A048654 -id:A048654 - cp's OEIS frontend

A102662 Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).

1, 1, 3, 1, 5, 3, 1, 7, 11, 3, 1, 9, 23, 17, 3, 1, 11, 39, 51, 23, 3, 1, 13, 59, 113, 91, 29, 3, 1, 15, 83, 211, 255, 143, 35, 3, 1, 17, 111, 353, 579, 489, 207, 41, 3, 1, 19, 143, 547, 1143, 1323, 839, 283, 47, 3, 1, 21, 179, 801, 2043, 3045, 2651, 1329, 371, 53, 3, 1, 23, 219

Offset: 1

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 03 2005

Generalization of A008288 (use initial terms 1,1,3). Triangle seen as lower triangular matrix: The absolute values of the coefficients of the characteristic polynomials of the n X n matrix are the (n+1)th row of A038763. Row sums give A048654.

			Triangle begins:
  1
  1 3
  1 5 3
  1 7 11 3
  1 9 23 17 3

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Reinhard Zumkeller, Rows n=0..149 of triangle, flattened
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 37.

Cf. A038763, A048654, A008288.

a102662 n k = a102662_tabl !! n !! k
a102662_row n = a102662_tabl !! n
a102662_tabl = [1] : [1,3] : f [1] [1,3] where
   f xs ys = zs : f ys zs where
     zs = zipWith (+) ([0] ++ xs ++ [0]) $
                      zipWith (+) ([0] ++ ys) (ys ++ [0])
-- Reinhard Zumkeller, Feb 23 2012

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207624 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A102662 *)
(* Clark Kimberling, Feb 20 2012 *)

T(k,r)=if(r>k,0,if(k==1,1,if(k==2,if(r==1,1,3),if(r==1,1,if(r==k,3,T(k-1,r-1)+T(k-1,r)+T(k-2,r-1))))))
BM(n) = M=matrix(n,n);for(i=1,n, for(j=1,n,M[i,j]=T(i,j)));M
M=BM(10)
for(i=1,10,s=0;for(j=1,i,s+=M[i,j]);print1(s,","))

From Clark Kimberling, Feb 20 2012: (Start)
A102662=v and A207624=u, defined together as follows:
u(n,x)=u(n-1,x)+v(n-1,x), v(n,x)=2*x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1; see the Mathematica section. (End)

A117584 Generalized Pellian triangle.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 7, 12, 1, 5, 9, 17, 29, 1, 6, 11, 22, 41, 70, 1, 7, 13, 27, 53, 99, 169, 1, 8, 15, 32, 65, 128, 239, 408, 1, 9, 17, 37, 77, 157, 309, 577, 985, 1, 10, 19, 42, 89, 186, 379, 746, 1393, 2378

Offset: 1

Gary W. Adamson, Mar 29 2006

			First few rows of the triangle are:
  1;
  1, 2;
  1, 3,  5;
  1, 4,  7, 12;
  1, 5,  9, 17, 29;
  1, 6, 11, 22, 41, 70;
  1, 7, 13, 27, 53, 99, 169;
  ...
The triangle rows are antidiagonals of the generalized Pellian array:
  1, 2,  5, 12, 29, ...
  1, 3,  7, 17, 41, ...
  1, 4,  9, 22, 53, ...
  1, 5, 11, 27, 65, ...
  ...
For example, in the row (1, 5, 11, 27, 65, ...), 65 = 2*27 + 11.

G. C. Greubel, Rows n = 1..100, flattened

Diagonals include A000129, A001333, A048654, A048655, A048693.

Cf. A117185.

P:= func< n | Round( ((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2)) ) >;
T:= func< n,k | P(k) + (n-1)*P(k-1) >;
[T(n-k+1, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 05 2021

T[n_, k_]:= Fibonacci[k, 2] + (n-1)*Fibonacci[k-1, 2];
Table[T[n-k+1, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 05 2021 *)

def T(n,k): return lucas_number1(k,2,-1) + (n-1)*lucas_number1(k-1,2,-1)
flatten([[T(n-k+1, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 05 2021

Antidiagonals of the generalized Pellian array. First row of the array = A000129: (1, 2, 5, 12, ...). n-th row of the array starts (1, n+1, ...); as a Pellian sequence.
From G. C. Greubel, Jul 05 2021: (Start)
T(n, k) = P(k) + (n-1)*P(k-1), where P(n) = A000129(n) (square array).
Sum_{k=1..n} T(n-k+1, k) = A117185(n). (End)

A209704 Triangle of coefficients of polynomials v(n,x) jointly generated with A209703; see the Formula section.

Original entry on oeis.org

1, 3, 1, 4, 3, 2, 5, 6, 8, 3, 6, 10, 18, 14, 5, 7, 15, 33, 38, 27, 8, 8, 21, 54, 81, 83, 49, 13, 9, 28, 82, 150, 197, 170, 89, 21, 10, 36, 118, 253, 401, 448, 342, 159, 34, 11, 45, 163, 399, 736, 999, 987, 671, 282, 55, 12, 55, 218, 598, 1253, 1988, 2387, 2106

Offset: 1

Clark Kimberling, Mar 12 2012

For n>1, row n starts with n+1, followed by the n-th
triangular number, and ends in F(n+1), where F=A000045
(Fibonacci numbers).
Column 3:  A166830.
Row sums:  A048654.
Alternating row sums: 1,2,3,4,5,6,7,8,9,...
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
3...1
4...3....2
5...6....8....3
6...10...18...14...5
First three polynomials v(n,x): 1, 3 + x , 4 + 3x + 2x^2.

Cf. A209703, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A209703 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A209704 *)

u(n,x)=x*u(n-1,x)+x*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A266505 a(n) = 2*a(n - 2) + a(n - 4) with a(0) = -1, a(1) = 1, a(2) = 3, a(3) = 5.

Original entry on oeis.org

-1, 1, 3, 5, 5, 11, 13, 27, 31, 65, 75, 157, 181, 379, 437, 915, 1055, 2209, 2547, 5333, 6149, 12875, 14845, 31083, 35839, 75041, 86523, 181165, 208885, 437371, 504293, 1055907, 1217471, 2549185, 2939235, 6154277, 7095941, 14857739, 17131117, 35869755, 41358175, 86597249, 99847467

Offset: 0

Raphie Frank, Dec 30 2015

a(n)/A266504(n) converges to sqrt(2).
Alternatively, bisection of A266506.
Alternatively, A135532(n) and A048655(n) interlaced.
Alternatively, A255236(n-1), A054490(n), A038762(n) and A101386(n) interlaced.
Let b(n) = (a(n) - (a(n) mod 2))/2, that is b(n) = {-1, 0, 1, 2, 2, 5, 6, 13, 15, 32, 37, 78, 90, ...}. Then:
A006451(n) = {b(4n+0) U b(4n+1)} gives n in N such that triangular(n) + 1 is square;
A216134(n) = {b(4n+2) U b(4n+3)} gives n in N such that triangular(n) follows form n^2 + n + 1 (twice a triangular number + 1).

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

Cf. A000129, A001333, A002203, A002965, A006451, A006452, A002965, A038761, A038762, A048654, A048655, A054490, A078343, A098586, A098790, A100525, A101386, A135532, A216134, A216162, A253811, A255236, A266504, A266505, A266507.

I:=[-1,1,3,5]; [n le 4 select I[n] else 2*Self(n-2)+Self(n-4): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015

a:=proc(n) option remember; if n=0 then -1 elif n=1 then 1 elif n=2 then 3 elif n=3 then 5 else 2*a(n-2)+a(n-4); fi; end:  seq(a(n), n=0..50); # Wesley Ivan Hurt, Jan 01 2016

LinearRecurrence[{0, 2, 0, 1}, {-1, 1, 3, 5}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[SeriesCoefficient[(-1 + 3 x) (1 + x)^2/(1 - 2 x^2 - x^4), {x, 0, n}], {n, 0, 42}] (* Michael De Vlieger, Dec 31 2015 *)

my(x='x+O('x^40)); Vec((-1+3*x)*(1+x)^2/(1-2*x^2-x^4)) \\ G. C. Greubel, Jul 26 2018

G.f.: (-1 + 3*x)*(1 + x)^2/(1 - 2*x^2 - x^4).
a(n) = (-(1+sqrt(2))^floor(n/2)*(-1)^n - sqrt(8)*(1-sqrt(2))^floor(n/2) - (1-sqrt(2))^floor(n/2)*(-1)^n + sqrt(8)*(1+sqrt(2))^floor(n/2))/2.
a(n) = 3*(((1+sqrt(2))^floor(n/2)-(1-sqrt(2))^floor(n/2))/sqrt(8)) - (-1)^n*(((1+sqrt(2))^(floor(n/2)-(-1)^n)-(1-sqrt(2))^(floor(n/2)-(-1)^n))/sqrt(8)).
a(n) = (3*A000129(floor(n/2)) - A000129(n-(-1)^n)), where A000129 gives the Pell numbers.
a(n) = sqrt(2*A266504(n)^2 - 7*(-1)^A266504(n))*sgn(2*n-1), where A266504 gives all x in N such that 2*x^2 - 7*(-1)^x = y^2. This sequence gives associated y values.
a(2n) = (-(1 + sqrt(2))^n - sqrt(8)*(1 - sqrt(2))^n - (1 - sqrt(2))^n + sqrt(8)*(1 + sqrt(2))^n)/2 = a(2n) = A135532(n).
a(2n) = 3*(((1+sqrt(2))^n-(1-sqrt(2))^n)/sqrt(8)) - (((1+sqrt(2))^(n-1)-(1-sqrt(2))^(n-1))/sqrt(8)) = A135532(n).
a(2n+1) = (+(1 + sqrt(2))^n - sqrt(8)*(1 - sqrt(2))^n + (1 - sqrt(2))^n + sqrt(8)*(1 + sqrt(2))^n)/2 = a(2n + 1) = A048655(n).
a(2n+1) = 3*(((1+sqrt(2))^n-(1-sqrt(2))^n)/sqrt(8)) + (((1+sqrt(2))^(n+1)-(1-sqrt(2))^(n+1))/sqrt(8)) = A048655(n).
a(4n + 0) = 6*a(4n - 4) - a(4n - 8) = A255236(n-1).
a(4n + 1) = 6*a(4n - 3) - a(4n - 7) = A054490(n).
a(4n + 2) = 6*a(4n - 2) - a(4n - 6) = A038762(n).
a(4n + 3) = 6*a(4n - 1) - a(4n - 5) = A101386(n).
(sqrt(2*(a(2n + 1) )^2 + 14*(-1)^floor(n/2)))/2 = A266504(n).
(a(2n + 1) + a(2n))/8 = A000129(n), where A000129 gives the Pell numbers.
a(2n + 1) - a(2n) = A002203(n), where A002203 gives the companion Pell numbers.
(a(2n + 2) + a(2n + 1))/2 = A000129(n+2).
(a(2n + 2) - a(2n + 1))/2 = A000129(n-1).

A048695 Generalized Pellian with second term equal to 8.

Original entry on oeis.org

1, 8, 17, 42, 101, 244, 589, 1422, 3433, 8288, 20009, 48306, 116621, 281548, 679717, 1640982, 3961681, 9564344, 23090369, 55745082, 134580533, 324906148, 784392829, 1893691806, 4571776441, 11037244688

Offset: 0

Barry E. Williams

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,1)

Cf. A001333, A000129, A048654, A048655.

with(combinat): a:=n->6*fibonacci(n-1,2)+fibonacci(n,2): seq(a(n), n=1..26); # Zerinvary Lajos, Apr 04 2008

a[n_]:=(MatrixPower[{{1,2},{1,1}},n].{{7},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{2,1},{1,8},30] (* Harvey P. Dale, May 01 2013 *)

a(n) = 2*a(n-1) + a(n-2); a(0)=1, a(1)=8.
a(n) = ((7+sqrt(2))(1+sqrt(2))^n - (7-sqrt(2))(1-sqrt(2))^n)/2*sqrt(2).
G.f.: (1+6*x)/(1 - 2*x - x^2). - Philippe Deléham, Nov 03 2008

A117895 Triangle T(n, k) = (k-n)A000129(k+1) + (3n-3k+1)A000129(k) with T(n,0) = 1, for 0 <= k <= n-1, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 4, 8, 1, 5, 5, 11, 19, 1, 6, 6, 14, 26, 46, 1, 7, 7, 17, 33, 63, 111, 1, 8, 8, 20, 40, 80, 152, 268, 1, 9, 9, 23, 47, 97, 193, 367, 647, 1, 10, 10, 26, 54, 114, 234, 466, 886, 1562, 1, 11, 11, 29, 61, 131, 275, 565, 1125, 2139, 3771, 1, 12, 12, 32, 68, 148, 316, 664, 1364, 2716, 5164, 9104

Offset: 0

Gary W. Adamson, Mar 30 2006

Successive deletions of the right borders of triangle A117894 produces triangles whose row sums = generalized Pell sequences starting (1, 2...), (1, 3...), (1, 4...); etc. Row sums of A117894 = A000129: (1, 2, 5...). Row sums of A117895 = A001333: (1, 3, 7...). Deletion of the border of A117895 would produce a triangle with row sums of the Pell sequence A048654 (1, 4, 9...); and so on.

			First few rows of the triangle are:
  1;
  1, 2;
  1, 3, 3;
  1, 4, 4,  8;
  1, 5, 5, 11, 19;
  1, 6, 6, 14, 26, 46;
  1, 7, 7, 17, 33, 63, 111;
  1, 8, 8, 20, 40, 80, 152, 268;
...
Row 4, (1, 4, 4, 8) is produced by adding (0, 1, 1, 3) to row 4 of A117894: (1, 3, 3, 5).

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

Cf. A000129, A001333, A048654, A048655, A048693, A117894.

Pell:= func< n | Round(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2))) >;
[k eq 0 select 1 else (k-n)*Pell(k+1) + (3*n-3*k+1)*Pell(k): k in [0..n-1], n in [0..12]]; // G. C. Greubel, Sep 27 2021

T[n_, k_]:= T[n, k]= If[k==0, 1, (k-n)*Fibonacci[k+1, 2] + (3*n-3*k +1)*Fibonacci[k, 2]]; Table[T[n, k], {n,0,12}, {k,0,n-1}]//Flatten (* G. C. Greubel, Sep 27 2021 *)

def P(n): return lucas_number1(n, 2, -1)
def A117895(n,k): return 1 if (k==0) else (k-n)*P(k+1) + (3*n-3*k+1)*P(k)
flatten([[A117895(n,k) for k in (0..n-1)] for n in (0..12)]) # G. C. Greubel, Sep 27 2021

Delete right border of triangle A117894. Alternatively, let row 1 = 1 and using the heading 0, 1, 1, 3, 7, 17, 41, 99, 239...(i.e. A001333 starting with 0, 1, 1, 3...); add the first n terms of the heading to n-th row of triangle A117894.
From G. C. Greubel, Sep 27 2021: (Start)
T(n, k) = (k-n)*A000129(k+1) + (3*n-3*k+1)*A000129(k) with T(n,0) = 1.
T(n, 1) = n+1 for n >= 1.
T(n, 2) = n+1 for n >= 2.
T(n, n) = 2*[n=0] + A078343(n). (End)

New name and more terms added by G. C. Greubel, Sep 27 2021

A182001 Riordan array ((2*x+1)/(1-x-x^2), x/(1-x-x^2)).

Original entry on oeis.org

1, 3, 1, 4, 4, 1, 7, 9, 5, 1, 11, 20, 15, 6, 1, 18, 40, 40, 22, 7, 1, 29, 78, 95, 68, 30, 8, 1, 47, 147, 213, 185, 105, 39, 9, 1, 76, 272, 455, 466, 320, 152, 49, 10, 1, 123, 495, 940, 1106, 891, 511, 210, 60, 11, 1, 199, 890, 1890, 2512, 2317, 1554, 770, 280, 72, 12, 1

Offset: 0

Philippe Deléham, Apr 05 2012

Subtriangle of the triangle given by (0, 3, -5/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Antidiagonal sums are A001045(n+2).

			Triangle begins :
    1;
    3,   1;
    4,   4,    1;
    7,   9,    5,    1;
   11,  20,   15,    6,    1;
   18,  40,   40,   22,    7,    1;
   29,  78,   95,   68,   30,    8,   1;
   47, 147,  213,  185,  105,   39,   9,   1;
   76, 272,  455,  466,  320,  152,  49,  10, 1;
  123, 495,  940, 1106,  891,  511, 210,  60, 11,  1;
  199, 890, 1890, 2512, 2317, 1554, 770, 280, 72, 12, 1;
(0, 3, -5/3, -1/3, 0, 0, ...) DELTA (1, 0, -2/3, 2/3, 0, 0, ...) begins:
  1;
  0,  1;
  0,  3,  1;
  0,  4,  4,  1;
  0,  7,  9,  5,  1;
  0, 11, 20, 15,  6, 1;
  0, 18, 40, 40, 22, 7, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. Columns : A000032, A023607, A152881

function T(n,k)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  elif k eq 0 then return Lucas(n+1);
  else return T(n-1,k) + T(n-1,k-1) + T(n-2,k);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020

with(combinat);
T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=n then 1
    elif k=0 then fibonacci(n+2) + fibonacci(n)
    else T(n-1,k) + T(n-1,k-1) + T(n-2,k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Feb 18 2020

With[{m = 10}, CoefficientList[CoefficientList[Series[(1+2*x)/(1-x-y*x-x^2), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 18 2020 *)
T[n_, k_]:= T[n, k]= If[k<0||k>n, 0, If[k==n, 1, If[k==0, LucasL[n+1], T[n-1, k] + T[n-1, k-1] + T[n-2, k] ]]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2020 *)

G.f.: (1+2*x)/(1-x-y*x-x^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = 3, T(n,k) = 0 if k<0 or if k>n.
Sum_{k=0..nn} T(n,k)*x^k = A000034(n), A000032(n+1), A048654(n), A108300(n), A048875(n) for x = -1, 0, 1, 2, 3 respectively.

a(29) corrected by and a(55)-a(65) from Georg Fischer, Feb 18 2020

A375726 a(n) = a(n-1) + 3*a(n-2) + a(n-3) with a(0) = 1, a(1) = 3, a(2) = 6.

Original entry on oeis.org

1, 3, 6, 16, 37, 91, 218, 528, 1273, 3075, 7422, 17920, 43261, 104443, 252146, 608736, 1469617, 3547971, 8565558, 20679088, 49923733, 120526555, 290976842, 702480240, 1695937321, 4094354883, 9884647086, 23863649056, 57611945197, 139087539451, 335787024098

Offset: 0

Yifan Xie, Aug 25 2024

a(n) is the number of subsets T of A = {1, 2, ..., 2*n} such that no pair of elements a, b of T satisfy |a-b| = 1 or n.

			For n = 2, the a(2) = 6 subsets of {1, 2, 3, 4} are {}, {1}, {2}, {3}, {4}, {1, 4}.

Yifan Xie, Table of n, a(n) for n = 0..2612
Yifan Xie, Proof for the comment
Index entries for linear recurrences with constant coefficients, signature (1,3,1).

Cf. A001333, A000129, A097076.

LinearRecurrence[{1, 3, 1}, {1, 3, 6}, 31] (* Hugo Pfoertner, Aug 26 2024 *)

my(a=1, b=3, c=6); for(n=1, 31, print1(a, ", "); my(d=a+3*b+c); a=b; b=c; c=d)

a(n) = (1/4)*((3 + sqrt(2))*(1+sqrt(2))^n + (3 - sqrt(2))*(1-sqrt(2))^n-2*(-1)^n).
For n >= 2, a(n) = 2*a(n-1) + a(n-2) - (-1)^n.
From Stefano Spezia, Aug 26 2024: (Start)
G.f.: (1 + 2*x)/((1 + x)*(1 - 2*x - x^2)).
E.g.f.: (sinh(x) - cosh(x) + exp(x)*(3*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)))/2. (End)
Let f = (3 - sqrt(2))*exp((1 - sqrt(2))*x) + (3 + sqrt(2))*exp((1 + sqrt(2))*x), then 4*a(n) + 2*(-1)^n = n! * [x^n] f. - Peter Luschny, Sep 10 2024
a(n)+a(n-1) = A048654(n). - R. J. Mathar, Sep 27 2024

A228894 Nodes of tree generated as follows: (2,1) is an edge, and if (x,y) is an edge, then (y,y+x) and (y,2y+x) are edges.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 22, 23, 24, 25, 26, 27, 29, 31, 32, 33, 35, 37, 40, 41, 43, 44, 47, 48, 51, 52, 53, 55, 56, 57, 58, 60, 61, 63, 64, 65, 67, 68, 69, 71, 75, 76, 78, 79, 80, 83, 84, 85, 87, 88, 89, 91, 92, 93, 97, 98, 99

Offset: 1

Clark Kimberling, Sep 08 2013

The tree has infinitely many branches which are essentially linear recurrence sequences (and infinitely many which are not). For example, the branch 2->1->3->4->7->11-> contributes the Lucas sequence, A000032. The other extreme branch, 1->4->9->22->53-> contributes A048654.

			Taking the first generation of edges to be G(1) = {(2,1)}, the edge (2,1) grows G(2) = {(1,3), (1,4)}, which grows G(3) = {(3,4), (3,7), (4,5), (4,9)}, ... Expelling duplicate nodes and sorting leave (1,2,3,4,5,7,9,...).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A228853.

f[x_, y_] := {{y, x + y}, {y, x + 2 y}}; x = 2; y = 1; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {12}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]

A048755 Partial sums of A048693.

Original entry on oeis.org

1, 7, 20, 52, 129, 315, 764, 1848, 4465, 10783, 26036, 62860, 151761, 366387, 884540, 2135472, 5155489, 12446455, 30048404, 72543268, 175134945, 422813163, 1020761276, 2464335720, 5949432721, 14363201167, 34675835060, 83714871292, 202105577649, 487926026595

Offset: 0

Barry E. Williams

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Cf. A048693, A048654, A048655.

Accumulate[LinearRecurrence[{2,1},{1,6},30]] (* or *) LinearRecurrence[ {3,-1,-1},{1,7,20},40] (* Harvey P. Dale, Mar 29 2013 *)

a(n)=2*a(n-1)+a(n-2)+5; a(0)=1, a(1)=6.
a(n)=[ {(6+(7/2)*sqrt(2))(1+sqrt(2))^n - (6-(7/2)*sqrt(2))(1-sqrt(2))^n}/ 2*sqrt(2) ]-5/2.
G.f. ( 1+4*x ) / ( (x-1)*(x^2+2*x-1) ). - R. J. Mathar, Nov 08 2012
a(0)=1, a(1)=7, a(2)=20, a(n)=3*a(n-1)-a(n-2)-a(n-3). - Harvey P. Dale, Mar 29 2013

More terms from Harvey P. Dale, Mar 29 2013

cp's OEIS Frontend

A102662 Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).

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A117584 Generalized Pellian triangle.

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A209704 Triangle of coefficients of polynomials v(n,x) jointly generated with A209703; see the Formula section.

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A266505 a(n) = 2*a(n - 2) + a(n - 4) with a(0) = -1, a(1) = 1, a(2) = 3, a(3) = 5.

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A048695 Generalized Pellian with second term equal to 8.

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A117895 Triangle T(n, k) = (k-n)*A000129(k+1) + (3*n-3*k+1)*A000129(k) with T(n,0) = 1, for 0 <= k <= n-1, read by rows.

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A182001 Riordan array ((2*x+1)/(1-x-x^2), x/(1-x-x^2)).

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A117895 Triangle T(n, k) = (k-n)A000129(k+1) + (3n-3k+1)A000129(k) with T(n,0) = 1, for 0 <= k <= n-1, read by rows.