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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; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..77.

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 ~~~: (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: A240729 A258259 A240182 * A275900 A054430 A276572

Adjacent sequences:  A115294 A115295 A115296 * A115298 A115299 A115300

KEYWORD

easy,nonn,tabl,uned

AUTHOR

André F. Labossière, Jan 19 2006

STATUS

approved

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Last modified December 7 07:15 EST 2016. Contains 278841 sequences.