A115298 -id:A115298 - cp's OEIS frontend

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

André F. Labossière, Jan 19 2006

			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)

Cf. A115298, A000040, A000166, A008276.

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

A120315 Combinatorial prime formulas. This sequence gives the coefficients a(n) of combinatorial sum formulas of n-th primes or lesser: prime(n) = 2^(n-5)/(n-1)! Sum_{i=1..n} a(i) * C(n-1,i-1) * (1-(n-i)/2).

Original entry on oeis.org

32, 8, 32, 52, 208, 508, 2672, 9278, 56048, 304132, 1654552, 12649198, 79342112, 615363002, 5010269828, 43213043413, 393086195632, 3633203615548, 38586294965048, 389261740224662, 4344329090764472, 51205748753742838

Offset: 1

André F. Labossière, Jun 20 2006

			prime(7) = [ 2^(7-5)/(7-1)! ] * [ 32*C(7-1,0)*(1-(7-1)/2) + 8*C(7-1,1)*(1-(7-2)/2) + 32*C(7-1,2)*(1-(7-3)/2) + 52*C(7-1,3)*(1-(7-4)/2) + 208*C(7-1,4)*(1-(7-5)/2) + 508*C(7-1,5)*(1-(7-6)/2) + 2672*C(7-1,6)*(1-(7-7)/2) ] = 17

Cf. A000040, A007504, A000166, A115298, A008275.

A169627 A conjectured sequence of bud numbers for eucalyptus flowers.

Original entry on oeis.org

1, 3, 7, 13, 21, 31

Offset: 0

N. J. A. Sloane, based on email from Omar E. Pol, Mar 07 2010

The Carr and Carr article says: "Pryor (1954) has postulated three developmental series of bud numbers, as follows : (a) 1, 3, 7, 11, 15, 23, 35 ... (b) 1, 3, 7, 13, 21, 31 ...; (c) 1, 3, 15, 31, 63, 127 ..."
It would be nice to have more information.
The given terms (corresponding to (b) above) would match the formula a(n)=n^2+n+1, as suggested (with offset corrected) on the Weidmann's web page, cf. LINKS. This would mean that a(n) = A002061(n+1). But based on the construction and illustration in the Pryor paper, it could as well be a duplicate of A161206. The first terms also coincide with the Treillis sequence A115298. - M. F. Hasler, Apr 14 2015

D. J. Carr and S. G. M. Carr, Developmental morphology of the floral organs of Eucalyptus. I. The inflorescence, Austr. J. Botany 7(2) (1959), 109-141. [2.6 MB, may need longer to load], (a link with less information. The main links are below).
L. D. Pryor, Illustration of initial terms (Fig. 2b)
L. D. Pryor, The inheritance of inflorescence characters in Eucalyptus, Proceedings of the Linnean Society of New South Wales, V. 79, (1954), p. 79-89. [From _Omar E. Pol_, Mar 07 2010]
P. E. Weidmann, The OEIS Sequencer survey, Apr 11 2015

Cf. A169626.

Added information to the Carr & Carr link. - R. J. Mathar, Mar 08 2010

A119362 Combinatorial twin prime formulas. This sequence gives the coefficients a(n) of combinatorial sum formulas of n-th twin primes or lesser: twin_prime(n) = 2^(n-6)/(n-1)! Sum_{i=1..n} a(i) * C(n-1,i-1) * (i+2-n).

Original entry on oeis.org

32, 8, 32, 52, 208, 508, 2672, 9278, 71168, 258772, 2448352, 11401798, 123001112, 660768362, 7257835148, 50721144013, 643561620832, 4610932367548, 57562797069608, 547637465534282, 7281278491404272, 71412114145523938

Offset: 1

André F. Labossière, Jul 24 2006

			twin_prime(10) = [ 2^(10-6)/(10-1)! ] * [ 32*C(10-1,0)*(-7) + 8*C(10-1,1)*(-6) + 32*C(10-1,2)*(-5) + ... + 9278*C(10-1,7)*(0) + 71168*C(10-1,8)*(1) + 258772*C(10-1,9)*(2) ] = 31

Cf. A001097, A000040, A120315, A000166, A115298, A008275.

Mathematica 5.2 - webMathematica 2 - http://library.wolfram.com/webMathematica/Education/LongDivide.jsp

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A120315 Combinatorial prime formulas. This sequence gives the coefficients a(n) of combinatorial sum formulas of n-th primes or lesser: prime(n) = 2^(n-5)/(n-1)! Sum_{i=1..n} a(i) * C(n-1,i-1) * (1-(n-i)/2).

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A169627 A conjectured sequence of bud numbers for eucalyptus flowers.

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A119362 Combinatorial twin prime formulas. This sequence gives the coefficients a(n) of combinatorial sum formulas of n-th twin primes or lesser: twin_prime(n) = 2^(n-6)/(n-1)! Sum_{i=1..n} a(i) * C(n-1,i-1) * (i+2-n).

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