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%I #34 Feb 15 2022 12:59:08
%S 1,1,4,1,1,4,9,4,1,1,4,9,16,9,4,1,1,4,9,16,25,16,9,4,1,1,4,9,16,25,36,
%T 25,16,9,4,1,1,4,9,16,25,36,49,36,25,16,9,4,1,1,4,9,16,25,36,49,64,49,
%U 36,25,16,9,4,1,1,4,9,16,25,36,49,64,81,64,49,36,25,16,9,4,1,1,4,9,16
%N Triangle whose rows are sequences of increasing and decreasing squares: 1; 1,4,1; 1,4,9,4,1; ...
%C The triangle A003983 with individual entries squared and each 2nd row skipped.
%C Analogous to A004737. - _Peter Bala_, Sep 25 2007
%C T(n,k) = min(n,k)^2. 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). - _Boris Putievskiy_, Jan 13 2013
%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)^2/((1-x)(1-qx)^2(1-q^2x)) = 1 + x(1 + 4q + q^2) + x^2(1 + 4q + 9q^2 + 4q^3 + q^4) + ... . - _Peter Bala_, Sep 25 2007
%F From _Boris Putievskiy_, Jan 13 2013: (Start)
%F a(n) = (A004737(n))^2.
%F a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^2. (End)
%e Triangle starts
%e 1;
%e 1, 4, 1;
%e 1, 4, 9, 4, 1:
%e 1, 4, 9, 16, 9, 4, 1:
%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...4...4...4...4...4...
%e 1...4...9...9...9...9...
%e 1...4...9..16..16..16...
%e 1...4...9..16..25..25...
%e 1...4...9..16..25..36...
%e ...
%e The start of the sequence as triangle array read by rows:
%e 1;
%e 1, 4, 1;
%e 1, 4, 9, 4, 1;
%e 1, 4, 9, 16, 9, 4, 1;
%e 1, 4, 9, 16, 25, 16, 9, 4, 1;
%e 1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 1;
%e ...
%e Row number k contains 2*k-1 numbers 1,4,...,(k-1)^2,k^2,(k-1)^2,...,4,1. (End)
%p A003983 := proc(n,k) min(n,k) ; end: A124258 := proc(n,k) A003983(n,k)^2 ; end: for d from 1 to 20 by 2 do for c from 1 to d do printf("%d, ",A124258(d+1-c,c)) ; od: od: # _R. J. Mathar_, Sep 21 2007
%p # second Maple program:
%p T:= n-> i^2$i=1..n, (n-i)^2$i=1..n-1:
%p seq(T(n), n=1..10); # _Alois P. Heinz_, Feb 15 2022
%t Flatten[Table[Join[Range[n]^2,Range[n-1,1,-1]^2],{n,10}]] (* _Harvey P. Dale_, Jun 14 2015 *)
%Y Cf. A005900 (row sums), A004737, A133819, A133823, A133824, A133825, A003983.
%K nonn,tabf,easy
%O 1,3
%A _Jonathan Vos Post_, Dec 16 2006
%E More terms from _R. J. Mathar_, Sep 21 2007
%E Edited by _N. J. A. Sloane_, Jun 30 at the suggestion of _R. J. Mathar_
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