cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A132025 Decimal expansion of Product_{k>=0} 1-1/(2*9^k).

Original entry on oeis.org

4, 6, 8, 9, 4, 5, 1, 7, 8, 3, 6, 7, 0, 2, 3, 6, 9, 3, 2, 8, 3, 2, 8, 0, 0, 3, 5, 4, 1, 8, 6, 5, 6, 3, 9, 4, 0, 6, 8, 0, 4, 5, 7, 5, 8, 6, 9, 8, 9, 8, 5, 6, 0, 1, 6, 7, 1, 9, 7, 9, 9, 2, 3, 2, 7, 4, 7, 5, 7, 3, 2, 8, 3, 4, 6, 7, 0, 4, 3, 8, 1, 7, 5, 4, 9, 5, 0, 9, 4, 2, 7, 5, 7, 0, 0, 0, 1, 5, 9, 1, 7, 1, 1
Offset: 0

Views

Author

Hieronymus Fischer, Aug 14 2007

Keywords

Examples

			0.4689451783670236932832800...

Crossrefs

Cf. A048651, A098844, A067080, A132019, A132026, A132033, A132037, A000217, A270369.

Programs

  • Mathematica
    digits = 103; NProduct[1-1/(2*9^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/9], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Formula

Equals lim inf_{n->oo} Product_{k=0..floor(log_9(n))} floor(n/9^k)*9^k/n.
Equals lim inf_{n->oo} A132033(n)/n^(1+floor(log_9(n)))*9^(1/2*(1+floor(log_9(n)))*floor(log_9(n))).
Equals lim inf_{n->oo} A132033(n)/n^(1+floor(log_9(n)))*9^A000217(floor(log_9(n))).
Equals (1/2)*exp(-Sum_{n>0} 9^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132033(n)/A132033(n+1).
Equals Product_{n>=1} (1 - 1/A270369(n)). - Amiram Eldar, May 08 2023

A335684 Array read by antidiagonals: T(m,n) = number of m-by-n hexagonal digraphs without oriented 3-cycles (for m >= 1, n >= 1).

Original entry on oeis.org

1, 2, 2, 4, 18, 4, 8, 162, 162, 8, 16, 1458, 6570, 1458, 16, 32, 13122, 266454, 266454, 13122, 32, 64, 118098, 10806354, 48697686, 10806354, 118098, 64, 128, 1062882, 438264342, 8900099046, 8900099046, 438264342, 1062882, 128, 256, 9565938, 17774323650, 1626602228838, 7330125946050, 1626602228838, 17774323650, 9565938, 256
Offset: 1

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Author

N. J. A. Sloane, Jul 03 2020, based on an email from Don Knuth

Keywords

Comments

More precisely, consider the directed graph with m*n vertices i,j
for 0<=i
(i+1),(j+1) when those vertices exist. [There are m(n-1)+(m-1)n=(m-1)(n-1) arcs.]
Each arc between neighboring vertices is directed, one way or the
other. We are not allowed to have vertices u,v,w with u->v->w->u.

Examples

			The array begins:
1, 2, 4, 8, 16, 32, 64, 128, 256, ...
2, 18, 162, 1458, 13122, 118098, 1062882, 9565938, 86093442, ...
4, 162, 6570, 266454, 10806354, 438264342, 17774323650, 720858511494, 29235261145554, ...
8, 1458, 266454, 48697686, 8900099046, 1626602228838, 297281501943462, 54331839604996902, 9929809879071710886, ...
16, 13122, 10806354, 8900099046, 7330125946050, 6037095692927862, 4972155285312413586, 4095069788483623200006, 3372701697814125393026946, ...
32, 118098, 438264342, 1626602228838, 6037095692927862, 22406540276117433798, 83161353485088190184022, 308651430745402593036755238, 1145552611990801975992739211382, ...
64, 1062882, 17774323650, 297281501943462, 4972155285312413586, 83161353485088190184022, 1390908038123039657933009250, 23263512314950157506021612227654, 389091866894670127046561452469612466, ...
128, 9565938, 720858511494, 54331839604996902, 4095069788483623200006, 308651430745402593036755238, 23263512314950157506021612227654, 1753405140846978937849992172443469926, 132156724503381398420197323509979254463366, ...
256, 86093442, 29235261145554, 9929809879071710886, 3372701697814125393026946, 1145552611990801975992739211382, 389091866894670127046561452469612466, 132156724503381398420197323509979254463366,
44887599350675449085445484460546360180897201250, ...
...
The initial antidiagonals are:
[1]
[2, 2]
[4, 18, 4]
[8, 162, 162, 8]
[16, 1458, 6570, 1458, 16]
[32, 13122, 266454, 266454, 13122, 32]
[64, 118098, 10806354, 48697686, 10806354, 118098, 64]
[128, 1062882, 438264342, 8900099046, 8900099046, 438264342, 1062882, 128]
[256, 9565938, 17774323650, 1626602228838, 7330125946050, 1626602228838, 17774323650, 9565938, 256]
[512, 86093442, 720858511494, 297281501943462, 6037095692927862, 6037095692927862, 297281501943462, 720858511494, 86093442, 512]
[1024, 774840978, 29235261145554, 54331839604996902, 4972155285312413586, 22406540276117433798, 4972155285312413586, 54331839604996902, 29235261145554, 774840978, 1024]
...

References

  • D. E. Knuth, The Art of Computer Programming, Section 7.2.2.3, in preparation.

Crossrefs

First two rows are A000079, A270369; main diagonal is A335685.

A384853 Squared length of interior diagonal of n-th (U, V)-crossbox, where U = (1, 0, 1) and V = (0, 1, 0), as in Comments.

Original entry on oeis.org

1, 5, 9, 21, 57, 165, 489, 1461, 4377, 13125, 39369, 118101, 354297, 1062885, 3188649, 9565941, 28697817, 86093445, 258280329, 774840981, 2324522937, 6973568805, 20920706409, 62762119221, 188286357657, 564859072965, 1694577218889, 5083731656661
Offset: 1

Views

Author

Clark Kimberling, Jul 02 2025

Keywords

Comments

Suppose that U and V are 3-dimensional vectors over the field of real numbers. Define f(1) = U, f(2) = V, f(3) = UxV, where x = cross product, and for n>=2, define f(n) = h(n - 1), g(n) = f(n - 1) + g(n - 1) - h(n - 1), h(n) = f(n) x g(n).
The parallelopiped having edge vectors f(n), g(n), h(n) is the n-th (U,V)-crossbox, with volume |f(n).(g(n) x h(n))|, where . = dot product, and interior diagonal length ||g(n)||. These two sequences, after removal of their first 2 terms, are given for selected U and V by the following table, except for the 3 initial terms:
U V volume squared diagonal length, ||g(n)||^2
(1, 0, 0) (0, 1, 0) A000079 A052548
(1, 0, 0) (0, 1, 1) A008776 3*A052919
(1, 0, 0) (1, 0, 1) A000351 A178676
(1, 0, 0) (1, 1, 1) A167747 5*A204061
(1, 0, 0) (0, 2, 0) A005054 4*A199215
(1, 0, 0) (1, 2, 0) A013731 8*A199552
(1, 0, 0) (2, 1, 0) A011557 10*A000533
(1, 0, 0) (1, 1, 2) A067403 18*A135423
(1, 0, 0) (2, 1, 1) A334603 11*A199750
(1, 0, 1) (0, 1, 0) A008776 this sequence
(1, 1, 0) (0, 1, 1) A081341 6*A199318
(1, 1, 0) (1, 1, 1) A270369 9*A199559
(1, 2, 3) (3, 2, 1) 2*A009992 48 + 96*A009992

Examples

			Taking U = (1, 0, 1) and V = (0, 1, 0), successive edge vectors are given by
(f(n)) = ( (1, 0, 1), (-1,0,1), (-1,2,-1), (3,0,-3), (3,-6,3), ...)
(g(n)) = ( (0,1,0), (2,1,0), (2,-1,2), (-2,1,4), (-2,7,-2), (10,1,-8), ...)
(h(n)) = ( (-1.0,1), (-1,2,-1), (3,0,-3), (3,-6,3), (-9,0,9),...)
The successive volumes are (2, 6, 18, 54, 162, 486, 1458, 4374, 13122,...).
The lengths of diagonals of the first five crossboxes are 1, sqrt(5), 3, sqrt(21), sqrt(57), so the first five squared lengths are 1, 5, 9, 21, 57.

Links

Crossrefs

Cf. A000079, A052548, A008776.

Programs

  • Mathematica
    f[1] = {1, 0, 1}; g[1] = {0, 1, 0}; h[1] = Cross[f[1], g[1]];
f[n_] := f[n] = h[n - 1];
g[n_] := g[n] = f[n - 1] + g[n - 1] - h[n - 1];
h[n_] := h[n] = Cross[f[n], g[n]];
v[n_] := f[n] . Cross[g[n], h[n]] (* signed volume of nth parallelopiped P(n) *)
d[n_] := Norm[g[n]] (* length of interior diagonal of P(n) *)
Column[Table[{f[n], g[n], h[n]}, {n, 1, 16}]]  (* edge vectors of P(n) *)
Table[v[n], {n, 1, 16}]  (* A008776 *)
u = Table[d[n]^2, {n, 1, 30}] (* A384853 *)
Join[{1},Table[1+2*(3^(n-1)+1),{n,40}]] (* or *) LinearRecurrence[{4,-3},{1,5,9},50] (* Harvey P. Dale, Jul 20 2025 *)

Formula

a(0) = 1, a(n) = 1 + 2 * (3^(n-1)+1) for n>=1.
a(n) = 4*a(n-1) - 3*a(n-2) for n>=4.
In general, suppose that U = (a,b,c) and V = (s,t,u), and let D = -(a^2 + b^2 + c^2 + s^2 + t^2 + u^2 + 2 (a s + b t + c u)). Then, linear recurrences are given for n>=3 by f(n) = D*f (n - 2), g(n) = g(n - 1) + D*g(n - 2) - D*g(n - 3), h(n) = D*h(n - 2). If w(n) denotes the volume of the n-th (U,V)-crossbox, then w(n) = D*w(n-1) for n>=2.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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