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%I #13 Jan 04 2019 03:19:18
%S 1,3,2,5,4,9,2,8,11,6,10,15,4,8,14,17,6,12,16,21,2,10,14,20,27,6,12,
%T 18,26,29,4,8,16,24,28,35,6,12,22,26,34,39,2,10,18,24,32,38,41,6,16,
%U 20,30,36,40,45,4,12,18,26,34,38,44,51,10,14,24,30,36,42,50,57,6,12,20,28,32
%N Treillis triangle: a triangle read by rows showing the coefficients of sum formulas of Treillis numbers (A115298). The k-th row (k>=1) contains a(n,k) for n=1 to (k+1)/2 (odd rows) and for n=1 to k/2 (even rows), where a(n,k) satisfies Sum_{n=1..[(k+1)/2_odd, k/2_even]} a(n,k). The last term of each row (and its only odd number) equals Prime(k+1)-2.
%F For odd rows:
%F a(1, k) = a(1, k-1) - a(1, k-2)
%F a(2, k) = a(1, k-1) + [ a(2, k-1) - a(2, k-2) ]
%F a(3, k) = a(2, k-1) + [ a(3, k-1) - a(3, k-2) ]
%F ...
%F a((k-1)/2, k) = a((k-3)/2, k-1) + [ a((k-1)/2, k-1) - a((k-1)/2, k-2) ]
%F a((k+1)/2, k) = Prime(k) - 2
%F and a((k-1)/2, k-1) = Prime(k-1) - 2
%F a((k-1)/2, k-2) = Prime(k-2) - 2
%F For even rows:
%F a(1, k) = a(1, k-1) + [ a(2, k-1) - a(1, k-2) ]
%F a(2, k) = a(2, k-1) + [ a(3, k-1) - a(2, k-2) ]
%F a(3, k) = a(3, k-1) + [ a(4, k-1) - a(3, k-2) ]
%F ...
%F a((k-2)/2, k) = a((k-2)/2, k-1) + [ a(k/2, k-1) - a((k-2)/2, k-2) ]
%F a(k/2, k) = Prime(k) - 2
%F and a(k/2, k-1) = Prime(k-1) - 2
%F a((k-2)/2, k-2) = Prime(k-2) - 2
%F The recurrent prime formulas for odd and even rows are the following : prime(k_odd) = A000040(k_odd) = A115298(k) + Sum_{n=1..(k-3)/2} [ a(n,k-2) -2*a(n,k-1) ] + A000040(k-2) - A000040(k-1) +2; prime(k_even) = A000040(k_even) = A115298(k) + Sum_{n=1..(k-2)/2} [ a(n,k-2) -a((k-2)/2,k-2) -2*a(n,k-1) +a(1,k-1) ] + A000040(k-2) - A000040(k-1) + 2
%e The computation for obtaining the coefficients of each row of the Treillis triangle are the paired differences between primes ascending and those descending. Only half-rows are to be considered for deducing such terms.
%e For the 13th row:
%e ...................19-17,.23-13,.29-11,.31-7,.37-5,.41-3,.43-2
%e .....................2,.....10,....18,....24,...32,...38,...41
%e For the 14th row:
%e ...................19-19,.23-17,.29-13,.31-11,.37-7,.41-5,.43-3,.47-2
%e .....................0,.....6,.....16,....20,....30,...36,...40,...45
%e From _Michael Somos_, Oct 17 2016: (Start)
%e Triangle:
%e 1: 1,
%e 2: 3,
%e 3: 2, 5,
%e 4: 4, 9,
%e 5: 2, 8, 11,
%e 6: 6, 10, 15,
%e 7: 4, 8, 14, 17,
%e 8: 6, 12, 16, 21,
%e ... (End)
%Y Cf. A115298, A000040, A000166, A008276.
%K easy,nonn,tabf,uned
%O 1,2
%A _André F. Labossière_, Jan 19 2006
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