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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. 3
 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 MATHEMATICA a[_, 0] = a[0, _] = 2; a[i_, 1] := i + 2; a[i_, j_] := a[i, j] =(i + 2) a[i, j - 1] - a[i, j - 2]; Table[a[i - j, j], {i, 0, 10}, {j, 0, i}] // Flatten (* Jean-François Alcover, Dec 07 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). Main diagonal gives A343261. Sequence in context: A128764 A324818 A233417 * A074589 A199800 A338094 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 June 3 18:44 EDT 2023. Contains 363116 sequences. (Running on oeis4.)