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A133824 Triangle whose rows are sequences of increasing and decreasing fourth powers: 1; 1,16,1; 1,16,81,16,1; ... .

%I A133824 #33 Feb 15 2022 14:00:48
%S A133824 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 A133824 1,1,16,81,256,625,1296,625,256,81,16,1,1,16,81,256,625,1296,2401,
%U A133824 1296,625,256,81,16,1,1,16,81,256,625,1296,2401,4096,2401,1296,625,256,81,16
%N A133824 Triangle whose rows are sequences of increasing and decreasing fourth powers: 1; 1,16,1; 1,16,81,16,1; ... .
%C A133824 Reading the triangle by rows produces the sequence 1,1,16,1,1,16,81,16,1,..., analogous to A004737.
%C A133824 From - _Boris Putievskiy_, Jan 13 2013: (Start)
%C A133824 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 A133824 Row number k contains 2*k-1 numbers 1,16,...,(k-1)^4,k^4,(k-1)^4,...,16,1. (End)
%H A133824 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F A133824 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 A133824 From _Boris Putievskiy_, Jan 13 2013: (Start)
%F A133824 T(n,k) = min(n,k)^4.
%F A133824 a(n) = (A004737(n))^4.
%F A133824 a(n) = (A124258(n))^2.
%F A133824 a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^4. (End)
%e A133824 Triangle starts:
%e A133824   1;
%e A133824   1, 16, 1;
%e A133824   1, 16, 81, 16, 1;
%e A133824   1, 16, 81, 256, 81, 16, 1;
%e A133824   ...
%e A133824 From _Boris Putievskiy_, Jan 13 2013: (Start)
%e A133824 The start of the sequence as table:
%e A133824   1...1...1...1...1.. .1...
%e A133824   1..16..16..16..16...16...
%e A133824   1..16..81..81..81...81...
%e A133824   1..16..81.256.256..256...
%e A133824   1..16..81.256.625..625...
%e A133824   1..16..81.256.625.1296...
%e A133824   ...
%e A133824 (End)
%t A133824 p4[n_]:=Module[{c=Range[n]^4},Join[c,Rest[Reverse[c]]]]; Flatten[p4/@ Range[10]] (* _Harvey P. Dale_, Dec 08 2014 *)
%Y A133824 Cf. A004737, A061803 (row sums), A133821, A124258, A133823, A003983.
%K A133824 easy,nonn,tabf
%O A133824 0,3
%A A133824 _Peter Bala_, Sep 25 2007