A130883 -id:A130883 - cp's OEIS frontend

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

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).

A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 7, 5, 1, 4, 14, 16, 7, 1, 5, 25, 41, 29, 9, 1, 6, 41, 91, 92, 46, 11, 1, 7, 63, 182, 246, 175, 67, 13, 1, 8, 92, 336, 582, 550, 298, 92, 15, 1, 9, 129, 582, 1254, 1507, 1079, 469, 121, 17, 1, 10, 175, 957, 2508, 3718, 3367, 1925, 696, 154

Offset: 0

Philippe Deléham, Jan 24 2014

Triangle T(n,k), read by rows, given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Row sums are A111282(n+1) = A025169(n-1).
Diagonal sums are A122391(n+1) = A003945(n-1).

			Triangle begins:
  1;
  1,  1;
  2,  3,   1;
  3,  7,   5,   1;
  4, 14,  16,   7,   1;
  5, 25,  41,  29,   9,  1;
  6, 41,  91,  92,  46, 11,  1;
  7, 63, 182, 246, 175, 67, 13, 1;

Cf. Columns: A028310, A004006.
Cf. Diagonals: A000012, A005408, A130883.
Cf. Similar sequences: A078812, A085478, A111125, A128908, A165253, A207606.
Cf. A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare(1+x/(1-x)^2, 8); # Peter Luschny, Mar 06 2022

CoefficientList[#, y] & /@
CoefficientList[
Series[(1 - x + x^2)/(1 - 2*x - x*y + x^2), {x, 0, 12}], x] (* Wouter Meeussen, Jan 25 2014 *)

G.f.: (1 - x + x^2)/(1 - 2*x - x*y + x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(2,2) = 1, T(n,k) = 0 if k < 0 or k > n.
The Riordan square (see A321620) of 1 + x/(1 - x)^2. - Peter Luschny, Mar 06 2022

A381534 A084849 interleaved with positive even numbers.

Original entry on oeis.org

1, 2, 4, 4, 11, 6, 22, 8, 37, 10, 56, 12, 79, 14, 106, 16, 137, 18, 172, 20, 211, 22, 254, 24, 301, 26, 352, 28, 407, 30, 466, 32, 529, 34, 596, 36, 667, 38, 742, 40, 821, 42, 904, 44, 991, 46, 1082, 48, 1177, 50, 1276, 52

Offset: 1

Ali Sada, Feb 26 2025

To construct the sequence, we start with two 1’s on separate lines:
  1,
  1,
Next, we zigzag natural numbers between the lines, leaving spaces:
  1,,3,,5,,7,,9,,11,...
  1,2,,4,,6,,8,,10,_...
To fill the spaces, we insert the sum of the numbers in the previous column:
  1, 2, 3, 7, 5, 16, 7, 29, 9, 46, 11, 67...
  1, 2, 4, 4, 11, 6, 22, 8, 37, 10, 56,...
a(n) is the second sequence. The first sequence is A354008(k), for k > 2.
The first sequence is odd numbers interleaved with A130883. (From M. F. Hasler via Seqfan.)
The numbers we find by adding the columns are: 2,4,7,11,16,22,29,37,46,56,67,…. which is A000124 (n >= 1). The sequence is constructed by alternating the even indexed terms of this sequence (1,4,11,22,37,56…) with the numbers (added by “zigzag” to the second row before we add the columns to get the missing numbers); namely the even numbers 2*n (n >= 1). Therefore, the sequence seems to be A000124(2n)  (n>=0), interleaved with A005843(n); (n>=1). (From David James Sycamore via Seqfan.)

			A084849(0) = 1, so a(1) = 1.
a(2) is the first positive even number, 2.

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

Cf. A354008, A005843, A000124.

LinearRecurrence[{0,3,0,-3,0,1},{1,2,4,4,11,6},60] (* Harvey P. Dale, May 09 2025 *)

G.f.: -x*(-2*x^4+2*x^3-x^2-2*x-1)/(-x^6+3*x^4-3*x^2+1). - Michel Marcus Feb 27 2025

A386481 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = binomial(n,2)k^2 + n(k-1) + 1 (k >= 1, n >= 0).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 4, 1, 1, 4, 14, 16, 7, 1, 1, 5, 23, 34, 29, 11, 1, 1, 6, 34, 58, 63, 46, 16, 1, 1, 7, 47, 88, 109, 101, 67, 22, 1, 1, 8, 62, 124, 167, 176, 148, 92, 29, 1, 1, 9, 79, 166, 237, 271, 259, 204, 121, 37, 1, 1, 10, 98, 214, 319, 386, 400, 358, 269, 154, 46, 1, 1, 11, 119, 268, 413, 521, 571, 554, 473, 343, 191, 56, 1

Offset: 0

N. J. A. Sloane, Aug 11 2025

T(k,n) is the maximum number of regions the plane can be divided into by drawing n k-armed long-legged V's.

			Array begins (the rows are T(0,n>=0),, T(1,n>=0), T(2,n>=0), ...):
   1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   1, 1, 2, 4, 7, 11, 16, 22, 29, ...
   1, 2, 7, 16, 29, 46, 67, 92, 121, ...
   1, 3, 14, 34, 63, 101, 148, 204, 269, ...
   1, 4, 23, 58, 109, 176, 259, 358, 473, ...
   1, 5, 34, 88, 167, 271, 400, 554, 733, ...
   1, 6, 47, 124, 237, 386, 571, 792, 1049, ...
   1, 7, 62, 166, 319, 521, 772, 1072, 1421, ...
    ...
The first few antidiagonals are:
    1,
    1, 1,
    1, 1, 1,
    1, 2, 2, 1,
    1, 3, 7, 4, 1,
    1, 4, 14, 16, 7, 1,
    1, 5, 23, 34, 29, 11, 1,
    1, 6, 34, 58, 63, 46, 16, 1,
    1, 7, 47, 88, 109, 101, 67, 22, 1,
     ...

David O. H. Cutler and Neil J. A. Sloane, Cutting a pancake with an exotic knife, Paper in preparation, Sep 05 2025

N. J. A. Sloane, Illustration for T(3,2) = 14.
N. J. A. Sloane, Illustration for T(3,3) = 34 (a 3-armed V is also known as a Wu).

This is a companion to the array A386478.

The rows and columns include A000124, A130883, A140064, A383464, and A008865.

cp's OEIS Frontend

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.

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

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A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

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A381534 A084849 interleaved with positive even numbers.

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A386481 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = binomial(n,2)*k^2 + n*(k-1) + 1 (k >= 1, n >= 0).

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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.

A386481 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = binomial(n,2)k^2 + n(k-1) + 1 (k >= 1, n >= 0).