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%I #35 Mar 28 2024 21:54:26
%S 0,1,1,0,2,4,3,3,7,15,0,6,12,26,56,9,9,21,45,97,209,0,18,36,78,168,
%T 362,780,27,27,63,135,291,627,1351,2911,0,54,108,234,504,1086,2340,
%U 5042,10864,81,81,189,405,873,1881,4053,8733,18817,40545
%N Array A(n,k) with all numbers m such that 3*m^2 +- 3^k is a square and their corresponding square roots, read downward by diagonals.
%C Array is analogous to A228405 in goal and structure, with key differences.
%C Left column is A001353. Top row (not in OEIS) interleaves 0 with the powers of 3, as: 0, 1, 0, 3, 0, 9, 0, 27, 0, 81.
%C Either or both may be used as initializing values. See Formula section.
%C The left column is the second binomial transform of the top row. The intermediate transform sequence is A002605, not present in this array.
%C The columns of the array hold all values, in sequential order, of numbers m such that 3*m^2 + 3^k or 3*m^2 - 3^k are squares, and their corresponding square roots in the next column, which then form the "next round" of m values for column k+1.
%C For example: A(n,0) are numbers such that 3*m^2 + 1 are squares, the integer square roots of each are in A(n,1), which are then numbers m such that 3*m^2 - 3 are squares, with those square roots in A(n,2), etc. The sign alternates for each increment of k, etc. No integer square roots exist for the opposite sign in a given column, regardless of n.
%C Also, A(n,1) are values of m such that floor(m^2/3) is square, with the corresponding square roots given by A(n,0).
%C A(n, k)/A(n,k-2) = 3; A(n,k)/A(n,k-1) converges to sqrt(3) for large n.
%C A(n,k)/A(n-1,k) converges to 2 + sqrt(3) for large n.
%C Several ways of combining the first few columns give OEIS sequences:
%C A(n,0) + A(n,1) = A001835; A(n,1) + A(n,2)= A001834; A(n,2) + A(n,3) = A082841;
%C A(n,0)*A(n,1)/2 = A007655(n); A(n+2,0)*A(n+1,1) = A001922(n);
%C A(n,0)*A(n+1,1) = A001921(n); A(n,0)^2 + A(n,1)^2 = A103974(n);
%C A(n,1)^2 - A(n,0)^2 = A011922(n); (A(n+2,0)^2 + A(n+1,1)^2)/2 = A122770(n) = 2*A011916(n).
%C The main diagonal (without initial 0) = 2*A090018. The first subdiagonal = abs(A099842). First superdiagonal = A141041.
%C A001353 (in left column) are the only initializing set of numbers where the recursive square root equation (see below) produces exclusively integer values, for all iterations of k. For any other initial values only even iterations (at k = 2, 4, ...) produce integers.
%H G. C. Greubel, <a href="/A227418/b227418.txt">Antidiagonals n = 0..50, flattened</a>
%F If using the left column and top row to initialize, then: A(n,k) = 2*A(n, k-1) - A(n-1, k-1).
%F If using only the top row to initialize, then: A(n,k) = 4*A(n-1,k) - A(n-2,k).
%F If using the left column to initialize, then: A(n,k) = sqrt(3*A(n,k-1) + (-3)^(k-1)), for all n, k > 0.
%F Other internal relationships that apply are: A(2*n-1, 2*k) = A(n,k)^2 - A(n-1,k)^2;
%F A(n+1,k) * A(n,k+1) - A(n+1, k+1) * A(n,k) = (-3)^k, for all n, k > 0.
%F A(n, 0) = A001353(n).
%F A(n, 1) = A001075(n).
%F A(n, 2) = A005320(n).
%F A(n, 3) = A151961(n).
%F A(1, k) = A038754(k).
%F A(n, n) = 2*A090018(n), for n > 0 (main diagonal).
%F A(n, n+1) = A141041(n-1) (superdiagonal).
%F A(n+1, n) = abs(A099842(n)) (subdiagonal).
%F From _G. C. Greubel_, Oct 09 2022: (Start)
%F T(n, 0) = (1/2)*(1-(-1)^n)*3^((n-1)/2).
%F T(n, 1) = A038754(n-1).
%F T(n, 2) = A228879(n-2).
%F T(2*n-1, n-1) = A141041(n-1).
%F T(2*n, n) = 2*A090018(n-1), n > 0.
%F T(n, n-4) = 3*A005320(n-4).
%F T(n, n-3) = 3*A001075(n-3).
%F T(n, n-2) = 3*A001353(n-2).
%F T(n, n-1) = A001075(n-1).
%F T(n, n) = A001353(n).
%F Sum_{k=0..n-1} T(n, k) = A084156(n).
%F Sum_{k=0..n} T(n, k) = A084156(n) + A001353(n). (End)
%e The array, A(n, k), begins as:
%e 0, 1, 0, 3, 0, 9, 0, 27, ... see A000244;
%e 1, 2, 3, 6, 9, 18, 27, 54, ... A038754;
%e 4, 7, 12, 21, 36, 63, 108, 189, ... A228879;
%e 15, 26, 45, 78, 135, 234, 405, 702, ...
%e 56, 97, 168, 291, 504, 873, 1512, 2619, ...
%e 209, 362, 627, 1086, 1881, 3258, 5643, 9774, ...
%e 780, 1351, 2340, 4053, 7020, 12159, 21060, 36477, ...
%e Antidiagonal triangle, T(n, k), begins as:
%e 0;
%e 1, 1;
%e 0, 2, 4;
%e 3, 3, 7, 15;
%e 0, 6, 12, 26, 56;
%e 9, 9, 21, 45, 97, 209;
%e 0, 18, 36, 78, 168, 362, 780;
%e 27, 27, 63, 135, 291, 627, 1351, 2911;
%e 0, 54, 108, 234, 504, 1086, 2340, 5042, 10864;
%e 81, 81, 189, 405, 873, 1881, 4053, 8733, 18817, 40545;
%t A[n_, k_]:= If[k<0, 0, If[k==0, ChebyshevU[n-1, 2], 2*A[n, k-1] - A[n-1, k-1]]];
%t T[n_, k_]:= A[k, n-k];
%t Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 09 2022 *)
%o (Magma)
%o function A(n,k)
%o if k lt 0 then return 0;
%o elif n eq 0 then return Round((1/2)*(1-(-1)^k)*3^((k-1)/2));
%o elif k eq 0 then return Evaluate(ChebyshevSecond(n), 2);
%o else return 2*A(n, k-1) - A(n-1, k-1);
%o end if; return A;
%o end function;
%o A227418:= func< n,k | A(k, n-k) >;
%o [A227418(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Oct 09 2022
%o (SageMath)
%o def A(n,k):
%o if (k<0): return 0
%o elif (k==0): return chebyshev_U(n-1,2)
%o else: return 2*A(n, k-1) - A(n-1, k-1)
%o def A227418(n, k): return A(k, n-k)
%o flatten([[A227418(n,k) for k in range(n+1)] for n in range(15)]) # _G. C. Greubel_, Oct 09 2022
%Y Cf. A001353, A001075, A005320, A038754, A090018, A099842, A141041, A151961.
%Y Cf. A000244, A084156, A228879.
%K nonn
%O 0,5
%A _Richard R. Forberg_, Sep 02 2013
%E Offset corrected by _G. C. Greubel_, Oct 09 2022
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