A059993 -id:A059993 - cp's OEIS frontend

A323222 A(n, k) = [x^k] (1 - 4x)^(-n/2)x/(1 - x), square array read by ascending antidiagonals with n >= 0 and k >= 0.

0, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 9, 1, 0, 1, 7, 21, 29, 1, 0, 1, 9, 37, 85, 99, 1, 0, 1, 11, 57, 177, 341, 351, 1, 0, 1, 13, 81, 313, 807, 1365, 1275, 1, 0, 1, 15, 109, 501, 1593, 3579, 5461, 4707, 1, 0, 1, 17, 141, 749, 2811, 7737, 15591, 21845, 17577, 1

Offset: 0

Peter Luschny, Jan 24 2019

General asymptotic formula for g.f. (1 - 4*x)^(-j/2)*x/(1 - x) and fixed j>0 is a(n) ~ n^(j/2 - 1) * 4^n / (3*Gamma(j/2)). - Vaclav Kotesovec, Jan 29 2019

			[n\k] 0  1   2    3     4      5       6       7        8         9
-------------------------------------------------------------------
[0]   0, 1,  1,   1,    1,     1,      1,      1,       1,        1, ... A057427
[1]   0, 1,  3,   9,   29,    99,    351,   1275,    4707,    17577, ... A006134
[2]   0, 1,  5,  21,   85,   341,   1365,   5461,   21845,    87381, ... A002450
[3]   0, 1,  7,  37,  177,   807,   3579,  15591,   67071,   285861, ... A277178
[4]   0, 1,  9,  57,  313,  1593,   7737,  36409,  167481,   757305, ... A014916
[5]   0, 1, 11,  81,  501,  2811,  14823,  74883,  366603,  1752273, ... A323223
[6]   0, 1, 13, 109,  749,  4589,  26093, 140781,  730605,  3679725, ...
[7]   0, 1, 15, 141, 1065,  7071,  43107, 247311, 1355847,  7175661, ...
[8]   0, 1, 17, 177, 1457, 10417,  67761, 411825, 2377905, 13191345, ...
[9]   0, 1, 19, 217, 1933, 14803, 102319, 656587, 3982195, 23104441, ...
Triangle given by antidiagonals:
0;
0, 1;
0, 1,  1;
0, 1,  3,   1;
0, 1,  5,   9,   1;
0, 1,  7,  21,  29,    1;
0, 1,  9,  37,  85,   99,    1;
0, 1, 11,  57, 177,  341,  351,    1;
0, 1, 13,  81, 313,  807, 1365, 1275,    1;
0, 1, 15, 109, 501, 1593, 3579, 5461, 4707, 1;

Sums of antidiagonals are A323217. Main diagonal is A323219.
Rows: A057427 (n=0), A006134 (n=1), A002450 (n=2), A277178 (n=3), A014916 (n=4), A323223 (n=5).
Columns: A005408 (k=2), A059993 (k=3), A323218 (k=4).
Similar array based on Catalan numbers is A323224.

Row := proc(n, len) local ogf, ser; ogf := (1 - 4*x)^(-n/2)*x/(1 - x);
ser := series(ogf, x, (n+1)*len+1); seq(coeff(ser, x, j), j=0..len) end:
for n from 0 to 9 do Row(n, 9) od;

BF[N_, K_] := Module[{}, r[n_, k_] := FrobeniusSolve[ConstantArray[1, n], k];
X[n_] := Flatten[Table[r[N, j], {j, 0, n - 1}], 1];
CentralBinomial[n_] := Binomial[2 n, n];
Sum[Product[CentralBinomial[m[[i]]], {i, 1, N}], {m , X[K]}]];
Trow[n_] := Table[BF[n, k], {k, 0, 9}]; Table[Trow[n], {n, 1, 9}]

For n>0 and k>0 let X(n, k) denote the set of all tuples of length n with elements from {0, ..., k-1} with sum < k. Let b(m) = binomial(2*m, m). Then A(n, k) = Sum_{(j1,...,jn) in X(n, k)} b(j1)*b(j2)*...*b(jn).

A211394 T(n,k) = (k+n)(k+n-1)/2-(k+n-1)(-1)^(k+n)-k+2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 6, 2, 3, 4, 12, 13, 14, 15, 7, 8, 9, 10, 11, 23, 24, 25, 26, 27, 28, 16, 17, 18, 19, 20, 21, 22, 38, 39, 40, 41, 42, 43, 44, 45, 29, 30, 31, 32, 33, 34, 35, 36, 37, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 80

Offset: 1

Boris Putievskiy, Feb 08 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). The order of the list:
T(1,1)=1;
T(1,3), T(2,2), T(3,1);
T(1,2), T(2,1);
. . .
T(1,n), T(2,n-1), T(3,n-2), ... T(n,1);
T(1,n-1), T(2,n-3), T(3,n-4),...T(n-1,1);
. . .
First row  matches with the elements antidiagonal {T(1,n), ... T(n,1)},
second row matches with the elements antidiagonal {T(1,n-1), ... T(n-1,1)}.
Table contains:
row  1 is alternation of elements A130883 and A096376,
row  2  accommodates elements A033816 in even places,
row  3  accommodates elements A100037 in odd  places,
row  5  accommodates elements A100038 in odd  places;
column 1 is alternation of elements A084849 and  A000384,
column 2 is alternation of elements A014106 and  A014105,
column 3 is alternation of elements A014107 and  A091823,
column 4 is alternation of elements A071355 and |A168244|,
column 5 accommodates elements A033537 in even places,
column 7 is alternation of elements A100040 and  A130861,
column 9 accommodates elements A100041 in even places;
the main diagonal is A058331,
diagonal  1, located above the main diagonal is  A001844,
diagonal  2, located above the main diagonal is  A001105,
diagonal  3, located above the main diagonal is  A046092,
diagonal  4, located above the main diagonal is  A056220,
diagonal  5, located above the main diagonal is  A142463,
diagonal  6, located above the main diagonal is  A054000,
diagonal  7, located above the main diagonal is  A090288,
diagonal  9, located above the main diagonal is  A059993,
diagonal 10, located above the main diagonal is |A147973|,
diagonal 11, located above the main diagonal is  A139570;
diagonal  1, located under the main diagonal is  A051890,
diagonal  2, located under the main diagonal is  A005893,
diagonal  3, located under the main diagonal is  A097080,
diagonal  4, located under the main diagonal is  A093328,
diagonal  5, located under the main diagonal is  A137882.

			The start of the sequence as table:
  1....5...2..12...7..23..16...
  6....3..13...8..24..17..39...
  4...14...9..25..18..40..31...
  15..10..26..19..41..32..60...
  11..27..20..42..33..61..50...
  28..21..43..34..62..51..85...
  22..44..35..63..52..86..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  5,6;
  2,3,4;
  12,13,14,15;
  7,8,9,10,11;
  23,24,25,26,27,28;
  16,17,18,19,20,21,22;
  . . .
Row number r matches with r numbers segment {(r+1)*r/2-r*(-1)^(r+1)-r+2,... (r+1)*r/2-r*(-1)^(r+1)+1}.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A130883, A096376, A033816, A100037, A100038, A084849, A000384, A014106, A014105, A014107, A091823, A071355, A168244, A033537, A100040, A130861, A100041, A058331, A001844, A001105, A046092, A056220, A142463, A054000, A090288, A059993, A147973, A139570, A051890, A005893, A097080, A093328, A137882.

T[n_, k_] := (n+k)(n+k-1)/2 - (-1)^(n+k)(n+k-1) - k + 2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
j=(t*t+3*t+4)/2-n
result=(t+2)*(t+1)/2-(t+1)*(-1)**t-j+2

T(n,k) = (k+n)*(k+n-1)/2-(k+n-1)*(-1)^(k+n)-k+2.
As linear sequence
a(n) = A003057(n)*A002024(n)/2- A002024(n)*(-1)^A003056(n)-A004736(n)+2.
a(n) = (t+2)*(t+1)/2 - (t+1)*(-1)^t-j+2, where j=(t*t+3*t+4)/2-n and t=int((math.sqrt(8*n-7) - 1)/ 2).

A213197 T(n,k) = (2(n+k)^2 - 2(n+k) - 4k + 6 + (2k-2)(-1)^n + (2k-1)(-1)^k + (-2n+1)*(-1)^(n+k))/4; n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 3, 4, 2, 6, 5, 8, 9, 11, 12, 7, 15, 10, 14, 13, 17, 18, 20, 21, 23, 24, 16, 28, 19, 27, 22, 26, 25, 30, 31, 33, 34, 36, 37, 39, 40, 29, 45, 32, 44, 35, 43, 38, 42, 41, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 46, 66, 49, 65, 52, 64, 55, 63, 58, 62, 61, 68

Offset: 1

Boris Putievskiy, Mar 01 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). Let m be natural number. The order of the list:
T(1,1)=1;
T(3,1), T(2,2), T(1,3);
T(2,1), T(1,2);
...
T(1,2*m+1), T(1,2*m), T(2, 2*m-1), T(3, 2*m-1),... T(2*m,1), T(2*m+1,1);
T(2*m,2), T(2*m-2,4), ...T(2,2*m);
...
Movement along two adjacent antidiagonals. The first row consists of phases: step to the west, step to the southwest, step to the south. The second row consists of phases: 2 steps to the north, 2 steps to the east. The length of each step is 1.

			The start of the sequence as a table:
   1,  3,  2,  8,  7, 17, 16, ...
   4,  6,  9, 15, 18, 28, 31, ...
   5, 11, 10, 20, 19, 33, 32, ...
  12, 14, 21, 27, 34, 44, 51, ...
  13, 23, 22, 36, 35, 53, 52, ...
  24, 26, 37, 43, 54, 64, 75, ...
  25, 39, 38, 56, 55, 77, 76, ...
  ...
The start of the sequence as a triangular array read by rows:
   1;
   3,  4;
   2,  6,  5;
   8,  9, 11, 12;
   7, 15, 10, 14, 13;
  17, 18, 20, 21, 23, 24;
  16, 28, 19, 27, 22, 26, 25;
  ...
The start of the sequence as an array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from row 2*r-2 of the triangular array above.
Last  2*r-1 numbers are from row 2*r-1 of the triangular array above.
   1;
   3,  4,  2,  6,  5;
   8,  9, 11, 12,  7, 15, 10, 14, 13;
  17, 18, 20, 21, 23, 24, 16, 28, 19, 27, 22, 26, 25;
  ...
Row r contains permutation of 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+5, 2*r*r-5*r+6, ..., 2*r*r-2*r+2, 2*r*r-2*r+1.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A002260, A004736, A003056, A003057.

Table T(n,k) contains: in rows A130883, A033816, A100037, A000384, A100038, A014106, A091823; in columns A001844, A142463, A090288, A139570, A046092, A051890, A059993, A097080, A181510, A137882, A152813.

T:=(n,k)->(2*(n+k)^2-2*(n+k)-4*k+6+(2*k-2)*(-1)^n+(2*k-1)*(-1)^k+(1-+2*n)*(-1)^(n+k))/4: seq(seq(T(k,n-k),k=1..n-1),n=1..13); # Muniru A Asiru, Dec 06 2018

T[n_, k_] := (2(n+k)^2 - 2(n+k) - 4k + 6 + (2k-2)(-1)^n + (2k-1)(-1)^k + (-2n+1)(-1)^(n+k))/4;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=(2*(t+2)**2-2*(t+2)-4*j+6 +(2*j-2)*(-1)**i+(2*j-1)*(-1)**j+(-2*i+1)*(-1)**t)/4

As a table:
T(n,k) = (2*(n+k)^2 - 2*(n+k) - 4*k + 6 + (2*k-2)*(-1)^n + (2*k-1)*(-1)^k + (-2*n+1)*(-1)^(n+k))/4.
As a linear sequence:
a(n) = (2*A003057(n)^2 - 2*A003057(n) - 4*A004736(n) + 6 + (2*A004736(n)-2)*(-1)^A002260(n) + (2*A004736(n)-1)*(-1)^A004736(n) + (-2*A002260(n)+1)*(-1)^A003056(n))/4;
a(n) = (2*(t+2)^2 - 2*(t+2) - 4*j + 6 + (2*j-2)*(-1)^i + (2*j-1)*(-1)^j + (-2*i+1)*(-1)^t)/4, where i = n - t*(t+1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1+sqrt(8*n-7))/2).

A166873 a(n) = a(n-1) + 12*n for n > 1; a(1) = 1.

Original entry on oeis.org

1, 25, 61, 109, 169, 241, 325, 421, 529, 649, 781, 925, 1081, 1249, 1429, 1621, 1825, 2041, 2269, 2509, 2761, 3025, 3301, 3589, 3889, 4201, 4525, 4861, 5209, 5569, 5941, 6325, 6721, 7129, 7549, 7981, 8425, 8881, 9349, 9829, 10321, 10825, 11341, 11869

Offset: 1

Klaus Brockhaus, Oct 22 2009

Binomial transform of 1,24,12,0,0,0,....

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

Cf. A008594 (multiples of 12).

A000217, A028387, A133694, A059993, A166137, A166143, A166146, A166147, A166148, A166150, A166144 have recurrence a(n-1)+k*n with a(1)=1 or a(0)=1 for k = 1..11 resp.

[ n eq 1 select 1 else Self(n-1)+12*n: n in [1..44] ];

LinearRecurrence[{3,-3,1},{1,25,61},50] (* G. C. Greubel, May 27 2016 *)

a(n)=6*n^2+6*n-11 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 6*n^2 + 6*n - 11.
a(n) = 2*a(n-1) - a(n-2) + 12.
G.f.: x*(1 + 22*x - 11*x^2)/(1-x)^3.
a(n) - a(n-1) = A008594(n) for n > 1.
From G. C. Greubel, May 27 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (-11 + 12*x + 6*x^2)*exp(x) + 11. (End)

A294774 a(n) = 2n^2 + 2n + 5.

Original entry on oeis.org

5, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 269, 317, 369, 425, 485, 549, 617, 689, 765, 845, 929, 1017, 1109, 1205, 1305, 1409, 1517, 1629, 1745, 1865, 1989, 2117, 2249, 2385, 2525, 2669, 2817, 2969, 3125, 3285, 3449, 3617, 3789, 3965, 4145, 4329, 4517, 4709, 4905

Offset: 0

Bruno Berselli, Nov 08 2017

This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:

2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.
Second column of Mathar's array in A016813 (Comments section).
Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).
Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).

seq(2*n^2 + 2*n + 5, n=0..100); # Robert Israel, Nov 10 2017

Table[2n^2+2n+5,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{5,9,17},50] (* Harvey P. Dale, Sep 18 2023 *)

Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Nov 13 2017

O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.
E.g.f.: (5 + 4*x + 2*x^2)*exp(x).
a(n) = a(-1-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).
n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.
a(n) = Integral_{x=0..2n+4} |3-x| dx. - Pedro Caceres, Dec 29 2020

A199855 Inverse permutation to A210521.

Original entry on oeis.org

1, 4, 2, 5, 3, 6, 11, 7, 12, 8, 13, 9, 14, 10, 15, 22, 16, 23, 17, 24, 18, 25, 19, 26, 20, 27, 21, 28, 37, 29, 38, 30, 39, 31, 40, 32, 41, 33, 42, 34, 43, 35, 44, 36, 45, 56, 46, 57, 47, 58, 48, 59, 49, 60, 50, 61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66, 79

Offset: 1

Boris Putievskiy, Feb 04 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). The order of the list:
  T(1,1)=1;
  T(2,1), T(2,2), T(1,2), T(1,3), T(3,1),
  ...
  T(2,n-1), T(4,n-3), T(6,n-5), ..., T(n,1),
  T(2,n),   T(4,n-2), T(6,n-4), ..., T(n,2),
  T(1,n),   T(3,n-2), T(5,n-4), ..., T(n-1,2),
  T(1,n+1), T(3,n-1), T(5,n-3), ..., T(n+1,1),
  ...
The order of the list elements of adjacent antidiagonals. Let m be a positive integer.
Movement by antidiagonal {T(1,2*m), T(2*m,1)}     from T(2,2*m-1) to T(2*m,1)   length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(2,2*m)   to T(2*m,2)   length of step is 2,
movement by antidiagonal {T(1,2*m), T(2*m,1)}     from T(1,2*m)   to T(2*m-1,2) length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(1,2*m+1) to T(2*m+1,1) length of step is 2.
Table contains:
  row  1 is alternation of elements A001844 and A084849,
  row  2 is alternation of elements A130883 and A058331,
  row  3 is alternation of elements A051890 and A096376,
  row  4 is alternation of elements A033816 and A005893,
  row  6 is alternation of elements A100037 and A093328;
  row  5  accommodates elements A097080 in odd places,
  row  7  accommodates elements A137882 in odd places,
  row 10  accommodates elements A100038 in odd places,
  row 14  accommodates elements A100039 in odd places;
  column 1 is A093005 and alternation of elements A000384 and A001105,
  column 2 is alternation of elements A046092 and A014105,
  column 3 is A105638 and alternation of elements A014106 and A056220,
  column 4 is alternation of elements A142463 and A014107,
  column 5 is alternation of elements A091823 and A054000,
  column 6 is alternation of elements A090288 and |A168244|,
  column 8 is alternation of elements A059993 and A033537;
  column  7 accommodates elements  A071355  in odd  places,
  column  9 accommodates elements |A147973| in even places,
  column 10 accommodates elements  A139570  in odd  places,
  column 13 accommodates elements  A130861  in odd  places.

			The start of the sequence as table:
   1,  4,  5,  11,  13,  22,  25,  37,  41,  56,  61, ...
   2,  3,  7,   9,  16,  19,  29,  33,  46,  51,  67, ...
   6, 12, 14,  23,  26,  38,  42,  57,  62,  80,  86, ...
   8, 10, 17,  20,  30,  34,  47,  52,  68,  74,  93, ...
  15, 24, 27,  39,  43,  58,  63,  81,  87, 108, 115, ...
  18, 21, 31,  35,  48,  53,  69,  75,  94, 101. 123, ...
  28, 40, 44,  59,  64,  82,  88, 109, 116, 140, 148, ...
  32, 36, 49,  54,  70,  76,  95, 102, 124, 132, 157, ...
  45, 60, 65,  83,  89, 110, 117, 141, 149, 176, 185, ...
  50, 55, 71,  77,  96, 103, 125, 133, 158, 167, 195, ...
  66, 84, 90, 111, 118, 142, 150, 177, 186, 216, 226, ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   4,  2;
   5,  3,  6;
  11,  7, 12,  8;
  13,  9, 14, 10, 15;
  22, 16, 23, 17, 24, 18;
  25, 19, 26, 20, 27, 21, 28;
  37, 29, 38, 30, 39, 31, 40, 32;
  41, 33, 42, 34, 43, 35, 44, 36, 45;
  56, 46, 57, 47, 58, 48, 59, 49, 60, 50;
  61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66;
  ...
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of  triangle array, located above.
Last  2*r-1 numbers are from the row number 2*r-1 of  triangle array, located above.
   1;
   4, 2, 5, 3, 6;
  11, 7,12, 8,13, 9,14,10,15;
  22,16,23,17,24,18,25,19,26,20,27,21,28;
  37,29,38,30,39,31,40,32,41,33,42,34,43,35,44,36,45;
  56,46,57,47,58,48,59,49,60,50,61,51,62,52,63,53,64,54,65,55,66;
  ...
Row number r contains permutation numbers 4*r-3 from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-3*r+2,2*r*r-5*r+4, 2*r*r-3*r+3, 2*r*r-5*r+5, 2*r*r-3*r+4, 2*r*r-5*r+6, ..., 2*r*r-3*r+1, 2*r*r-r.
...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A210521, A001844, A084849, A130883, A058331, A051890, A096376, A033816, A005893, A100037, A093328, A097080, A137882, A100038, A100039, A093005, A000384, A001105, A046092, A014105, A105638, A014106, A056220, A142463, A014107, A091823, A054000, A090288, A168244, A059993, A033537, A071355, A147973, A139570, A130861.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=(2*j**2+(4*i-5)*j+2*i**2-3*i+2+(2+(-1)**j)*((1-(t+1)*(-1)**i)))/4

T(n,k) = (2*k^2+(4*n-5)*k+2*n^2-3*n+2+(2+(-1)^k)*((1-(k+n-1)*(-1)^i)))/4.

a(n) = (2*j^2+(4*i-5)*j+2*i^2-3*i+2+(2+(-1)^j)*((1-(t+1)*(-1)^i)))/4, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((sqrt(8*n-7) - 1)/2).

A221216 T(n,k) = ((n+k)^2-2(n+k)+4-(3n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 6, 4, 3, 2, 12, 13, 14, 15, 11, 10, 9, 8, 7, 23, 24, 25, 26, 27, 28, 22, 21, 20, 19, 18, 17, 16, 38, 39, 40, 41, 42, 43, 44, 45, 37, 36, 35, 34, 33, 32, 31, 30, 29, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 80

Offset: 1

Boris Putievskiy, Feb 22 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). Let m be natural number. The order of the list:
  T(1,1)=1;
  T(3,1), T(2,2), T(1,3);
  T(1,2), T(2,1);
  . . .
  T(2*m+1,1), T(2*m,2), T(2*m-1,3),...T(2,2*m), T(1,2*m+1);
  T(1,2*m), T(2,2*m-1), T(3,2*m-2),...T(2*m-1,2),T(2*m,1);
  . . .
First row  contains antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read upwards.
Second row contains antidiagonal {T(1,2*m), ... T(2*m,1)},  read downwards.

			The start of the sequence as table:
  1....5...4..12..11..23..22...
  6....3..13..10..24..21..39...
  2...14...9..25..20..40..35...
  15...8..26..19..41..34..60...
  7...27..18..42..33..61..52...
  28..17..43..32..62..51..85...
  16..44..31..63..50..86..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  5,6;
  4,3,2;
  12,13,14,15;
  11,10,9,8,7;
  23,24,25,26,27,28;
  22,21,20,19,18,17,16;
  . . .
Row number r consecutive contains r numbers.
If r is odd,  row is decreasing.
If r is even, row is increasing.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A211394, A221215, A002260, A004736, A003057;
table T(n,k) contains: in rows A084849, A096376, A014105, A014107, A168244, A033537, A100040, A100041;
in columns A130883, A000384, A014106, A033816, A100037, A091823, A071355, A100038, A100039,A130861;
main diagonal and parallel diagonals are  A058331, A001844, A005893,A046092, A093328, A142463, A090288, A059993, A051890, A001105, A097080, A056220, A137882, A054000.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-2*(t+2)+4-(3*i+j-2)*(-1)**t)/2

As table
T(n,k) = ((n+k)^2-2*(n+k)+4-(3*n+k-2)*(-1)^(n+k))/2.
As linear sequence
a(n) = (A003057(n)^2-2*A003057(n)+4-(3*A002260(n)+A004736(n)-2)*(-1)^A003056(n))/2;  a(n) = ((t+2)^2-2*(t+2)+4-(i+3*j-2)*(-1)^t)/2,
  where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A221217 T(n,k) = ((n+k)^2-2n+3-(n+k-1)(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 6, 5, 4, 3, 2, 15, 14, 13, 12, 11, 10, 9, 8, 7, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 91

Offset: 1

Boris Putievskiy, Feb 22 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). Let m be natural number. The order of the list:
  T(1,1)=1;
  T(3,1), T(2,2), T(1,3);
  T(2,1), T(1,2);
  . . .
  T(2*m+1,1), T(2*m,2), T(2*m-1,3),...T(1,2*m+1);
  T(2*m,1), T(2*m-1,2), T(2*m-2,3),...T(1,2*m);
  . . .
First row  contains antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read upwards.
Second row contains antidiagonal {T(1,2*m), ... T(2*m,1)},  read upwards.

			The start of the sequence as table:
  1....6...4..15..11..28..22...
  5....3..14..10..27..21..44...
  2...13...9..26..20..43..35...
  12...8..25..19..42..34..63...
  7...24..18..41..33..62..52...
  23..17..40..32..61..51..86...
  16..39..31..60..50..85..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  6,5;
  4,3,2;
  15,14,13,12;
  11,10,9,8,7;
  28,27,26,25,24,23;
  22,21,20,19,18,17,16;
  . . .
Row number r consecutive contains r numbers in decreasing order.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A211394, A221215, A002260, A004736, A003057, A002024;
table T(n,k) contains: in rows A084849, A000384, A014106, A014105, A014107, A091823, A071355, A091823, A071355,  A100040, A130861, A100041;
in columns A130883, A096376, A033816, A100037, A100038, A100039;
main diagonal and parallel diagonals are A058331, A051890, A005893, A097080, A093328, A137882, A001844, A001105, A056220, A142463, A054000, A090288, A059993.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-2*i+3-(t+1)*(1+2*(-1)**t))/2

As table
T(n,k) = ((n+k)^2-2*n+3-(n+k-1)*(1+2*(-1)^(n+k)))/2.
As linear sequence
a(n) = (A003057(n)^2-2*A002260(n)+3-A002024(n)*(1+2*(-1)^A003056(n)))/2;
a(n) = ((t+2)^2-2*i+3-(t+1)*(1+2*(-1)**t))/2, where i=n-t*(t+1)/2,
  j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A336858 Triangle read by rows: T(n,k) = T(n,k-1) + T(n-1, k) + T(n-1,k-1) with T(n,0) = T(n, n) = 1 (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 9, 1, 1, 7, 21, 31, 1, 1, 9, 37, 89, 121, 1, 1, 11, 57, 183, 393, 515, 1, 1, 13, 81, 321, 897, 1805, 2321, 1, 1, 15, 109, 511, 1729, 4431, 8557, 10879, 1, 1, 17, 141, 761, 3001, 9161, 22149, 41585, 52465, 1, 1, 19, 177, 1079, 4841, 17003, 48313, 112047, 206097, 258563, 1

Offset: 0

Petros Hadjicostas, Aug 05 2020

This is J. M. Bergot's triangular array described in A104858 with the top vertex of the triangle shifted from (1,1) to (0,0).

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   9,   1;
  1,  7,  21,  31,    1;
  1,  9,  37,  89,  121,    1;
  1, 11,  57, 183,  393,  515,    1;
  1, 13,  81, 321,  897, 1805, 2321,     1;
  1, 15, 109, 511, 1729, 4431, 8557, 10879, 1;
  ...

Cf. A001003, A005408, A006318, A008288, A035011, A059993, A086616, A104858, A333090.

A336858row := proc(n) option remember; local T, k, row;
row := Array(0..n, fill=1);
if n = 0 then return row fi; T := procname(n-1);
for k from 1 to n-1 do row[k] := T[k] + T[k-1] + row[k-1] od; row end:
T := (n, k) -> A336858row(n)[k]:
seq(print(seq(T(n, k), k=0..n)), n=0..8); # Peter Luschny, Aug 06 2020

T[, 0] = 1; T[n, n_] = 1;
T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, k] + T[n-1, k-1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023 *)

T(n,k) = T(n, k-1) + T(n-1, k) + T(n-1, k-1) for 1 <= k <= n-1 with T(n,0) = 1 = T(n,n) for n >= 0.
T(n,k) = D(n,k) - Sum_{m=1..k} b(m-1)*D(n-m, k-m) - Sum_{m=0..k-1} D(n-m, k-m-1), where D(n,k) = A008288(n,k) (square array of Delannoy numbers) and b(n) = A086616(n).
T(n,1) = A005408(n-1) = 2*n - 1 for n >= 1.
T(n,2) = A059993(n-2) = 2*n^2 - 2*n - 3 for n >= 2.
T(n,n-1) = A086616(n-1) for n >= 1.
T(n,n-2) = A035011(n-1) = A006318(n-1) - 1 for n >= 2.
Sum_{k=0..n} T(n,k) = A104858(n) for n >= 0.
Bivariate o.g.f.: (1 - y - x*y*(1 + g(x*y)))/((1 - x*y)*(1 - x - y - x*y)), where g(w) = 2/(1 - w + sqrt(1 - 6*w + w^2)) = o.g.f. of A006318 (large Schroeder numbers).
Bivariate o.g.f.: (1 - y - 2*x*y*q(x*y))/((1 - x*y)*(1 - x - y - x*y)), where q(w) = 2/(1 + w + sqrt(1 - 6*w + w^2)) = o.g.f. of A001003 (little Schroeder numbers).
T(2*n,n) = A333090(n). - Peter Luschny, Aug 06 2020

A374584 Numbers k such that 7*k + 2 is a square.

Original entry on oeis.org

1, 2, 14, 17, 41, 46, 82, 89, 137, 146, 206, 217, 289, 302, 386, 401, 497, 514, 622, 641, 761, 782, 914, 937, 1081, 1106, 1262, 1289, 1457, 1486, 1666, 1697, 1889, 1922, 2126, 2161, 2377, 2414, 2642, 2681, 2921, 2962, 3214, 3257, 3521, 3566, 3842, 3889

Offset: 1

Juri-Stepan Gerasimov, Aug 12 2024

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

The numbers k such that (m + (9-m)*k) is a square: A000217 (m = 1), this sequence (m = 2), A003154 (m = 3), A195162 (m = 4), A028387 (m = 5), A100536 (m = 6), A059993 (m = 7), A028884 (m = 8).

Cf. A047341.

[k: k in [0..4000] | IsSquare(7*k + 2)];

((Table[7*n + {3, 4}, {n, 0, 23}] // Flatten)^2 - 2)/7 (* Amiram Eldar, Aug 12 2024 *)

a(n) = (A047341(n)^2 - 2)/7. - Amiram Eldar, Aug 12 2024

cp's OEIS Frontend

A323222 A(n, k) = [x^k] (1 - 4*x)^(-n/2)*x/(1 - x), square array read by ascending antidiagonals with n >= 0 and k >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A211394 T(n,k) = (k+n)*(k+n-1)/2-(k+n-1)*(-1)^(k+n)-k+2; n , k > 0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A213197 T(n,k) = (2*(n+k)^2 - 2*(n+k) - 4*k + 6 + (2*k-2)*(-1)^n + (2*k-1)*(-1)^k + (-2*n+1)*(-1)^(n+k))/4; n, k > 0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A166873 a(n) = a(n-1) + 12*n for n > 1; a(1) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A294774 a(n) = 2*n^2 + 2*n + 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A199855 Inverse permutation to A210521.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

A221216 T(n,k) = ((n+k)^2-2*(n+k)+4-(3*n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A323222 A(n, k) = [x^k] (1 - 4x)^(-n/2)x/(1 - x), square array read by ascending antidiagonals with n >= 0 and k >= 0.

A211394 T(n,k) = (k+n)(k+n-1)/2-(k+n-1)(-1)^(k+n)-k+2; n , k > 0, read by antidiagonals.

A213197 T(n,k) = (2(n+k)^2 - 2(n+k) - 4k + 6 + (2k-2)(-1)^n + (2k-1)(-1)^k + (-2n+1)*(-1)^(n+k))/4; n, k > 0, read by antidiagonals.

A294774 a(n) = 2n^2 + 2n + 5.

A221216 T(n,k) = ((n+k)^2-2(n+k)+4-(3n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

A221217 T(n,k) = ((n+k)^2-2n+3-(n+k-1)(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.