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A154325 Triangle with interior all 2's and borders 1.

%I A154325 #26 Jul 17 2025 00:48:57
%S A154325 1,1,1,1,2,1,1,2,2,1,1,2,2,2,1,1,2,2,2,2,1,1,2,2,2,2,2,1,1,2,2,2,2,2,
%T A154325 2,1,1,2,2,2,2,2,2,2,1,1,2,2,2,2,2,2,2,2,1,1,2,2,2,2,2,2,2,2,2,1,1,2,
%U A154325 2,2,2,2,2,2,2,2,2,1,1,2,2,2,2,2,2,2,2,2,2,2,1
%N A154325 Triangle with interior all 2's and borders 1.
%C A154325 This triangle follows a general construction method as follows: Let a(n) be an integer sequence with a(0)=1, a(1)=1. Then T(n,k,r):=[k<=n](1+r*a(k)*a(n-k)) defines a symmetrical triangle.
%C A154325 Row sums are n + 1 + r*Sum_{k=0..n} a(k)*a(n-k) and central coefficients are 1+r*a(n)^2.
%C A154325 Here a(n)=1-0^n and r=1. Row sums are A004277.
%C A154325 Eigensequence of the triangle = A000129, the Pell sequence. - _Gary W. Adamson_, Feb 12 2009
%C A154325 Inverse has general element T(n,k)*(-1)^(n-k). - _Paul Barry_, Oct 06 2010
%F A154325 Number triangle T(n,k) = [k<=n](2-0^(n-k)-0^k+0^(n+k)) = [k<=n](2-0^(k*(n-k))).
%F A154325 a(n) = 2 - A103451(n). - _Omar E. Pol_, Jan 18 2009
%e A154325 Triangle begins
%e A154325   1;
%e A154325   1, 1;
%e A154325   1, 2, 1;
%e A154325   1, 2, 2, 1;
%e A154325   1, 2, 2, 2, 1;
%e A154325   1, 2, 2, 2, 2, 1;
%e A154325   1, 2, 2, 2, 2, 2, 1;
%e A154325 From _Paul Barry_, Oct 06 2010: (Start)
%e A154325 Production matrix is
%e A154325   1,  1;
%e A154325   0,  1, 1;
%e A154325   0, -1, 0, 1;
%e A154325   0,  1, 0, 0, 1;
%e A154325   0, -1, 0, 0, 0, 1;
%e A154325   0,  1, 0, 0, 0, 0, 1;
%e A154325   0, -1, 0, 0, 0, 0, 0, 1;
%e A154325   0,  1, 0, 0, 0, 0, 0, 0, 1; (End)
%t A154325 a[n_] :=
%t A154325  If[Length@
%t A154325     NestWhileList[# -
%t A154325        Floor[(Sqrt[8 # + 1] - 1)/2] (Floor[(Sqrt[8 # + 1] - 1)/2] + 1)/
%t A154325 2 &, n, # > 1 &] <= 2, 1, 2] (* _David Naccache_, Jul 13 2025 *)
%o A154325 (PARI) row(n) = vector(n+1, k, k--; (2-0^(k*(n-k)))); \\ _Michel Marcus_, Jul 13 2025
%Y A154325 Cf. A000129, A103451, A129765.
%K A154325 easy,nonn,tabl
%O A154325 0,5
%A A154325 _Paul Barry_, Jan 07 2009
%E A154325 More terms from _Michel Marcus_, Jul 13 2025