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A299741 Array read by antidiagonals upwards: a(i,0) = 2, i >= 0; a(i,1) = i+2, i >= 0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), for i >= 0, j > 1. 2
2, 2, 2, 2, 3, 2, 2, 4, 7, 2, 2, 5, 14, 18, 2, 2, 6, 23, 52, 47, 2, 2, 7, 34, 110, 194, 123, 2, 2, 8, 47, 198, 527, 724, 322, 2, 2, 9, 62, 322, 1154, 2525, 2702, 843, 2, 2, 10, 79, 488, 2207, 6726, 12098, 10084, 2207, 2, 2, 11, 98, 702, 3842, 15127, 39202, 57965, 37634, 5778, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Note the similarity in form of the recursive steps in the array definition above and the polynomial definition under FORMULA.

LINKS

Table of n, a(n) for n=0..65.

William W. Collier, a(i,j) = f(i+2,j)

William W. Collier, Experimental Mathematics on Wisteria Tables, Talk to Poughkeepsie ACM Chapter.

OEIS Wiki, The (1,2) Pascal Triangle.

FORMULA

Let k be an integer, and let r1 and r2 be the roots of x + 1/x = k. Then f(k,n) = r1^n + r2^n is an integer, for integer n >= 0. Theorem: a(i,j) = f(i+2,j), for i,j >= 0. Proof: See the Collier link.

Define polynomials recursively by:

    p[0](n) = 2, for n >= 0 ( [ and ] demark subscripts).

    p[1](n) = n + 2, for n >= 0.

    p[j](n) = p[j-1](n) * p[1](n) - p[j-2](n), for j > 1, n >= 0. The coefficients of these polynomials occur as the even numbered, upward diagonals in the OEIS Wiki link. Conjecture: a(i,j) = p[j](i), i,j >= 0.

EXAMPLE

i\j |0  1   2    3      4       5        6          7           8            9

----+-------------------------------------------------------------------------

   0|2  2   2    2      2       2        2          2           2            2

   1|2  3   7   18     47     123      322        843        2207         5778

   2|2  4  14   52    194     724     2702      10084       37634       140452

   3|2  5  23  110    527    2525    12098      57965      277727      1330670

   4|2  6  34  198   1154    6726    39202     228486     1331714      7761798

   5|2  7  47  322   2207   15127   103682     710647     4870847     33385282

   6|2  8  62  488   3842   30248   238142    1874888    14760962    116212808

   7|2  9  79  702   6239   55449   492802    4379769    38925119    345946302

   8|2 10  98  970   9602   95050   940898    9313930    92198402    912670090

   9|2 11 119 1298  14159  154451  1684802   18378371   200477279   2186871698

  10|2 12 142 1692  20162  240252  2862862   34114092   406506242   4843960812

  11|2 13 167 2158  27887  360373  4656962   60180133   777684767  10049721838

  12|2 14 194 2702  37634  524174  7300802  101687054  1416317954  19726764302

  13|2 15 223 3330  49727  742575 11088898  165590895  2472774527  36926027010

  14|2 16 254 4048  64514 1028176 16386302  261152656  4162056194  66331746448

  15|2 17 287 4862  82367 1395377 23639042  400468337  6784322687 114933017342

  16|2 18 322 5778 103682 1860498 33385282  599074578 10749957122 192900153618

  17|2 19 359 6802 128879 2441899 46267202  876634939 16609796639 314709501202

  18|2 20 398 7940 158402 3160100 63043598 1257711860 25091193602 500566160180

  19|2 21 439 9198 192719 4037901 84603202 1772629341 37140612959 778180242798

MAPLE

A:= proc(i, j) option remember; `if`(min(i, j)=0, 2,

      `if`(j=1, i+2, (i+2)*A(i, j-1)-A(i, j-2)))

    end:

seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Mar 05 2019

CROSSREFS

The array first appeared in A298675.

Rows 1 through 29 of the array appear in these OEIS entries: A005248, A003500, A003501, A003499, A056854, A086903, A056918, A087799, A057076, A087800, A078363, A067902, A078365, A090727, A078367, A087215, A078369, A090728, A090729, A090730, A090731, A090732, A090733, A090247, A090248, A090249, A090251. Also entries occur for rows 45, 121, and 320:  A087265, A065705, A089775. Each of these entries asserts that a(i,j)=f(i+2,j) is true for that row.

A few of the columns appear in the OEIS: A008865 (for column 2), A058794 and A007754 (for column 3), and A230586 (for column 5).

Sequence in context: A128764 A324818 A233417 * A074589 A199800 A165035

Adjacent sequences:  A299738 A299739 A299740 * A299742 A299743 A299744

KEYWORD

easy,nonn,tabl

AUTHOR

William W. Collier, Feb 18 2018

EXTENSIONS

Edited by N. J. A. Sloane, Apr 04 2018

STATUS

approved

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Last modified May 19 22:32 EDT 2019. Contains 323411 sequences. (Running on oeis4.)