A201630 -id:A201630 - cp's OEIS frontend

A062092 a(n) = 2*a(n-1) - (-1)^n for n > 0, a(0)=2.

2, 5, 9, 19, 37, 75, 149, 299, 597, 1195, 2389, 4779, 9557, 19115, 38229, 76459, 152917, 305835, 611669, 1223339, 2446677, 4893355, 9786709, 19573419, 39146837, 78293675, 156587349, 313174699, 626349397, 1252698795, 2505397589

Offset: 0

Amarnath Murthy, Jun 16 2001

Let A be the Hessenberg matrix of order n, defined by: A[1,j] = A[i,i] = 1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n>=1, a(n-1) = charpoly(A,3). - Milan Janjic, Jan 24 2010

T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 98.

Harry J. Smith, Table of n, a(n) for n = 0..200
Petro Kosobutskyy, The Collatz problem as a reverse n->0 problem on a graph tree formed from theta*2^n Jacobsthal-type numbers, arXiv:2306.14635 [math.GM], 2023.
Petro Kosobutskyy and Dariia Rebot, Collatz conjecture 3n+/-1 as a Newton binomial problem, Comp. Des. Sys. Theor. Prac., Lviv Nat'l Polytech. Univ. (Ukraine 2023) Vol. 5, No. 1, 137-145. See p. 140.
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000032, A001045, A002487, A005009, A078008, A154879, A201630.

Cf. A171160 (first differences).

[(7*2^n-(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Apr 04 2023

LinearRecurrence[{1, 2}, {2, 5}, 40] (* Jean-François Alcover, Aug 02 2021 *)

a(n) = (7*2^n - (-1)^n)/3; \\ Harry J. Smith, Aug 01 2009

[(7*2^n-(-1)^n)/3 for n in range(41)] # G. C. Greubel, Apr 04 2023

a(n) = a(n-1) + 2*a(n-2).
a(n) = (7*2^n - (-1)^n)/3.
a(n) = 2^(n+1) + A001045(n).
A002487(a(n)) = A000032(n+1).
G.f.: (2+3*x)/(1-x-2*x^2).
E.g.f.: (7*exp(2*x) - exp(-x))/3.
a(n) = Sum_{j=0..2} A001045(n-j) (sum of 3 consecutive elements of the Jacobsthal sequence). - Alexander Adamchuk, May 16 2006
From Paul Curtz, Jun 03 2022: (Start)
a(n) = A001045(n+3) - A078008(n).
a(n) = A078008(n+3) - A001045(n).
a(n) = A005009(n-1) - a(n-1) for n >= 1.
a(n) = a(n-2) + A005009(n-2) for n >= 2.
a(n) = A154879(n-2) + 3*A201630(n-2) for n >= 2. (End)

More terms from Jason Earls, Jun 18 2001

Additional comments from Michael Somos, Jun 24 2002

A199116 a(n) = 6*4^n + 1.

Original entry on oeis.org

7, 25, 97, 385, 1537, 6145, 24577, 98305, 393217, 1572865, 6291457, 25165825, 100663297, 402653185, 1610612737, 6442450945, 25769803777, 103079215105, 412316860417, 1649267441665, 6597069766657, 26388279066625, 105553116266497, 422212465065985, 1688849860263937

Offset: 0

Vincenzo Librandi, Nov 04 2011

Bisection (odd part) of A181565 and A201630. - Bruno Berselli, Dec 04 2011

First differences of A221130, a(n) = A221130(n+2) - A221130(n+1). - Jaroslav Krizek, Jan 02 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A140529, A181565, A201630, A221130.

Magma
```
[6*4^n+1: n in [0..30]];
```

6*4^Range[0,30]+1 (* or *) LinearRecurrence[{5,-4},{7,25},30] (* Harvey P. Dale, Apr 18 2024 *)

a(n) = 4*a(n-1) - 3.
a(n) = 5*a(n-1) - 4*a(n-2).
G.f.: (7-10*x)/((1-x)*(1-4*x)). - Bruno Berselli, Nov 04 2011
From Elmo R. Oliveira, May 08 2025: (Start)
E.g.f.: exp(x)*(6*exp(3*x) + 1).
a(n) = A140529(n) + 2. (End)

A259713 a(n) = 32^n - 2(-1)^n.

Original entry on oeis.org

1, 8, 10, 26, 46, 98, 190, 386, 766, 1538, 3070, 6146, 12286, 24578, 49150, 98306, 196606, 393218, 786430, 1572866, 3145726, 6291458, 12582910, 25165826, 50331646, 100663298, 201326590, 402653186, 805306366, 1610612738, 3221225470, 6442450946, 12884901886

Offset: 0

Paul Curtz, Jul 03 2015

Inverse binomial transform of 3^n, with 3 (second term) excluded.

a(n) mod 9 gives A010689.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000244, A005010, A007283, A010689, A096045, A140788, A182460, A201630.

[3*2^n-2*(-1)^n: n in [0..40]]; // Vincenzo Librandi, Jul 04 2015

Table[3 2^n - 2 (-1)^n, {n, 0, 50}] (* Vincenzo Librandi, Jul 04 2015 *)
LinearRecurrence[{1,2},{1,8},40] (* Harvey P. Dale, Aug 19 2020 *)

Vec(-(7*x+1)/((x+1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Jul 03 2015

a(n) = a(n-1) + 2*a(n-2) for n>1, a(0)=1, a(1)=8.
a(n) = 2*a(n-1) - 6*(-1)^n for n>0, a(0)=1.
a(4n+2) = 10*A182460(n); a(2n) = A096045(n), a(2n+1) = A140788(n).
a(n) = 3*A014551(n+1) - A201630(n).
a(n+2) - a(n) = a(n) + a(n+1) = A005010(n).
G.f.: -(7*x+1) / ((x+1)*(2*x-1)). - Colin Barker, Jul 03 2015

Typo in data fixed by Colin Barker, Jul 03 2015

A124040 Triangle read by rows: characteristic polynomials of certain matrices, see Mathematica program.

Original entry on oeis.org

3, 3, -1, 8, -6, 1, 20, -24, 9, -1, 45, -84, 50, -12, 1, 125, -275, 225, -85, 15, -1, 320, -864, 900, -468, 129, -18, 1, 845, -2639, 3339, -2219, 840, -182, 21, -1, 2205, -7896, 11756, -9528, 4610, -1368, 244, -24, 1, 5780, -23256, 39825, -38121, 22518, -8532, 2079, -315, 27, -1, 15125, -67650, 130975, -144660, 101065, -46746, 14525, -3000, 395, -30, 1

Offset: 1

Gary W. Adamson and Roger L. Bagula, Nov 04 2006

Matrices:
X 1 : {{3}},
X 2 : {{3, 1}, {1, 3}},
X 3 : {{3, 1, 1}, {1, 3, 1}, {1, 1, 3}},
X 4 : {{3, 1, 0, 1}, {1, 3, 1, 0}, {0, 1, 3, 1}, {1, 0, 1, 3}},
X 5 : {{3, 1, 0, 0, 1}, {1, 3, 1, 0, 0}, {0, 1, 3, 1, 0}, {0, 0, 1, 3, 1}, {1, 0, 0, 1, 3}}.

			Triangle begins:
     3;
     3,     -1;
     8,     -6,     1;
    20,    -24,     9,     -1;
    45,    -84,    50,    -12,     1;
   125,   -275,   225,    -85,    15,    -1;
   320,   -864,   900,   -468,   129,   -18,    1;
   845,  -2639,  3339,  -2219,   840,  -182,   21,   -1;
  2205,  -7896, 11756,  -9528,  4610, -1368,  244,  -24,  1;
  5780, -23256, 39825, -38121, 22518, -8532, 2079, -315, 27, -1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A010675, A099921, A201630, A370280.

T[n_, m_, d_]:= If[n==m, 3, If[n==m-1 || n==m+1, 1, If[(n==1 && m==d) || (n==d && m==1), 1, 0]]];
M[d_]:= Table[T[n, m, d], {n,d}, {m,d}];
Join[{M[1]}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d,12}]]

From G. C. Greubel, Feb 03 2025: (Start)
T(n, 1) = A099921(n-1) + 3*[n=1] - 2*[n=2] + 3*[n=3].
T(n, 2) = -(n-1)*Fibonacci(2*n-2).
T(n, 3) = (1/10)*(n-1)*(2*(n-1)*Fibonacci(2*n-1) - (n+2)*Fibonacci(2*n-2)).
T(n, 4) = (1/150)*(n-1)*(18*(n-1)*Fibonacci(2*n-1) - (5*n^2 - n + 18)*Fibonacci(2*n-2)).
T(n, 5) = (1/600)*(n-1)*(2*(n-1)*(n^2-2*n+24)*Fibonacci(2*n-1) - (n^3+15*n^2 -10*n+48)*Fibonacci(2*n-2)).
T(n, n) = (-1)^(n-1) + 2*[n=1].
T(n, n-1) = 3*(-1)^n*(n-1).
T(n, n-2) = (1/2)*(-1)^(n+1)*(n-1)*(9*n-20) + [n=3].
T(n, n-3) = (3/2)*(-1)^n*(n-1)*(n-3)*(3*n-8) + 2*[n=4].
T(n, n-4) = (1/8)*(-1)^(n-1)*n*(n-3)*(27*n^2-117*n+130) - 2*[n=5].
T(n, n-5) = (3/40)*(-1)^n*(n-1)*(n-4)*(n-5)*(27*n^2-195*n+362) + 2*[n=6].
T(n, n-6) = (1/240)*(-1)^(n-1)*(n-1)*(n-5)*(n-6)(243*n^3-2997*n^2+12528*n -17752) -2*[n=7].
T(2*n-1, n) = 2*(-1)^(n-1)*A370280(n-1) + [n=1].
Sum_{k=1..n} T(n, k) = A010675(n-1) + 3*[n=1] -2*[n=2] +3*[n=3].
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A201630(n-2) - A201630(n-1) + (1/4)*[n=1] + (7/2)*[n=2] + 2*[n=3].
(End)

cp's OEIS Frontend

A062092 a(n) = 2*a(n-1) - (-1)^n for n > 0, a(0)=2.

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A199116 a(n) = 6*4^n + 1.

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A259713 a(n) = 3*2^n - 2*(-1)^n.

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A124040 Triangle read by rows: characteristic polynomials of certain matrices, see Mathematica program.

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A259713 a(n) = 32^n - 2(-1)^n.