A083086 -id:A083086 - cp's OEIS frontend

A139818 Squares of Jacobsthal numbers.

0, 1, 1, 9, 25, 121, 441, 1849, 7225, 29241, 116281, 466489, 1863225, 7458361, 29822521, 119311929, 477204025, 1908903481, 7635439161, 30542106169, 122167725625, 488672300601, 1954686406201, 7818751217209, 31274993684025

Offset: 0

Paul Curtz, May 17 2008

Run length transform gives A246035. - N. J. A. Sloane, Feb 26 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249, 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (3,6,-8).

Cf. A001045, A246035. First differences give (apart from signs) A083086.

[1/9-(2/9)*(-2)^n+(1/9)*4^n: n in [0..35]]; // Vincenzo Librandi, Aug 09 2011

LinearRecurrence[{3, 6, -8}, {0, 1, 1}, 25] (* Jean-François Alcover, Jan 09 2019 *)

concat (0, Vec(x*(1-2*x)/((1-x)*(1+2*x)*(1-4*x)) + O(x^30))) \\ Michel Marcus, Mar 04 2015

a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3).
a(n) = (A001045(n))^2.
G.f.: x*(1-2*x)/((1-x)*(1+2*x)*(1-4*x)).

More terms from R. J. Mathar, Dec 12 2009

A192382 Coefficient of x in the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

0, 2, 4, 24, 80, 352, 1344, 5504, 21760, 87552, 349184, 1398784, 5591040, 22372352, 89473024, 357924864, 1431633920, 5726666752, 22906404864, 91626143744, 366503526400, 1466016202752, 5864060616704, 23456250855424, 93824986644480

Offset: 1

Clark Kimberling, Jun 30 2011

nonn

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+2). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+2, see A192232.

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0, x) = 1 -> 1.
  p(1, x) = 2*x -> 2*x.
  p(2, x) = 2 + x + 3*x^2 -> 8 + 4*x.
  p(3, x) = 8*x + 4*x^2 + 4*x^3 -> 16 + 24*x.
  p(4, x) = 4 + 4*x + 21*x^2 + 10*x^3 + 5*x^4 -> 96 + 80*x.
From these, read A083086 = (1, 0, 9, 16, 96, ...) and A192382 =(0, 2, 4, 24, 80, ...).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A003683, A083086, A162517, A192232.

[(4^(n-1) - (-2)^(n-1))/3: n in [1..40]]; // G. C. Greubel, Feb 19 2023

seq(4^n*(1-(-1/2)^n)/3, n=0..24); # Peter Luschny, Oct 02 2019

q[x_]:= x+2; d= Sqrt[x+2];
p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2 d); (* suggested by A162517 *)
Table[Expand[p[n, x]], {n, 6}]
reductionRules= {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x*q[x]^((y- 1)/2)};
t = Table[FixedPoint[Expand[#1/. reductionRules] &, p[n,x]], {n,30}];
Table[Coefficient[Part[t,n], x, 0], {n,30}] (* abs value of A083086 *)
Table[Coefficient[Part[t,n], x, 1], {n,30}] (* 2*A003683 *)
Table[Coefficient[Part[t,n]/2, x, 1], {n,30}] (* A003683 *)
LinearRecurrence[{2,8}, {0,2}, 40] (* G. C. Greubel, Feb 19 2023 *)

[(4^(n-1) - (-2)^(n-1))/3 for n in range(1,41)] # G. C. Greubel, Feb 19 2023

Conjectures from Colin Barker, May 12 2014: (Start)
a(n) = 2^(n-2)*(2*(-1)^n + 2^n)/3 = 2*A003683(n-1).
a(n) = 2*a(n-1) + 8*a(n-2).
G.f.: 2*x^2 / ((1+2*x)*(1-4*x)). (End).
a(n) = 4^n*(1 - (-1/2)^n)/3. - Peter Luschny, Oct 02 2019
E.g.f: (1/3)*(2 + exp(2*x))*(sinh(x))^2. - G. C. Greubel, Feb 19 2023

A140944 Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.

Original entry on oeis.org

0, 1, 0, -1, 2, 0, 3, -2, 4, 0, -5, 6, -4, 8, 0, 11, -10, 12, -8, 16, 0, -21, 22, -20, 24, -16, 32, 0, 43, -42, 44, -40, 48, -32, 64, 0, -85, 86, -84, 88, -80, 96, -64, 128, 0, 171, -170, 172, -168, 176, -160, 192, -128, 256, 0, -341, 342, -340, 344, -336, 352, -320, 384, -256, 512, 0

Offset: 0

Paul Curtz, Jul 24 2008

A variant of the triangle A140503, now including the diagonal.

Since the diagonal contains zeros, rows sums are those of A140503.

			Triangle begins as:
    0;
    1,   0;
   -1,   2,   0;
    3,  -2,   4,  0;
   -5,   6,  -4,  8,   0;
   11, -10,  12, -8,  16,  0;
  -21,  22, -20, 24, -16, 32,  0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000079, A001045, A016131, A045883, A052992, A083086, A084175.

Cf. A091056, A097038, A109765, A115489, A140503, A192382, A246036.

[2^k*(1-(-2)^(n-k))/3: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 18 2023

A001045:= n -> (2^n-(-1)^n)/3;
A140944:= proc(n,k) if n = 0 then A001045(k); else procname(n-1,k+1)-procname(n-1,k) ; fi; end:
seq(seq(A140944(n,k),k=0..n),n=0..10); # R. J. Mathar, Sep 07 2009

T[0, 0]=0; T[1, 0]= T[0, 1]= 1; T[0, k_]:= T[0, k]= T[0, k-1] + 2*T[0, k-2]; T[n_, n_]=0; T[n_, k_]:= T[n, k] = T[n-1, k+1] - T[n-1, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Dec 17 2014 *)
Table[2^k*(1-(-2)^(n-k))/3, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2023 *)

T(n, k) = (2^k - 2^n*(-1)^(n+k))/3 \\ Jianing Song, Aug 11 2022

def A140944(n,k): return 2^k*(1 - (-2)^(n-k))/3
flatten([[A140944(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Feb 18 2023

T(n, k) = T(n-1, k+1) - T(n-1, k). T(0, k) = A001045(k).
T(n, k) = (2^k - 2^n*(-1)^(n+k))/3, for n >= k >= 0. - Jianing Song, Aug 11 2022
From G. C. Greubel, Feb 18 2023: (Start)
T(n, n-1) = A000079(n).
T(2*n, n) = (-1)^(n+1)*A192382(n+1).
T(2*n, n-1) = (-1)^n*A246036(n-1).
T(2*n, n+1) = A083086(n).
T(3*n, n) = -A115489(n).
Sum_{k=0..n} T(n, k) = A052992(n)*[n>0] + 0*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = A045883(n).
Sum_{k=0..n} 2^k*T(n, k) = A084175(n).
Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^(n+1)*A109765(n).
Sum_{k=0..n} 3^k*T(n, k) = A091056(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^(n+1)*A097038(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^(n+1)*A138495(n). (End)

Edited and extended by R. J. Mathar, Sep 07 2009

A083085 a(n) = (2+(-5)^n)/3.

Original entry on oeis.org

1, -1, 9, -41, 209, -1041, 5209, -26041, 130209, -651041, 3255209, -16276041, 81380209, -406901041, 2034505209, -10172526041, 50862630209, -254313151041, 1271565755209, -6357828776041, 31789143880209, -158945719401041, 794728597005209, -3973642985026041

Offset: 0

Paul Barry, Apr 23 2003

Inverse binomial transform of A083086.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (-4,5).

Cf. A083086.

[(2+(-5)^n)/3: n in [0..30]]; // Vincenzo Librandi, Nov 12 2011

A083085:=n->(2+(-5)^n)/3: seq(A083085(n), n=0..30); # Wesley Ivan Hurt, Nov 02 2014

Table[(2 + (-5)^n)/3, {n, 0, 30}] (* Wesley Ivan Hurt, Nov 02 2014 *)

a(n) = (2+(-5)^n)/3.
G.f.: (1+3*x)/((1+5*x)*(1-x)).
E.g.f.: (2*exp(x)+exp(-5*x))/3.
a(n) = -5*a(n-1) + 4. - Vincenzo Librandi, Nov 12 2011

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A139818 Squares of Jacobsthal numbers.

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A192382 Coefficient of x in the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.

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A140944 Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.

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A083085 a(n) = (2+(-5)^n)/3.

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