|
| |
|
|
A303273
|
|
Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.
|
|
1
|
|
|
|
1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares.
Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
|
|
|
REFERENCES
|
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,10) = A051942(n+10) + 1, n > 0.
T(n,n+2) = T(n+1,n) = A005448(n+1).
T(2*n,n+4) = A186353(4*n+1) (conjectured).
T(2*n,n+5) = A184103(8*n+1) (conjectured).
T(n+2,2*n) = A249013(5*n+4) (conjectured).
T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
floor(T(n,n/2)) = A267682(n) (conjectured).
floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
ceiling(T(n,n)/n) = A007494(n), n > 0.
ceiling(T(n,n^2)/n) = A171769(n), n > 0.
ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.
|
|
|
EXAMPLE
|
The array T(n,k) begins
1 1 1 1 1 1 1 1 1 1 1 1 1 ... A000012
1 2 3 4 5 6 7 8 9 10 11 12 13 ... A000027
2 4 6 8 10 12 14 16 18 20 22 24 26 ... A005843
4 7 10 13 16 19 22 25 28 31 34 37 40 ... A016777
7 11 15 19 23 27 31 35 39 43 47 51 55 ... A004767
11 16 21 26 31 36 41 46 51 56 61 66 71 ... A016861
16 22 28 34 40 46 52 58 64 70 76 82 88 ... A016957
22 29 36 43 50 57 64 71 78 85 92 99 106 ... A016993
29 37 45 53 61 69 77 85 93 101 109 117 125 ... A004770
37 46 55 64 73 82 91 100 109 118 127 136 145 ... A017173
46 56 66 76 86 96 106 116 126 136 146 156 166 ... A017341
56 67 78 89 100 111 122 133 144 155 166 177 188 ... A017401
67 79 91 103 115 127 139 151 163 175 187 199 211 ... A017605
79 92 105 118 131 144 157 170 183 196 209 222 235 ... A190991
...
The inverse binomial transforms of the columns are
1 1 1 1 1 1 1 1 1 1 1 1 1 ...
0 1 2 3 4 5 6 7 8 9 10 11 12 ...
1 1 1 1 1 1 1 1 1 1 1 1 1 ...
0 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 0 0 0 0 0 0 0 0 0 0 0 0 ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle
1
1 1
1 2 2
1 3 4 4
1 4 6 7 7
1 5 8 10 11 11
1 6 10 13 15 16 16
1 7 12 16 19 21 22 22
1 8 14 19 23 26 28 29 29
1 9 16 22 27 31 34 36 37 37
1 10 18 25 31 36 40 43 45 46 46
...
|
|
|
MAPLE
|
T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;
|
|
|
MATHEMATICA
|
Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)
|
|
|
PROG
|
(Maxima)
T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
print(makelist(T(n, k), k, 0, 20));
(PARI) T(n, k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
