This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A115297 #13 Jan 04 2019 03:19:18
%S A115297 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 A115297 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 A115297 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 A115297 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 A115297 For odd rows:
%F A115297 a(1, k) = a(1, k-1) - a(1, k-2)
%F A115297 a(2, k) = a(1, k-1) + [ a(2, k-1) - a(2, k-2) ]
%F A115297 a(3, k) = a(2, k-1) + [ a(3, k-1) - a(3, k-2) ]
%F A115297 ...
%F A115297 a((k-1)/2, k) = a((k-3)/2, k-1) + [ a((k-1)/2, k-1) - a((k-1)/2, k-2) ]
%F A115297 a((k+1)/2, k) = Prime(k) - 2
%F A115297 and a((k-1)/2, k-1) = Prime(k-1) - 2
%F A115297 a((k-1)/2, k-2) = Prime(k-2) - 2
%F A115297 For even rows:
%F A115297 a(1, k) = a(1, k-1) + [ a(2, k-1) - a(1, k-2) ]
%F A115297 a(2, k) = a(2, k-1) + [ a(3, k-1) - a(2, k-2) ]
%F A115297 a(3, k) = a(3, k-1) + [ a(4, k-1) - a(3, k-2) ]
%F A115297 ...
%F A115297 a((k-2)/2, k) = a((k-2)/2, k-1) + [ a(k/2, k-1) - a((k-2)/2, k-2) ]
%F A115297 a(k/2, k) = Prime(k) - 2
%F A115297 and a(k/2, k-1) = Prime(k-1) - 2
%F A115297 a((k-2)/2, k-2) = Prime(k-2) - 2
%F A115297 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 A115297 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 A115297 For the 13th row:
%e A115297 ...................19-17,.23-13,.29-11,.31-7,.37-5,.41-3,.43-2
%e A115297 .....................2,.....10,....18,....24,...32,...38,...41
%e A115297 For the 14th row:
%e A115297 ...................19-19,.23-17,.29-13,.31-11,.37-7,.41-5,.43-3,.47-2
%e A115297 .....................0,.....6,.....16,....20,....30,...36,...40,...45
%e A115297 From _Michael Somos_, Oct 17 2016: (Start)
%e A115297 Triangle:
%e A115297 1: 1,
%e A115297 2: 3,
%e A115297 3: 2, 5,
%e A115297 4: 4, 9,
%e A115297 5: 2, 8, 11,
%e A115297 6: 6, 10, 15,
%e A115297 7: 4, 8, 14, 17,
%e A115297 8: 6, 12, 16, 21,
%e A115297 ... (End)
%Y A115297 Cf. A115298, A000040, A000166, A008276.
%K A115297 easy,nonn,tabf,uned
%O A115297 1,2
%A A115297 _André F. Labossière_, Jan 19 2006