A081341 -id:A081341 - cp's OEIS frontend

A261399 a(1)=1; thereafter a(n) = (2/5)(96^(n-2)+1).

1, 4, 22, 130, 778, 4666, 27994, 167962, 1007770, 6046618, 36279706, 217678234, 1306069402, 7836416410, 47018498458, 282110990746, 1692665944474, 10155995666842, 60935974001050, 365615844006298, 2193695064037786, 13162170384226714, 78973022305360282, 473838133832161690

Offset: 1

N. J. A. Sloane, Aug 19 2015

Partial sums of A081341. - Klaus Purath, Jul 28 2020

K. Hong, H. Lee, H. J. Lee and S. Oh, Small knot mosaics and partition matrices, J. Phys. A: Math. Theor. 47 (2014) 435201; arXiv:1312.4009 [math.GT], 2013-2014. See Cor. 2.
Index entries for linear recurrences with constant coefficients, signature (7,-6).
Index entries for sequences related to knots

The number 22, the third term here, is the same 22 seen in A261400 and illustrated in a link in that entry.

Cf. A199412.

G.f.: x-2*x^2*(-2+3*x) / ( (6*x-1)*(x-1) ). - R. J. Mathar, Aug 19 2015
a(n) = 2*A199412(n-2), n>1. - R. J. Mathar, Aug 19 2015
From Klaus Purath, Jul 28 2020: (Start)
a(n) = 7*a(n-1) - 6*a(n-2), n > 2.
a(n) = 6*a(n-1) - 2, n > 1.
a(n) = 3*6^(n-2) + a(n-1), n > 1.
(End)

A382491 a(n) is the numerator of the asymptotic density of the numbers whose number of 3-smooth divisors is n.

Original entry on oeis.org

1, 5, 13, 71, 97, 1355, 793, 19163, 53473, 292355, 60073, 13102907, 535537, 78584915, 790859641, 3523099499, 43112257, 99646519235, 387682633, 2764285630427, 7604811750289, 7337148996275, 31385253913, 2226944658077771, 3656440886376673, 2341258386360995, 80539587570991081

Offset: 1

Amiram Eldar, Mar 29 2025

The denominator that corresponds to a(n) is 3*6^(n-1) = A169604(n-1) = A081341(n).

			Fractions begin with 1/3, 5/18, 13/108, 71/648, 97/3888, 1355/23328, 793/139968, 19163/839808, 53473/5038848, 292355/30233088, 60073/181398528, 13102907/1088391168, ...
a(1) = 1 since a(1)/A081341(1) = 1/3 is the asymptotic density of the numbers with a single 3-smooth divisor, 1, i.e., the numbers that are congruent to 1 or 5 mod 6 (A007310).
a(2) = 5 since a(2)/A081341(2) = 5/18 is the asymptotic density of the numbers with exactly two 3-smooth divisors, either 1 and 2 or 1 and 3, i.e., A171126.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A007310, A072078, A081341 (denominators), A169604, A171126.

a[n_] := DivisorSum[n, 2^(n-#) * 3^(n-n/#) &]; Array[a, 30]

PARI
```
a(n) = sumdiv(n, d, 2^(n-d)*3^(n-n/d));
```

a(n) = Sum_{d|n} 2^(n-d) * 3^(n-n/d).
a(p) = 2^(p-1) + 3^(p-1).
Let f(n) = a(n)/A081341(n). Then:
f(n) = (1/3) * Sum_{d|n} (1/2)^(d-1) * (1/3)^(n/d-1).
Sum_{n>=1} f(n) = 1.
Sum_{n>=1} n * f(n) = 3 (the asymptotic mean of A072078).
Sum_{n>=1} n^2 * f(n) = 18, and therefore, the asymptotic variance of A072078 is 18 - 3^2 = 9, and its asymptotic standard deviation is 3.

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

Clark Kimberling, Jul 02 2025

nonn

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

			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.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000079, A052548, A008776.

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 *)

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.

A191687 Table T(n,k) = ceiling((1/2)*((k+1)^n+(1+(-1)^k)/2)) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 14, 8, 3, 1, 1, 16, 41, 32, 13, 3, 1, 1, 32, 122, 128, 63, 18, 4, 1, 1, 64, 365, 512, 313, 108, 25, 4, 1, 1, 128, 1094, 2048, 1563, 648, 172, 32, 5, 1

Offset: 1

Adi Dani, Jun 11 2011

T(n,k) is the number of compositions of even natural numbers into n parts <= k.

			Top left corner:
  1, 1, 1,  1,  1,...
  1, 1, 2,  2,  3,...
  1, 2, 5,  8, 13,...
  1, 4,14, 32, 63,...
  1, 8,41,128,313,...
T(2,4)=13: there are 13 compositions of even natural numbers into 2 parts <=4
0: (0,0);
2: (0,2), (2,0), (1,1);
4: (0,4), (4,0), (1,3), (3,1), (2,2);
6: (2,4), (4,2), (3,3);
8: (4,4).

Adi Dani, Restricted compositions of natural numbers

Rows sums gives: A000012, A004526, A000982, A036486, A171714, A191484, A191489, A191494, A191495, A191496

Columns sums gives: A000012, A000079, A007051, A004171, A034478, A081341, A034494, A092811, A083884, A093143

Table[Table[Ceiling[1/2*((k+1)^n+(1+(-1)^k)/2)],{n,0,9},{k,0,9}]]

cp's OEIS Frontend

A261399 a(1)=1; thereafter a(n) = (2/5)*(9*6^(n-2)+1).

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A382491 a(n) is the numerator of the asymptotic density of the numbers whose number of 3-smooth divisors is n.

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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.

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Formula

A191687 Table T(n,k) = ceiling((1/2)*((k+1)^n+(1+(-1)^k)/2)) read by antidiagonals.

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Mathematica

A261399 a(1)=1; thereafter a(n) = (2/5)(96^(n-2)+1).