A135423 -id:A135423 - cp's OEIS frontend

A057198 a(n) = (5*3^(n-1)+1)/2.

3, 8, 23, 68, 203, 608, 1823, 5468, 16403, 49208, 147623, 442868, 1328603, 3985808, 11957423, 35872268, 107616803, 322850408, 968551223, 2905653668, 8716961003, 26150883008, 78452649023, 235357947068, 706073841203, 2118221523608

Offset: 1

Colin Mallows and N. J. A. Sloane, Sep 16 2000

It appears that if s(n) is a first-order rational sequence of the form s(0)=4, s(n) = (2*s(n-1)+1)/(s(n-1)+2), n > 0, then s(n) = a(n)/(a(n)-1), n > 0.

			G.f. = 3*x + 8*x^2 + 23*x^3 + 68*x^4 + 203*x^5 + 608*x^6 + 1823*x^7 + 5468*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv:1611.08244 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Related to A046901.
Equals A060816 + 1.
Cf. A135423 (bisection), A191450 (2nd row).

[(5*3^(n-1) + 1)/2: n in [1..30]]; // Vincenzo Librandi, Oct 11 2012

Table[(5*3^(n-1) + 1)/2, {n, 30}] (* T. D. Noe, Oct 11 2012 *)

a(n)=(5*3^(n-1)+1)/2 \\ Charles R Greathouse IV, Oct 11 2012

a(n+1) = 3*a(n) - 1 for n > 1. - Reinhard Zumkeller, Jan 22 2011
G.f.: (5/2)*U(0) where U(k) = 1 + 2/(5*3^k + 5*3^k/(1 - 30*x*3^k/(15*x*3^k - 1/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 01 2012
E.g.f.: (5/2)*U(0) where U(k) = 1 + 2/(5*3^k + 5*3^k/(1 - 30*x*3^k/(15*x*3^k - (k+1)/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 01 2012
G.f.: x*(3-4*x) / ( (3*x-1)*(x-1) ). - R. J. Mathar, Jan 25 2015
E.g.f.: (5*exp(3*x) + 3*exp(x) - 8)/6. - Stefano Spezia, Aug 28 2023

Incorrect zeroth term removed by Jon Perry, Oct 11 2012

A138894 Expansion of (1+x)/(1-10x+9x^2).

Original entry on oeis.org

1, 11, 101, 911, 8201, 73811, 664301, 5978711, 53808401, 484275611, 4358480501, 39226324511, 353036920601, 3177332285411, 28595990568701, 257363915118311, 2316275236064801, 20846477124583211, 187618294121248901

Offset: 0

Paul Barry, Apr 02 2008

Orbit starting at 1 of A138893: a(n)=A138893^(n)(1). Partial sums of A003952.

Sum of n-th row of triangle of powers of 9: 1; 1 9 1; 1 9 81 9 1; 1 9 81 729 81 9 1; ... - Philippe Deléham, Feb 22 2014

			a(0) = 1;
a(1) = 1 + 9 + 1 = 11;
a(2) = 1 + 9 + 81 + 9 + 1 = 101;
a(3) = 1 + 9 + 81 + 729 + 81 + 9 + 1 = 911; etc. - _Philippe Deléham_, Feb 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A096053 ((3*9^n-1)/2), a(n+1)=9a(n)-4 in A135423.

Cf. A002452, A003952, A135522, A138893.

[(5/4)*9^n-1/4: n in [0..20]]; // Vincenzo Librandi, Aug 09 2011

Table[(5*9^n - 1)/4, {n, 0, 18}] (* L. Edson Jeffery, Feb 13 2015 *)

Vec((1+x)/(1-10*x+9*x^2) + O(x^30)) \\ Michel Marcus, Feb 13 2015

G.f.: (1+x)/((1-x)*(1-9x)).
a(n) = (5/4)*9^n - 1/4.
a(n) = A002452(n) + A002452(n+1).
Bisection of A135522/3. a(n+1)=9*a(n)+2. - Paul Curtz, Apr 22 2008
a(n) = Sum_{k=0..n} A112468(n,k)*10^k. - Philippe Deléham, Feb 22 2014

A199563 5*9^n+1.

Original entry on oeis.org

6, 46, 406, 3646, 32806, 295246, 2657206, 23914846, 215233606, 1937102446, 17433922006, 156905298046, 1412147682406, 12709329141646, 114383962274806, 1029455660473246, 9265100944259206, 83385908498332846, 750473176484995606

Offset: 0

Vincenzo Librandi, Nov 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Magma
```
[5*9^n+1: n in [0..30]];
```

A199563:=n->5*9^n+1; seq(A199563(k), k=0..50); # Wesley Ivan Hurt, Oct 24 2013

Table[5*9^n+1, {n,0,50}] (* Wesley Ivan Hurt, Oct 24 2013 *)

a(n) = 2*A135423(n).
a(n) = 9*a(n-1)-8.
a(n) = 10*a(n-1)-9*a(n-2).
G.f.: 2*(3-7*x)/((1-x)*(1-9*x)).

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.

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A057198 a(n) = (5*3^(n-1)+1)/2.

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A138894 Expansion of (1+x)/(1-10*x+9*x^2).

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A199563 5*9^n+1.

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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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A138894 Expansion of (1+x)/(1-10x+9x^2).