|
|
A133824
|
|
Triangle whose rows are sequences of increasing and decreasing fourth powers: 1; 1,16,1; 1,16,81,16,1; ... .
|
|
3
|
|
|
1, 1, 16, 1, 1, 16, 81, 16, 1, 1, 16, 81, 256, 81, 16, 1, 1, 16, 81, 256, 625, 256, 81, 16, 1, 1, 16, 81, 256, 625, 1296, 625, 256, 81, 16, 1, 1, 16, 81, 256, 625, 1296, 2401, 1296, 625, 256, 81, 16, 1, 1, 16, 81, 256, 625, 1296, 2401, 4096, 2401, 1296, 625, 256, 81, 16
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Reading the triangle by rows produces the sequence 1,1,16,1,1,16,81,16,1,..., analogous to A004737.
The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1).
Row number k contains 2*k-1 numbers 1,16,...,(k-1)^4,k^4,(k-1)^4,...,16,1. (End)
|
|
LINKS
|
|
|
FORMULA
|
O.g.f.: (1+qx)(1+11qx+11q^2x^2+q^3x^3)/((1-x)(1-qx)^4(1-q^2x)) = 1 + x(1 + 16q + q^2) + x^2(1 + 16q + 81q^2 + 16q^3 + q^4) + ... . Cf. 4th row of A008292.
T(n,k) = min(n,k)^4.
a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^4. (End)
|
|
EXAMPLE
|
Triangle starts:
1;
1, 16, 1;
1, 16, 81, 16, 1;
1, 16, 81, 256, 81, 16, 1;
...
The start of the sequence as table:
1...1...1...1...1.. .1...
1..16..16..16..16...16...
1..16..81..81..81...81...
1..16..81.256.256..256...
1..16..81.256.625..625...
1..16..81.256.625.1296...
...
(End)
|
|
MATHEMATICA
|
p4[n_]:=Module[{c=Range[n]^4}, Join[c, Rest[Reverse[c]]]]; Flatten[p4/@ Range[10]] (* Harvey P. Dale, Dec 08 2014 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|