n^2 - 984n - 696.

A338369 Triangle read by rows: T(n,k) = (Sum_{i=0..n-k}(1+ki)^3)/(Sum_{i=0..n-k} (1+ki)) for 0 <= k <= n.