A056023 -id:A056023 - cp's OEIS frontend

A373663 a(n) = (1 + (n+2)^2 + (n-3)*(-1)^n)/2.

6, 8, 13, 19, 24, 34, 39, 53, 58, 76, 81, 103, 108, 134, 139, 169, 174, 208, 213, 251, 256, 298, 303, 349, 354, 404, 409, 463, 468, 526, 531, 593, 598, 664, 669, 739, 744, 818, 823, 901, 906, 988, 993, 1079, 1084, 1174, 1179, 1273, 1278, 1376, 1381, 1483, 1488, 1594

Offset: 1

Wesley Ivan Hurt, Jun 12 2024

Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 3 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=3.

			       [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
  [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
  [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
  [ 3]   6    8   13   19   24   34   39   53   58   76   ...
  [ 4]   7   14   18   25   33   40   52   59   75   ...
  [ 5]  15   17   26   32   41   51   60   74   ...
  [ 6]  16   27   31   42   50   61   73   ...
  [ 7]  28   30   43   49   62   72   ...
  [ 8]  29   44   48   63   71   ...
  [ 9]  45   47   64   70   ...
  [10]  46   65   69   ...
  [11]  66   68   ...
  [12]  67   ...
        ...

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

For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), this sequence (k=3), A374004 (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).

Row 3 of the example in A056011, Column 3 of the rectangular array in A056023.

[(1 + (n+2)^2 + (n-3)*(-1)^n)/2: n in [1..80]];

k := 3; Table[(1 + (n+k-1)^2 + (n-k) (-1)^(n+k-1))/2, {n, 80}]

def A373663(n): return ((n+1)*(n+2)+6 if n&1 else (n+2)*(n+3)-4)>>1 # Chai Wah Wu, Jun 23 2024

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = A373662(n+1) - (-1)^n.
G.f.: -x*(x^4+2*x^3-7*x^2+2*x+6)/((x+1)^2*(x-1)^3).

A383419 a(n) = A378762(A381968(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, May 01 2025

This sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is 4n-3 = A016813(n-1), n > 0.
The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.
A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A383590 (the inverse permutation), A381968, A381662, A382499, A380817, A382679, A376214, A382680, A378762, A383589, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy link. - Boris Putievskiy, Jun 02 2025

			Triangle array begins:
  k=    1   2   3  4   5  6   7  8   9
  n=1:  1;
  n=2:  5,  3,  6, 2,  4;
  n=3: 12, 10, 14, 8, 15, 9, 13, 7, 11;
(1, 5, 3, ..., 7, 11) = (1, 2, 3, ..., 12, 11) (1, 5, 3, ..., 9, 15). The first permutation on the right-hand side is from Example A378762 and the second from Example A381968.
Ord(1, 5, 3, ..., 7, 11) = 4.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  4,  8, 11, ...
   5,  2, 14,  7, 27, ...
   6, 10, 13, 19, 24, ...
  12,  9, 25, 18, 42, ...
  15, 21, 26, 34, 41, ...
  ...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
  1;
  4, 2, 5, 1, 3;
  6, 4, 8, 2, 9, 3, 7, 1, 5.

Boris Putievskiy, Table of n, a(n) for n = 1..9730
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table.
Groupprops, Subgroup structure of direct product of D8 and Z2.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

ord(a(1), a(2), ..., a(A000384(n+1))) = 4, where ord is the order of the permutation.

T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = m + k if k < m and k == 1 (mod 2), P(n, k) = m + 1 - k if k < m and k == 0 (mod 2), P(n, k) = 3m - 1 - k if k >= m and k == 1 (mod 2), P(n, k) = 2m - 1 - k if k >= m and k == 0 (mod 2), where m = 2n - 1.

A383590 a(n) = A378762(A382499(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, May 01 2025

This sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is 4n-3 = A016813(n-1), n > 0.
The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.
A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A383419 (the inverse permutation), A381968, A381662, A382499, A380817, A382679, A376214, A382680, A378762, A383589, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy link. - Boris Putievskiy, Jun 02 2025

			Triangle array begins:
  k=    1   2   3  4   5  6   7  8   9
  n=1:  1;
  n=2:  5,  3,  6, 2,  4;
  n=3: 14, 10, 12, 8, 15, 7, 13, 9, 11;
(1, 5, 3, ..., 9, 11) = (1, 2, 3, ..., 12, 11) (1, 5, 3, ..., 7, 15). The first permutation on the right-hand side is from Example A378762 and the second from Example A382499.
Ord(1, 5, 3, ... , 9, 11) = 4.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  4,  8, 11, ...
   5,  2, 12,  9, 23, ...
   6, 10, 13, 19, 24, ...
  14,  7, 25, 18, 40, ...
  15, 21, 26, 34, 41, ...
  ...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
  1;
  4, 2, 5, 1, 3;
  8, 4, 6, 2, 9, 1, 7, 3, 5.

Boris Putievskiy, Table of n, a(n) for n = 1..9730
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table.
Groupprops, Subgroup structure of direct product of D8 and Z2.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680, A383419, A383589, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

ord(a(1), a(2), ..., a(A000384(n+1))) = 4, where ord is the order of the permutation.

T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = 2m - 1 - k if k < m and k == 1 (mod 2), P(n, k) = m + 1 - k if k < m and k == 0 (mod 2), P(n, k) = 3m - 1 - k if k >= m and k == 1 (mod 2), P(n, k) = - m + k if k >= m and k == 0 (mod 2), where m = 2n - 1.

A374004 a(n) = (1 + (n+3)^2 - (n-4)*(-1)^n)/2.

Original entry on oeis.org

7, 14, 18, 25, 33, 40, 52, 59, 75, 82, 102, 109, 133, 140, 168, 175, 207, 214, 250, 257, 297, 304, 348, 355, 403, 410, 462, 469, 525, 532, 592, 599, 663, 670, 738, 745, 817, 824, 900, 907, 987, 994, 1078, 1085, 1173, 1180, 1272, 1279, 1375, 1382, 1482, 1489, 1593

Offset: 1

Wesley Ivan Hurt, Jun 24 2024

Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 4 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=4.

			       [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
  [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
  [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
  [ 3]   6    8   13   19   24   34   39   53   58   76   ...
  [ 4]   7   14   18   25   33   40   52   59   75   ...
  [ 5]  15   17   26   32   41   51   60   74   ...
  [ 6]  16   27   31   42   50   61   73   ...
  [ 7]  28   30   43   49   62   72   ...
  [ 8]  29   44   48   63   71   ...
  [ 9]  45   47   64   70   ...
  [10]  46   65   69   ...
  [11]  66   68   ...
  [12]  67   ...
        ...

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

For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), A373663 (k=3), this sequence (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).
Row 4 of the table in A056011.
Column 4 of the rectangular array in A056023.

[(1 + (n+3)^2 - (n-4)*(-1)^n)/2: n in [1..80]];

CoefficientList[Series[-(7*x^4 - 7*x^3 - 10*x^2 + 7 x + 7)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
k := 4; Table[(1 + (n+k-1)^2 + (n-k) (-1)^(n+k-1))/2, {n, 80}]

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(7*x^4-7*x^3-10*x^2+7x+7)/((x+1)^2*(x-1)^3).
a(n) = A373663(n+1) + (-1)^n.

A374005 a(n) = (1 + (n+4)^2 + (n-5)*(-1)^n)/2.

Original entry on oeis.org

15, 17, 26, 32, 41, 51, 60, 74, 83, 101, 110, 132, 141, 167, 176, 206, 215, 249, 258, 296, 305, 347, 356, 402, 411, 461, 470, 524, 533, 591, 600, 662, 671, 737, 746, 816, 825, 899, 908, 986, 995, 1077, 1086, 1172, 1181, 1271, 1280, 1374, 1383, 1481, 1490, 1592, 1601

Offset: 1

Wesley Ivan Hurt, Jun 24 2024

Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 5 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=5.

			       [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
  [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
  [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
  [ 3]   6    8   13   19   24   34   39   53   58   76   ...
  [ 4]   7   14   18   25   33   40   52   59   75   ...
  [ 5]  15   17   26   32   41   51   60   74   ...
  [ 6]  16   27   31   42   50   61   73   ...
  [ 7]  28   30   43   49   62   72   ...
  [ 8]  29   44   48   63   71   ...
  [ 9]  45   47   64   70   ...
  [10]  46   65   69   ...
  [11]  66   68   ...
  [12]  67   ...
        ...

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

For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), A373663 (k=3), A374004 (k=4), this sequence (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).
Row 5 of the table in A056011.
Column 5 of the rectangular array in A056023.

[(1 + (n+4)^2 + (n-5)*(-1)^n)/2: n in [1..80]];

CoefficientList[Series[-(6*x^4 + 2*x^3 - 21*x^2 + 2*x + 15)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
k := 5; Table[(1 + (n + k - 1)^2 + (n - k) (-1)^(n + k - 1))/2, {n, 80}]

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(6*x^4+2*x^3-21*x^2+2*x+15)/((x+1)^2*(x-1)^3).
a(n) = A374004(n+1) - (-1)^n.

A374007 a(n) = (1 + (n+5)^2 - (n-6)*(-1)^n)/2.

Original entry on oeis.org

16, 27, 31, 42, 50, 61, 73, 84, 100, 111, 131, 142, 166, 177, 205, 216, 248, 259, 295, 306, 346, 357, 401, 412, 460, 471, 523, 534, 590, 601, 661, 672, 736, 747, 815, 826, 898, 909, 985, 996, 1076, 1087, 1171, 1182, 1270, 1281, 1373, 1384, 1480, 1491, 1591, 1602

Offset: 1

Wesley Ivan Hurt, Jun 24 2024

Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 6 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=6.

			       [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
  [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
  [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
  [ 3]   6    8   13   19   24   34   39   53   58   76   ...
  [ 4]   7   14   18   25   33   40   52   59   75   ...
  [ 5]  15   17   26   32   41   51   60   74   ...
  [ 6]  16   27   31   42   50   61   73   ...
  [ 7]  28   30   43   49   62   72   ...
  [ 8]  29   44   48   63   71   ...
  [ 9]  45   47   64   70   ...
  [10]  46   65   69   ...
  [11]  66   68   ...
  [12]  67   ...
        ...

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

For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), A373663 (k=3), A374004 (k=4), A374005 (k=5), this sequence (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).
Row 6 of the table in A056011.
Column 6 of the rectangular array in A056023.

[(1 + (n+5)^2 - (n-6)*(-1)^n)/2: n in [1..80]];

CoefficientList[Series[-(16*x^4 - 11*x^3 - 28*x^2 + 11*x + 16)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
k := 6; Table[(1 + (n + k - 1)^2 + (n - k) (-1)^(n + k - 1))/2, {n, 80}]

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(16*x^4-11*x^3-28*x^2+11*x+16)/((x+1)^2*(x-1)^3).
a(n) = A374005(n+1) + (-1)^n.

A374008 a(n) = (1 + (n+6)^2 + (n-7)*(-1)^n)/2.

Original entry on oeis.org

28, 30, 43, 49, 62, 72, 85, 99, 112, 130, 143, 165, 178, 204, 217, 247, 260, 294, 307, 345, 358, 400, 413, 459, 472, 522, 535, 589, 602, 660, 673, 735, 748, 814, 827, 897, 910, 984, 997, 1075, 1088, 1170, 1183, 1269, 1282, 1372, 1385, 1479, 1492, 1590, 1603, 1705

Offset: 1

Wesley Ivan Hurt, Jun 24 2024

Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 7 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=7.

			       [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
  [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
  [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
  [ 3]   6    8   13   19   24   34   39   53   58   76   ...
  [ 4]   7   14   18   25   33   40   52   59   75   ...
  [ 5]  15   17   26   32   41   51   60   74   ...
  [ 6]  16   27   31   42   50   61   73   ...
  [ 7]  28   30   43   49   62   72   ...
  [ 8]  29   44   48   63   71   ...
  [ 9]  45   47   64   70   ...
  [10]  46   65   69   ...
  [11]  66   68   ...
  [12]  67   ...
        ...

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

For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), A373663 (k=3), A374004 (k=4), A374005 (k=5), A374007 (k=6), this sequence (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).
Row 7 of the table in A056011.
Column 7 of the rectangular array in A056023.

[(1 + (n+6)^2 + (n-7)*(-1)^n)/2: n in [1..80]];

CoefficientList[Series[-(15*x^4 + 2*x^3 - 43*x^2 + 2 x + 28)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
k := 7; Table[(1 + (n + k - 1)^2 + (n - k) (-1)^(n + k - 1))/2, {n, 80}]

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(15*x^4+2*x^3-43*x^2+2x+28)/((x+1)^2*(x-1)^3).
a(n) = A374007(n+1) - (-1)^n.

A374009 a(n) = (1 + (n+7)^2 - (n-8)*(-1)^n)/2.

Original entry on oeis.org

29, 44, 48, 63, 71, 86, 98, 113, 129, 144, 164, 179, 203, 218, 246, 261, 293, 308, 344, 359, 399, 414, 458, 473, 521, 536, 588, 603, 659, 674, 734, 749, 813, 828, 896, 911, 983, 998, 1074, 1089, 1169, 1184, 1268, 1283, 1371, 1386, 1478, 1493, 1589, 1604, 1704, 1719

Offset: 1

Wesley Ivan Hurt, Jun 24 2024

Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 8 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=8.

			       [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
  [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
  [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
  [ 3]   6    8   13   19   24   34   39   53   58   76   ...
  [ 4]   7   14   18   25   33   40   52   59   75   ...
  [ 5]  15   17   26   32   41   51   60   74   ...
  [ 6]  16   27   31   42   50   61   73   ...
  [ 7]  28   30   43   49   62   72   ...
  [ 8]  29   44   48   63   71   ...
  [ 9]  45   47   64   70   ...
  [10]  46   65   69   ...
  [11]  66   68   ...
  [12]  67   ...
        ...

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

For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), A373663 (k=3), A374004 (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), this sequence (k=8), A374010 (k=9), A374011 (k=10).
Row 8 of the table in A056011.
Column 8 of the rectangular array in A056023.

[(1 + (n+7)^2 - (n-8)*(-1)^n)/2: n in [1..80]];

CoefficientList[Series[-(29*x^4 - 15*x^3 - 54*x^2 + 15*x + 29)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
k := 8; Table[(1 + (n + k - 1)^2 + (n - k) (-1)^(n + k - 1))/2, {n, 80}]

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(29*x^4-15*x^3-54*x^2+15*x+29)/((x+1)^2*(x-1)^3).
a(n) = A374008(n+1) + (-1)^n.

A374010 a(n) = (1 + (n+8)^2 + (n-9)*(-1)^n)/2.

Original entry on oeis.org

45, 47, 64, 70, 87, 97, 114, 128, 145, 163, 180, 202, 219, 245, 262, 292, 309, 343, 360, 398, 415, 457, 474, 520, 537, 587, 604, 658, 675, 733, 750, 812, 829, 895, 912, 982, 999, 1073, 1090, 1168, 1185, 1267, 1284, 1370, 1387, 1477, 1494, 1588, 1605, 1703, 1720, 1822

Offset: 1

Wesley Ivan Hurt, Jun 24 2024

Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 9 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=9.

			       [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
  [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
  [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
  [ 3]   6    8   13   19   24   34   39   53   58   76   ...
  [ 4]   7   14   18   25   33   40   52   59   75   ...
  [ 5]  15   17   26   32   41   51   60   74   ...
  [ 6]  16   27   31   42   50   61   73   ...
  [ 7]  28   30   43   49   62   72   ...
  [ 8]  29   44   48   63   71   ...
  [ 9]  45   47   64   70   ...
  [10]  46   65   69   ...
  [11]  66   68   ...
  [12]  67   ...
        ...

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

For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), A373663 (k=3), A374004 (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), this sequence (k=9), A374011 (k=10).
Row 9 of the table in A056011.
Column 9 of the rectangular array in A056023.

[(1 + (n+8)^2 + (n-9)*(-1)^n)/2: n in [1..80]];

CoefficientList[Series[-(28*x^4 + 2*x^3 - 73*x^2 + 2*x + 45)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
k := 9; Table[(1 + (n + k - 1)^2 + (n - k) (-1)^(n + k - 1))/2, {n, 80}]

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(28*x^4+2*x^3-73*x^2+2*x+45)/((x+1)^2*(x-1)^3).
a(n) = A374009(n+1) - (-1)^n.

A374011 a(n) = (1 + (n+9)^2 - (n-10)*(-1)^n)/2.

Original entry on oeis.org

46, 65, 69, 88, 96, 115, 127, 146, 162, 181, 201, 220, 244, 263, 291, 310, 342, 361, 397, 416, 456, 475, 519, 538, 586, 605, 657, 676, 732, 751, 811, 830, 894, 913, 981, 1000, 1072, 1091, 1167, 1186, 1266, 1285, 1369, 1388, 1476, 1495, 1587, 1606, 1702, 1721, 1821

Offset: 1

Wesley Ivan Hurt, Jun 24 2024

Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 10 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=10.

			       [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
  [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
  [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
  [ 3]   6    8   13   19   24   34   39   53   58   76   ...
  [ 4]   7   14   18   25   33   40   52   59   75   ...
  [ 5]  15   17   26   32   41   51   60   74   ...
  [ 6]  16   27   31   42   50   61   73   ...
  [ 7]  28   30   43   49   62   72   ...
  [ 8]  29   44   48   63   71   ...
  [ 9]  45   47   64   70   ...
  [10]  46   65   69   ...
  [11]  66   68   ...
  [12]  67   ...
        ...

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

For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), A373663 (k=3), A374004 (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), this sequence (k=10).
Row 10 of the table in A056011.
Column 10 of the rectangular array in A056023.

[(1 + (n+9)^2 - (n-10)*(-1)^n)/2: n in [1..80]];

CoefficientList[Series[-(46*x^4 - 19*x^3 - 88*x^2 + 19*x + 46)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
k := 10; Table[(1 + (n + k - 1)^2 + (n - k) (-1)^(n + k - 1))/2, {n, 80}]

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(46*x^4-19*x^3-88*x^2+19*x+46)/((x+1)^2*(x-1)^3).
a(n) = A374010(n+1) + (-1)^n.

cp's OEIS Frontend

A373663 a(n) = (1 + (n+2)^2 + (n-3)*(-1)^n)/2.

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A383419 a(n) = A378762(A381968(n)).

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A383590 a(n) = A378762(A382499(n)).

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A374004 a(n) = (1 + (n+3)^2 - (n-4)*(-1)^n)/2.

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A374005 a(n) = (1 + (n+4)^2 + (n-5)*(-1)^n)/2.

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A374007 a(n) = (1 + (n+5)^2 - (n-6)*(-1)^n)/2.

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A374008 a(n) = (1 + (n+6)^2 + (n-7)*(-1)^n)/2.

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