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 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. 1
 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, 18, 26, 29, 4, 8, 16, 24, 28, 35, 6, 12, 22, 26, 34, 39, 2, 10, 18, 24, 32, 38, 41, 6, 16, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS FORMULA For odd rows: a(1, k) = a(1, k-1) - a(1, k-2) a(2, k) = a(1, k-1) + [ a(2, k-1) - a(2, k-2) ] a(3, k) = a(2, k-1) + [ a(3, k-1) - a(3, k-2) ] ... a((k-1)/2, k) = a((k-3)/2, k-1) + [ a((k-1)/2, k-1) - a((k-1)/2, k-2) ] a((k+1)/2, k) = Prime(k) - 2 and a((k-1)/2, k-1) = Prime(k-1) - 2 a((k-1)/2, k-2) = Prime(k-2) - 2 For even rows: a(1, k) = a(1, k-1) + [ a(2, k-1) - a(1, k-2) ] a(2, k) = a(2, k-1) + [ a(3, k-1) - a(2, k-2) ] a(3, k) = a(3, k-1) + [ a(4, k-1) - a(3, k-2) ] ... a((k-2)/2, k) = a((k-2)/2, k-1) + [ a(k/2, k-1) - a((k-2)/2, k-2) ] a(k/2, k) = Prime(k) - 2 and a(k/2, k-1) = Prime(k-1) - 2 a((k-2)/2, k-2) = Prime(k-2) - 2 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 EXAMPLE 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. For the 13th row: ...................19-17,.23-13,.29-11,.31-7,.37-5,.41-3,.43-2 .....................2,.....10,....18,....24,...32,...38,...41 For the 14th row: ...................19-19,.23-17,.29-13,.31-11,.37-7,.41-5,.43-3,.47-2 .....................0,.....6,.....16,....20,....30,...36,...40,...45 From Michael Somos, Oct 17 2016: (Start) Triangle: 1: 1, 2: 3, 3: 2, 5, 4: 4, 9, 5: 2, 8, 11, 6: 6, 10, 15, 7: 4, 8, 14, 17, 8: 6, 12, 16, 21, ... (End) CROSSREFS Cf. A115298, A000040, A000166, A008276. Sequence in context: A339371 A258259 A240182 * A275900 A296007 A277437 Adjacent sequences:  A115294 A115295 A115296 * A115298 A115299 A115300 KEYWORD easy,nonn,tabf,uned AUTHOR André F. Labossière, Jan 19 2006 STATUS approved

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Last modified September 27 10:28 EDT 2021. Contains 347689 sequences. (Running on oeis4.)