A131179 -id:A131179 - cp's OEIS frontend

A056011 Enumeration of natural numbers by the boustrophedonic diagonal method.

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

Offset: 1

Clark Kimberling, Aug 01 2000

A triangle such that (1) every positive integer occurs exactly once; (2) row n consists of n consecutive numbers; (3) odd-numbered rows are increasing; and (4) even-numbered rows are decreasing.
Self-inverse permutation of the natural numbers.
Mirror image of triangle in A056023. - Philippe Deléham, Apr 04 2009
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. - Boris Putievskiy, Dec 24 2012
For generalizations see A218890, A213927. - Boris Putievskiy, Mar 10 2013

			The start of the sequence as a table:
   1,  3,  4, 10, 11, 21, ...
   2,  5,  9, 12, 20, 23, ...
   6,  8, 13, 19, 24, 34, ...
   7, 14, 18, 25, 33, 40, ...
  15, 17, 26, 32, 41, 51, ...
  ...
Enumeration by boustrophedonic ("ox-plowing") diagonal method. - _Boris Putievskiy_, Dec 24 2012
The start of the sequence as triangle array read by rows:
   1;
   3,  2;
   4,  5,  6;
  10,  9,  8,  7;
  11, 12, 13, 14, 15;
  ...

Reinhard Zumkeller, Rows n = 1..125 of table, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A079826, A131179 (first column), A218890, A213927.

Cf. A000027, A038722.

a056011 n = a056011_tabl !! (n-1)
a056011_list = concat a056011_tabl
a056011_tabl = ox False a000027_tabl where
  ox turn (xs:xss) = (if turn then reverse xs else xs) : ox (not turn) xss
a056011_row n = a056011_tabl !! (n-1)
-- Reinhard Zumkeller, Nov 08 2013

A056011 := proc(n,k)
        if type(n,'even') then
                A131179(n)-k+1 ;
        else
                A131179(n)+k-1 ;
        end if;
end proc: # R. J. Mathar, Sep 05 2012

Flatten[If[EvenQ[Length[#]],Reverse[#],#]&/@Table[c=(n(n+1))/2;Range[ c-n+1,c],{n,20}]] (* Harvey P. Dale, Mar 25 2012 *)
With[{nn=20},{#[[1]],Reverse[#[[2]]]}&/@Partition[ TakeList[ Range[ (nn(nn+1))/2],Range[nn]],2]//Flatten] (* Harvey P. Dale, Oct 05 2021 *)

a(n) = ((i+j-1)*(i+j-2)+((-1)^t+1)*i - ((-1)^t-1)*j)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 24 2012

New name from Peter Luschny, Apr 15 2023, based on Boris Putievskiy's comment

A373662 a(n) = (1 + (n+1)^2 - (n-2)*(-1)^n)/2.

Original entry on oeis.org

2, 5, 9, 12, 20, 23, 35, 38, 54, 57, 77, 80, 104, 107, 135, 138, 170, 173, 209, 212, 252, 255, 299, 302, 350, 353, 405, 408, 464, 467, 527, 530, 594, 597, 665, 668, 740, 743, 819, 822, 902, 905, 989, 992, 1080, 1083, 1175, 1178, 1274, 1277, 1377, 1380, 1484, 1487, 1595

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 2 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=2.

			       [ 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, this sequence (k=2), A373663 (k=3), A374004 (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).
Cf. A001844, A038722.
Row n=2 of A056011.
Column k=2 of A056023.

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

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

def A373662(n): return ((n+1)*(n+2)-1 if n&1 else n*(n+1)+5)>>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) = A131179(n+1) + (-1)^n.
G.f.: -x*(2*x^4-3*x^3+3*x+2)/((x+1)^2*(x-1)^3). - Alois P. Heinz, Jun 12 2024

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

Original entry on oeis.org

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).

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

A056011 Enumeration of natural numbers by the boustrophedonic diagonal method.

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A373662 a(n) = (1 + (n+1)^2 - (n-2)*(-1)^n)/2.

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

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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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