

A115297


Treillis triangle: a triangle read by rows showing the coefficients of sum formulas of Treillis numbers (A115298). The kth 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
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OFFSET

1,2


LINKS

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


FORMULA

For odd rows:
a(1, k) = a(1, k1)  a(1, k2)
a(2, k) = a(1, k1) + [ a(2, k1)  a(2, k2) ]
a(3, k) = a(2, k1) + [ a(3, k1)  a(3, k2) ]
...
a((k1)/2, k) = a((k3)/2, k1) + [ a((k1)/2, k1)  a((k1)/2, k2) ]
a((k+1)/2, k) = Prime(k)  2
and a((k1)/2, k1) = Prime(k1)  2
a((k1)/2, k2) = Prime(k2)  2
For even rows:
a(1, k) = a(1, k1) + [ a(2, k1)  a(1, k2) ]
a(2, k) = a(2, k1) + [ a(3, k1)  a(2, k2) ]
a(3, k) = a(3, k1) + [ a(4, k1)  a(3, k2) ]
...
a((k2)/2, k) = a((k2)/2, k1) + [ a(k/2, k1)  a((k2)/2, k2) ]
a(k/2, k) = Prime(k)  2
and a(k/2, k1) = Prime(k1)  2
a((k2)/2, k2) = Prime(k2)  2
The recurrent prime formulas for odd and even rows are the following : prime(k_odd) = A000040(k_odd) = A115298(k) + Sum_{n=1..(k3)/2} [ a(n,k2) 2*a(n,k1) ] + A000040(k2)  A000040(k1) +2; prime(k_even) = A000040(k_even) = A115298(k) + Sum_{n=1..(k2)/2} [ a(n,k2) a((k2)/2,k2) 2*a(n,k1) +a(1,k1) ] + A000040(k2)  A000040(k1) + 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 halfrows are to be considered for deducing such terms.
For the 13th row:
...................1917,.2313,.2911,.317,.375,.413,.432
.....................2,.....10,....18,....24,...32,...38,...41
For the 14th row:
...................1919,.2317,.2913,.3111,.377,.415,.433,.472
.....................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



