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Triangle whose rows are sequences of increasing and decreasing fourth powers: 1; 1,16,1; 1,16,81,16,1; ... .
3

%I #33 Feb 15 2022 14:00:48

%S 1,1,16,1,1,16,81,16,1,1,16,81,256,81,16,1,1,16,81,256,625,256,81,16,

%T 1,1,16,81,256,625,1296,625,256,81,16,1,1,16,81,256,625,1296,2401,

%U 1296,625,256,81,16,1,1,16,81,256,625,1296,2401,4096,2401,1296,625,256,81,16

%N Triangle whose rows are sequences of increasing and decreasing fourth powers: 1; 1,16,1; 1,16,81,16,1; ... .

%C Reading the triangle by rows produces the sequence 1,1,16,1,1,16,81,16,1,..., analogous to A004737.

%C From - _Boris Putievskiy_, Jan 13 2013: (Start)

%C 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).

%C Row number k contains 2*k-1 numbers 1,16,...,(k-1)^4,k^4,(k-1)^4,...,16,1. (End)

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.

%F 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.

%F From _Boris Putievskiy_, Jan 13 2013: (Start)

%F T(n,k) = min(n,k)^4.

%F a(n) = (A004737(n))^4.

%F a(n) = (A124258(n))^2.

%F a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^4. (End)

%e Triangle starts:

%e 1;

%e 1, 16, 1;

%e 1, 16, 81, 16, 1;

%e 1, 16, 81, 256, 81, 16, 1;

%e ...

%e From _Boris Putievskiy_, Jan 13 2013: (Start)

%e The start of the sequence as table:

%e 1...1...1...1...1.. .1...

%e 1..16..16..16..16...16...

%e 1..16..81..81..81...81...

%e 1..16..81.256.256..256...

%e 1..16..81.256.625..625...

%e 1..16..81.256.625.1296...

%e ...

%e (End)

%t p4[n_]:=Module[{c=Range[n]^4},Join[c,Rest[Reverse[c]]]]; Flatten[p4/@ Range[10]] (* _Harvey P. Dale_, Dec 08 2014 *)

%Y Cf. A004737, A061803 (row sums), A133821, A124258, A133823, A003983.

%K easy,nonn,tabf

%O 0,3

%A _Peter Bala_, Sep 25 2007