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, 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
Offset: 0
Examples
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
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.
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.
Programs
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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 *)
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.
Extensions
Edited by N. J. A. Sloane, Apr 04 2018
Comments