A249327 -id:A249327 - cp's OEIS frontend

A144555 a(n) = 14*n^2.

0, 14, 56, 126, 224, 350, 504, 686, 896, 1134, 1400, 1694, 2016, 2366, 2744, 3150, 3584, 4046, 4536, 5054, 5600, 6174, 6776, 7406, 8064, 8750, 9464, 10206, 10976, 11774, 12600, 13454, 14336, 15246, 16184, 17150, 18144, 19166, 20216, 21294, 22400, 23534, 24696

Offset: 0

N. J. A. Sloane, Jan 01 2009

Sequence found by reading the line from 0, in the direction 0, 14, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same line and direction in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 10 2011

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

See also A033428, A033429, A033581, A033582, A033583, A033584, ... and A249327 for the whole table.
Cf. A000290, A001105, A064761, A118277, A152742, A195019, A195020.
Cf. A008596, A195145.

Table[14*n^2, {n, 0, 45}] (* Amiram Eldar, Feb 03 2021 *)

A144555(n)=14*n^2 \\ M. F. Hasler, Oct 31 2014

a(n) = 14*A000290(n) = 7*A001105(n) = 2*A033582(n). - Omar E. Pol, Jan 01 2009
a(n) = a(n-1) + 14*(2*n-1), with a(0) = 0. - Vincenzo Librandi, Nov 25 2010
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/84.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/168.
Product_{n>=1} (1 + 1/a(n)) = sqrt(14)*sinh(Pi/sqrt(14))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(14)*sin(Pi/sqrt(14))/Pi. (End)
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 14*x*(1 + x)/(1-x)^3.
E.g.f.: 14*x*(1 + x)*exp(x).
a(n) = n*A008596(n) = A195145(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A360633 Square array A(n, k), n, k > 0, read by antidiagonals upwards; A(n, k) = A360613(2n-1) A360613(2*k).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 12, 10, 8, 9, 21, 20, 16, 11, 13, 27, 35, 32, 22, 15, 14, 39, 45, 56, 44, 30, 17, 18, 42, 65, 72, 77, 60, 34, 19, 23, 54, 70, 104, 99, 105, 68, 38, 24, 25, 69, 90, 112, 143, 135, 119, 76, 48, 29, 26, 75, 115, 144, 154, 195, 153, 133, 96, 58, 31

Offset: 1

Rémy Sigrist, Feb 14 2023

This sequence can be seen as a greedy multiplication table where we alternately add rows and columns so that all products are distinct.

Conjecture: all integers appear in this sequence.

			Array A(n, k) begins:
  n\k |  1   2    3    4    5    6    7    8    9   10
  ----+-----------------------------------------------
    1 |  1   3    5    8   11   15   17   19   24   29
    2 |  2   6   10   16   22   30   34   38   48   58
    3 |  4  12   20   32   44   60   68   76   96  116
    4 |  7  21   35   56   77  105  119  133  168  203
    5 |  9  27   45   72   99  135  153  171  216  261
    6 | 13  39   65  104  143  195  221  247  312  377
    7 | 14  42   70  112  154  210  238  266  336  406
    8 | 18  54   90  144  198  270  306  342  432  522
    9 | 23  69  115  184  253  345  391  437  552  667
   10 | 25  75  125  200  275  375  425  475  600  725

Rémy Sigrist, Table of n, a(n) for n = 1..10011
Rémy Sigrist, C program

Cf. A249327, A360613, A360627, A360628.

C
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See Links section.
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A(n, k) = A360627(n) * A360628(k).
A(n, 1) = A360627(n).
A(1, k) = A360628(k).

A360646 Square array A(n, k), n, k > 0, read by antidiagonals upwards; A(n, k) = A066208(n) * A066207(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 5, 12, 14, 9, 8, 15, 28, 18, 13, 10, 24, 35, 36, 26, 19, 11, 30, 56, 45, 52, 38, 21, 16, 33, 70, 72, 65, 76, 42, 27, 17, 48, 77, 90, 104, 95, 84, 54, 29, 20, 51, 112, 99, 130, 152, 105, 108, 58, 37, 22, 60, 119, 144, 143, 190, 168, 135, 116, 74, 39

Offset: 1

Rémy Sigrist, Feb 15 2023

Every positive integer occurs exactly once.

			Array A(n, k) begins:
  n\k |  1   2    3    4    5    6    7    8    9   10
  ----+-----------------------------------------------
    1 |  1   3    7    9   13   19   21   27   29   37
    2 |  2   6   14   18   26   38   42   54   58   74
    3 |  4  12   28   36   52   76   84  108  116  148
    4 |  5  15   35   45   65   95  105  135  145  185
    5 |  8  24   56   72  104  152  168  216  232  296
    6 | 10  30   70   90  130  190  210  270  290  370
    7 | 11  33   77   99  143  209  231  297  319  407
    8 | 16  48  112  144  208  304  336  432  464  592
    9 | 17  51  119  153  221  323  357  459  493  629
   10 | 20  60  140  180  260  380  420  540  580  740

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A249327, A066207, A066208, A247503, A248101.

PARI
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See Links section.
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A(n, 1) = A066208(n) = A247503(A(n,k)).

A(1, k) = A066207(k) = A248101(A(n,k)).

A374600 If n = i^2 * A005117(j) for some i, j > 0 then a(n) = j^2 * A005117(i).

Original entry on oeis.org

1, 4, 9, 2, 16, 25, 36, 8, 3, 49, 64, 18, 81, 100, 121, 5, 144, 12, 169, 32, 196, 225, 256, 50, 6, 289, 27, 72, 324, 361, 400, 20, 441, 484, 529, 7, 576, 625, 676, 98, 729, 784, 841, 128, 48, 900, 961, 45, 10, 24, 1024, 162, 1089, 75, 1156, 200, 1225, 1296

Offset: 1

Rémy Sigrist, Jul 13 2024

nonn

This sequence is a self-inverse permutation of the positive integers with infinitely many fixed points.

			For n = 84: 84 = 2^2 * A005117(14), so a(84) = 14^2 * A005117(2) = 392.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

See A374611 for a similar sequence.

Cf. A000188, A005117, A007913, A249327.

PARI
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\\ See Links section.
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a(A249327(n, k)) = A249327(k, n).

a(n) = n iff n = k^2 * A005117(k) for some k > 0.

cp's OEIS Frontend

A144555 a(n) = 14*n^2.

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A360633 Square array A(n, k), n, k > 0, read by antidiagonals upwards; A(n, k) = A360613(2*n-1) * A360613(2*k).

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C

Formula

A360646 Square array A(n, k), n, k > 0, read by antidiagonals upwards; A(n, k) = A066208(n) * A066207(k).

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

PARI

Formula

A374600 If n = i^2 * A005117(j) for some i, j > 0 then a(n) = j^2 * A005117(i).

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

PARI

Formula

A360633 Square array A(n, k), n, k > 0, read by antidiagonals upwards; A(n, k) = A360613(2n-1) A360613(2*k).