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, 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

Offset: 1

Author

André F. Labossière, Jan 19 2006

Keywords

• easy
• nonn
• tabf
• uned

Examples

			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.

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