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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 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.)