A004737 -id:A004737 - cp's OEIS frontend

A004739 Concatenation of sequences (1,2,2,...,n-1,n-1,n,n,n-1,n-1,...,2,2,1) for n >= 1.

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

Offset: 1

R. Muller

From Artur Jasinski, Mar 07 2010: (Start)
Zeta(2, k/p) + Zeta(2, (p-k)/p) = (Pi/sin((Pi*a(n))/p))*2, where p=2,3,4, k=1..p-1.
This sequence is the odd subset of A003983 for odd p=3,5,7,9,....
For the even subset of A003983 see A004737. (End)
Table T(n,k) n, k > 0, T(n,k) = n-k+1, if n >= k, T(n,k) = k-n, if n < k.  Table read by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). General case A209301. Let m be a natural number. The first column of the table T(n,1) is the sequence of the natural numbers A000027. In all columns with number k (k > 1) the segment with the length of (k-1): {m+k-2, m+k-3, ..., m} shifts the sequence A000027. For m=1 the result is A004739, for m=2 the result is A004738, for m=3 the result is A209301. - Boris Putievskiy, Jan 24 2013

			From _Boris Putievskiy_, Jan 24 2013: (Start)
The start of the sequence as table:
  1, 1, 2, 3, 4, 5, 6, ...
  2, 1, 1, 2, 3, 4, 5, ...
  3, 2, 1, 1, 2, 3, 4, ...
  4, 3, 2, 1, 1, 2, 3, ...
  5, 4, 3, 2, 1, 1, 2, ...
  6, 5, 4, 3, 2, 1, 1, ...
  7, 6, 5, 4, 3, 2, 1, ...
  ...
The start of the sequence as triangle array read by rows:
  1;
  1, 1, 2;
  2, 1, 1, 2, 3;
  3, 2, 1, 1, 2, 3, 4;
  4, 3, 2, 1, 1, 2, 3, 4, 5;
  5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6;
  6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7;
  ...
Row number r contains 2*r - 1 numbers: r-1, r-2, ..., 1, 1, 2, ..., r. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
F. Smarandache, Collected Papers, Vol. II
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A004737, A004738, A004731, A003983, A079813, A187760, A004738, A209301.

a004739 n = a004739_list !! (n-1)
a004739_list = concat $ map (\n -> [1..n] ++ [n,n-1..1]) [1..]
-- Reinhard Zumkeller, Mar 26 2011

aa = {}; Do[Do[AppendTo[aa, (p/Pi) ArcSin[Sqrt[1/((1/Pi^2) (Zeta[2, k/p] + Zeta[2, (p - k)/p]))]]], {k, 1, p - 1}], {p, 3, 50, 2}]; Round[N[aa, 50]] (* Artur Jasinski, Mar 07 2010 *)

From Boris Putievskiy, Jan 24 2013: (Start)
For the general case,
a(n) = m*v + (2*v-1)*(t*t-n) + t, where t = floor(sqrt(n) - 1/2) + 1 and v = floor((n-1)/t) - t + 1.
For m=1,
a(n) = v + (2*v-1)*(t*t-n) + t, where t = floor(sqrt(n) - 1/2) + 1 and v = floor((n-1)/t) - t + 1. (End)

More terms from Patrick De Geest, Jun 15 1998

A106255 Triangle composed of triangular numbers, row sums = A006918.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 6, 3, 1, 1, 3, 6, 6, 3, 1, 1, 3, 6, 10, 6, 3, 1, 1, 3, 6, 10, 10, 6, 3, 1, 1, 3, 6, 10, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 15, 10, 6, 3, 1

Offset: 1

Gary W. Adamson, Apr 28 2005

Perform the operation Q * R; Q = infinite lower triangular matrix with 1, 2, 3, ... in each column (offset, fill in spaces with zeros). Q = upper right triangular matrix of the form:
  1, 1, 1, 1, ...
  0, 1, 1, 1, ...
  0, 0, 1, 1, ...
  0, 0, 0, 1, ...
Q * R generates an array:
  1, 1, 1,  1, ...
  1, 3, 3,  3, ...
  1, 3, 6,  6, ...
  1, 3, 6, 10, ...
  ...
... from which we take antidiagonals forming the rows of this triangle.

			From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1, 1, 1,  1,  1,  1, ...
  1, 3, 3,  3,  3,  3, ...
  1, 3, 6,  6,  6,  6, ...
  1, 3, 6, 10, 10, 10, ...
  1, 3, 6, 10, 15, 15, ...
  1, 3, 6, 10, 15, 21, ...
  1, 3, 6, 10, 15, 21, ...
  ...
(End)
Triangle rows or columns can be generated by following the triangle format:
  1;
  1, 1;
  1, 3, 1;
  1, 3, 3,  1;
  1, 3, 6,  3,  1;
  1, 3, 6,  6,  3,  1;
  1, 3, 6, 10,  6,  3,  1;
  1, 3, 6, 10, 10,  6,  3,  1;
  1, 3, 6, 10, 15, 10,  6,  3, 1;
  1, 3, 6, 10, 15, 15, 10,  6, 3, 1;
  1, 3, 6, 10, 15, 21, 15, 10, 6, 3, 1;
  ...

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A006918 (row sums, without the zero), A002260, A004736.

p[x_, n_] = Sum[x^i*If[i ==Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], 2*(i + 1), -(2*((n + 1) - i))]], {i, 0, n}]/(2*(1 - x));
Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];
Flatten[%]

From Boris Putievskiy, Jan 13 2013: (Start)
T(n,k) = min(n*(n+1)/2,k*(k+1)/2), read by antidiagonals.
a(n) = min(A002260(n)*(A002260(n)+1)/2, A004737(n)*(A004737(n)+1)/2).
a(n) = min(i*(i+1)/2, j*(j+1)/2), where
i = n-t*(t+1)/2,
j = (t*t+3*t+4)/2-n,
t = floor((-1+sqrt(8*n-7))/2). (End)

Additional comments from Roger L. Bagula and Gary W. Adamson, Apr 02 2009

A133823 Triangle whose rows are sequences of increasing and decreasing cubes:1; 1,8,1; 1,8,27,8,1; ... .

Original entry on oeis.org

1, 1, 8, 1, 1, 8, 27, 8, 1, 1, 8, 27, 64, 27, 8, 1, 1, 8, 27, 64, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 512, 343, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 512, 729

Offset: 0

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,8,1,1,8,27,8,1,..., analogous to A004737.

T(n,k) = min(n,k)^3. The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

			Triangle starts
  1;
  1, 8, 1;
  1, 8, 27, 8, 1;
  1, 8, 27, 64, 27, 8, 1;
From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1...1...
  1...8...8...8...8...8...
  1...8..27..27..27..27...
  1...8..27..64..64..64...
  1...8..27..64.125.125...
  1...8..27..64.125.216...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  1,8,1;
  1,8,27,8,1;
  1,8,27,64,27,8,1;
  1,8,27,64,125,64,27,8,1;
  1,8,27,64,125,216,125,64,27,8,1;
  . . .
Row number k contains 2*k-1 numbers 1,8,...,(k-1)^3,k^3,(k-1)^3,...,8,1. (End)

Harvey P. Dale, Table of n, a(n) for n = 0..9999
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A004737, A037270 (row sums), A133820, A124258, A133824, A003983.

Table[Join[Range[n]^3,Range[n-1,1,-1]^3],{n,10}]//Flatten (* Harvey P. Dale, May 29 2019 *)

O.g.f.: (1+qx)(1+4qx+q^2x^2)/((1-x)(1-qx)^3(1-q^2x)) = 1 + x(1 + 8q + q^2) + x^2(1 + 8q + 27q^2 + 8q^3 + q^4) + ... .
From Boris Putievskiy, Jan 13 2013: (Start)
a(n) = (A004737(n))^3.
a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^3. (End)

A133824 Triangle whose rows are sequences of increasing and decreasing fourth powers: 1; 1,16,1; 1,16,81,16,1; ... .

Original entry on oeis.org

1, 1, 16, 1, 1, 16, 81, 16, 1, 1, 16, 81, 256, 81, 16, 1, 1, 16, 81, 256, 625, 256, 81, 16, 1, 1, 16, 81, 256, 625, 1296, 625, 256, 81, 16, 1, 1, 16, 81, 256, 625, 1296, 2401, 1296, 625, 256, 81, 16, 1, 1, 16, 81, 256, 625, 1296, 2401, 4096, 2401, 1296, 625, 256, 81, 16

Offset: 0

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,16,1,1,16,81,16,1,..., analogous to A004737.
From - Boris Putievskiy, Jan 13 2013: (Start)
The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1).
Row number k contains 2*k-1 numbers 1,16,...,(k-1)^4,k^4,(k-1)^4,...,16,1. (End)

			Triangle starts:
  1;
  1, 16, 1;
  1, 16, 81, 16, 1;
  1, 16, 81, 256, 81, 16, 1;
  ...
From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1.. .1...
  1..16..16..16..16...16...
  1..16..81..81..81...81...
  1..16..81.256.256..256...
  1..16..81.256.625..625...
  1..16..81.256.625.1296...
  ...
(End)

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A004737, A061803 (row sums), A133821, A124258, A133823, A003983.

p4[n_]:=Module[{c=Range[n]^4},Join[c,Rest[Reverse[c]]]]; Flatten[p4/@ Range[10]] (* Harvey P. Dale, Dec 08 2014 *)

O.g.f.: (1+qx)(1+11qx+11q^2x^2+q^3x^3)/((1-x)(1-qx)^4(1-q^2x)) = 1 + x(1 + 16q + q^2) + x^2(1 + 16q + 81q^2 + 16q^3 + q^4) + ... . Cf. 4th row of A008292.
From Boris Putievskiy, Jan 13 2013: (Start)
T(n,k) = min(n,k)^4.
a(n) = (A004737(n))^4.
a(n) = (A124258(n))^2.
a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^4. (End)

A277949 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^4.

Original entry on oeis.org

1, 1, 4, 6, 4, 1, 1, 4, 10, 16, 19, 16, 10, 4, 1, 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1, 1, 4, 10, 20, 35, 52, 68, 80, 85, 80, 68, 52, 35, 20, 10, 4, 1, 1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1

Offset: 1

Juan Pablo Herrera P., Nov 05 2016

Sum of n-th row is n^4. The n-th row contains 4n-3 entries. Largest coefficients of each row are listed in A005900.
The n-th row is the fourth row of the n-nomial triangle. For example, row 2 (1,4,6,4,1) is the fourth row in the binomial triangle.
T(n,k) gives the number of possible ways of randomly selecting k cards from n-1 sets, each with four different playing cards. It is also the number of lattice paths from (0,0) to (4,k) using steps (1,0), (1,1), (1,2), ..., (1,n-1).

			Triangle starts:
1;
1, 4, 6, 4, 1;
1, 4, 10, 16, 19, 16, 10, 4, 1;
1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1;
1, 4, 10, 20, 35, 52, 68, 80, 85, 80, 68, 52, 35, 20, 10, 4, 1;
1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1.
...
There are T(3,2) = 10 ways to select 2 cards from two sets of four playing cards ABCD, namely, {AA}, {AB}, {AC}, {AD}, {BB}, {BC}, {BD}, {CC}, {CD}, and {DD}.

Juan Pablo Herrera P., Rows n=1..60 of the triangle, flattened

Cf. A000292, A004737, A005900, A109439, A210440, A277950, A277951.

Mentioned in: A273975

Table[CoefficientList[Series[((x^n - 1)/(x - 1))^4, {x, 0, 4 n}], x], {n, 6}] // Flatten (* Michael De Vlieger, Nov 10 2016 *)

row(n) = Vec(((1 - x^n)/(1 - x))^4);
tabf(nn) = for (n=1, nn, print(row(n)));

T(n,k) = Sum_{i=k-n+1..k} A109439(T(n,i)).
T(n,k) = A000292(k+1) = (k+3)!/(k!*6) if 0 =< k < n,
T(n,k) = ((k+3)*(k+2)*(k+1)-4*(k-n+3)*(k-n+2)*(k-n+1))/6 if n =< k < 2*n,
T(n,k) = ((4*n-1-k)*(4*n-2-k)*(4*n-3-k)-4*(3*n-1-k)*(3*n-2-k)*(3*n-3-k))/6 if 2*n-3 =< k < 3*n-3,
T(n,k) = A000292(4*n-3-k) = (4*n-1-k)!/((4*n-4-k)!*6) if 3*n-3  =< k < 4n-3.

A277950 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^5.

Original entry on oeis.org

1, 1, 5, 10, 10, 5, 1, 1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1, 1, 5, 15, 35, 65, 101, 135, 155, 155, 135, 101, 65, 35, 15, 5, 1, 1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1

Offset: 1

Juan Pablo Herrera P., Nov 05 2016

Sum of n-th row is n^5. The n-th row contains 5n-4 entries. Largest coefficients of each row are listed in A077044.
The n-th row is the fifth row of the n-nomial triangle.  For example, row 2 (1,5,10,10,5,1) is the fifth row in the binomial triangle.
T(n,k) gives the number of possible ways of randomly selecting k cards from n-1 sets, each with five different playing cards. It is also the number of lattice paths from (0,0) to (5,k) using steps (1,0), (1,1), (1,2), ..., (1,n-1).

			Triangle starts:
1;
1, 5, 10, 10, 5, 1;
1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1;
1, 5, 15, 35, 65, 101, 135, 155, 155, 135, 101, 65, 35, 15, 5, 1;
1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1;

Juan Pablo Herrera P., Rows n=1..60 of the triangle, flattened

Cf. A000332, A004737, A077044, A109439, A154286, A277949, A277951.

Mentioned in: A273975.

Table[CoefficientList[Series[((x^n - 1)/(x - 1))^5, {x, 0, 5 n}], x], {n, 10}] // Flatten

row(n) = Vec(((1 - x^n)/(1 - x))^5); tabf(nn) = for (n=1, nn, print(row(n)));

T(n,k) = Sum_{i=k-n+1..k} A277949(T(n,i)).
From Juan Pablo Herrera P., Dec 20 2016: (Start)
T(n,k) = A000332(k+4) = (k+4)!/(k!*24) if 0 =< k < n.
T(n,k) = ((k+4)!/k!-5*(k-n+4)!/(k-n)!)/24 if n =< k < 2*n.
T(n,k) = ((k+4)!/k!-5*(k-n+4)!/(k-n)!+10*(k-2*n+4)!/(k-2*n)!)/24 if 2*n =< k < 3*n.
T(n,k) = ((5*n-k-1)!/(5*n-k-5)!-5*(4*n-k-1)!/(4*n-k-5)!)/24 if 3*n-4 =< k < 4*n-4.
T(n,k) = A000332(5*n-k-1) = (5*n-k-1)!/(5*n-k-5)!*24 4*n-4 =< k < 5*n-4. (End)

A277951 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^6.

Original entry on oeis.org

1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 21, 50, 90, 126, 141, 126, 90, 50, 21, 6, 1, 1, 6, 21, 56, 120, 216, 336, 456, 546, 580, 546, 456, 336, 216, 120, 56, 21, 6, 1, 1, 6, 21, 56, 126, 246, 426, 666, 951, 1246, 1506, 1686, 1751, 1686, 1506, 1246, 951, 666, 426, 246, 126, 56, 21, 6, 1

Offset: 1

Juan Pablo Herrera P., Nov 18 2016

Sum of n-th row is n^6. The n-th row contains 6n-5 entries. Largest coefficients of each row are listed in A071816.
The n-th row is the sixth row of the n-nomial triangle.  For example, row 2 (1,6,15,20,15,6,1) is the sixth row in the binomial triangle
T(n,k) gives the number of possible ways of randomly selecting k cards from n-1 sets, each with six different playing cards. It is also the number of lattice paths from (0,0) to (6,k) using steps (1,0), (1,1), (1,2), ..., (1,n-1).

			Triangle starts:
1;
1, 6, 15, 20, 15, 6, 1;
1, 6, 21, 50, 90, 126, 141, 126, 90, 50, 21, 6, 1;
1, 6, 21, 56, 120, 216, 336, 456, 546, 580, 546, 456, 336, 216, 120, 56, 21, 6, 1.

Juan Pablo Herrera P., Rows n=1..50 of the triangle, flattened

Cf. A000332, A004737, A071816, A109439, A154286, A277949, A277950.

Mentioned in: A273975

Table[CoefficientList[Series[((x^n - 1)/(x - 1))^6, {x, 0, 6 n}], x], {n, 10}] // Flatten

row(n) = Vec(((1 - x^n)/(1 - x))^6);
tabf(nn) = for (n=1, nn, print(row(n)));

T(n,k) = Sum_{i=k-n+1..k} A277950(T(n,i))

A366130 Number of subsets of {1..n} with a subset summing to n + 1.

Original entry on oeis.org

0, 0, 1, 2, 7, 15, 38, 79, 184, 378, 823, 1682, 3552, 7208, 14948, 30154, 61698, 124302, 252125, 506521, 1022768, 2051555, 4127633, 8272147, 16607469, 33258510, 66680774, 133467385, 267349211, 535007304, 1071020315, 2142778192, 4288207796

Offset: 0

Gus Wiseman, Oct 07 2023

			The subset S = {1,2,4} has subset {1,4} with sum 4+1 and {2,4} with sum 5+1 and {1,2,4} with sum 6+1, so S is counted under a(4), a(5), and a(6).
The a(0) = 0 through a(5) = 15 subsets:
  .  .  {1,2}  {1,3}    {1,4}      {1,5}
               {1,2,3}  {2,3}      {2,4}
                        {1,2,3}    {1,2,3}
                        {1,2,4}    {1,2,4}
                        {1,3,4}    {1,2,5}
                        {2,3,4}    {1,3,5}
                        {1,2,3,4}  {1,4,5}
                                   {2,3,4}
                                   {2,4,5}
                                   {1,2,3,4}
                                   {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {2,3,4,5}
                                   {1,2,3,4,5}

For pairs summing to n + 1 we have A167762, complement A038754.
For n instead of n + 1 we have A365376, for pairs summing to n A365544.
The complement is counted by A365377 shifted.
The complement for pairs summing to n is counted by A365377.
A068911 counts subsets of {1..n} w/o two distinct elements summing to n.
A093971/A088809/A364534 count certain types of sum-full subsets.
Cf. A004526, A004737, A008967, A046663, A238628, A365381, A365541.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#],n+1]&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A366130(n):
    a = tuple(set(p.keys()) for p in partitions(n+1,k=n) if max(p.values(),default=0)==1)
    return sum(1 for k in range(2,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if any(s<=w for s in a)) # Chai Wah Wu, Nov 24 2023

Diagonal k = n + 1 of A365381.

a(20)-a(32) from Chai Wah Wu, Nov 24 2023

A051275 Expansion of (1+x^2)/((1-x^2)*(1-x^3)).

Original entry on oeis.org

1, 0, 2, 1, 2, 2, 3, 2, 4, 3, 4, 4, 5, 4, 6, 5, 6, 6, 7, 6, 8, 7, 8, 8, 9, 8, 10, 9, 10, 10, 11, 10, 12, 11, 12, 12, 13, 12, 14, 13, 14, 14, 15, 14, 16, 15, 16, 16, 17, 16, 18, 17, 18, 18, 19, 18, 20, 19, 20, 20, 21, 20, 22, 21, 22, 22, 23, 22, 24, 23, 24, 24

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 12 ).
Diagonal sums of A117567. - Paul Barry, Mar 29 2006
First differences of A156040. - Bob Selcoe, Feb 07 2014
Also first difference of diagonal sums of the triangle formed by rows T(2,k) k=0,1...,2m of ascending m-nomial triangles (see A004737). - Bob Selcoe, Feb 07 2014

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Luke James and Ben Salisbury, The weight function for monomial crystals of affine type, arXiv:1707.03159 [math.CO], 2017, p. 20 (sequence b_k).
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))
William A. Stein, The modular forms database
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Cf. A051274, A117567, A156040.

CoefficientList[Series[(1+x^2)/((1-x^2)(1-x^3)),{x,0,100}],x] (* or *) LinearRecurrence[{0,1,1,0,-1},{1,0,2,1,2},100] (* Harvey P. Dale, Dec 10 2024 *)

Vec((1+x^2)/((1-x^2)*(1-x^3))+ O(x^80)) \\ Michel Marcus, Nov 26 2019

From Paul Barry, Mar 29 2006: (Start)
a(n) = a(n-2) + a(n-3) - a(n-5);
a(n) = cos(2*Pi*n/3 + Pi/3)/3 - sqrt(3)*sin(2*Pi*n/3 + Pi/3)/9 + (-1)^n/2 + (2n+3)/6;
a(n) = Sum_{k=0..floor(n/2)} F(L((n-2k+2)/3)) where L(j/p) is the Legendre symbol of j and p. (End)
a(n) = 2*floor(n/2) + floor((n+4)/3) - n. - Ridouane Oudra, Nov 26 2019

A133825 Triangle whose rows are sequences of increasing and decreasing triangular numbers: 1; 1,3,1; 1,3,6,3,1; ... .

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 6, 3, 1, 1, 3, 6, 10, 6, 3, 1, 1, 3, 6, 10, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 28, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 28, 36, 28, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10

Offset: 0

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,3,1,1,3,6,3,1,..., analogous to A004737.

T(n,k) = min(n*(n+1)/2,k*(k+1)/2), n, k >0. The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

			Triangle starts
  1;
  1, 3, 1;
  1, 3, 6, 3, 1;
  1, 3, 6, 10, 6, 3, 1;
From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1...1...
  1...3...3...3...3...3...
  1...3...6...6...6...6...
  1...3...6..10..10..10...
  1...3...6..10..15..15...
  1...3...6..10..15..21...
  1...3...6..10..15..21...
  . . .
The start of the sequence as triangle array read by rows:
  1,
  1, 3, 1,
  1, 3, 6, 3, 1,
  1, 3, 6, 10, 6, 3, 1,
  1, 3, 6, 10, 15, 10, 6, 3, 1,
  1, 3, 6, 10, 15, 21, 15, 10, 6, 3, 1,
  1, 3, 6, 10, 15, 21, 28, 21, 15, 10, 6, 3, 1,
  . . .
Row number k contains 2*k-1 numbers 1,3,...,k*(k-1)/2,k*(k+1)/2,k*(k-1)/2,...,3,1. (End)

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A000330 (row sums), A004737, A124258, A133826, A106255.

Module[{nn=10,ac},ac=Accumulate[Range[nn]];Table[Join[Take[ ac,n],Reverse[ Take[ac,n-1]]],{n,nn}]]//Flatten (* Harvey P. Dale, Apr 18 2019 *)

O.g.f.: (1+qx)/((1-x)(1-qx)^2(1-q^2x)) = 1 + x(1 + 3q + q^2) + x^2(1 + 3q + 6q^2 + 3q^3 + q^4) + ... .
From Boris Putievskiy, Jan 13 2013: (Start)
a(n) = A004737(n)*(A004737(n)+1)/2.
a(n) = z*(z+1)/2, where z = floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1. (End)

cp's OEIS Frontend

A004739 Concatenation of sequences (1,2,2,...,n-1,n-1,n,n,n-1,n-1,...,2,2,1) for n >= 1.

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A106255 Triangle composed of triangular numbers, row sums = A006918.

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A133823 Triangle whose rows are sequences of increasing and decreasing cubes:1; 1,8,1; 1,8,27,8,1; ... .

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A133824 Triangle whose rows are sequences of increasing and decreasing fourth powers: 1; 1,16,1; 1,16,81,16,1; ... .

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A277949 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^4.

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A277950 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^5.

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A277951 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^6.

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