A099035 -id:A099035 - cp's OEIS frontend

A193649 Q-residue of the (n+1)st Fibonacci polynomial, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

1, 1, 3, 5, 15, 33, 91, 221, 583, 1465, 3795, 9653, 24831, 63441, 162763, 416525, 1067575, 2733673, 7003971, 17938661, 45954543, 117709185, 301527355, 772364093, 1978473511

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

Suppose that p=p(0)*x^n+p(1)*x^(n-1)+...+p(n-1)*x+p(n) is a polynomial of positive degree and that Q is a sequence of polynomials: q(k,x)=t(k,0)*x^k+t(k,1)*x^(k-1)+...+t(k,k-1)*x+t(k,k), for k=0,1,2,... The Q-downstep of p is the polynomial given by D(p)=p(0)*q(n-1,x)+p(1)*q(n-2,x)+...+p(n-1)*q(0,x)+p(n).

Since degree(D(p))

Example: let p(x)=2*x^3+3*x^2+4*x+5 and q(k,x)=(x+1)^k.

D(p)=2(x+1)^2+3(x+1)+4(1)+5=2x^2+7x+14

D(D(p))=2(x+1)+7(1)+14=2x+23

D(D(D(p)))=2(1)+23=25;

the Q-residue of p is 25.

We may regard the sequence Q of polynomials as the triangular array formed by coefficients:

t(0,0)

t(1,0)....t(1,1)

t(2,0)....t(2,1)....t(2,2)

t(3,0)....t(3,1)....t(3,2)....t(3,3)

and regard p as the vector (p(0),p(1),...,p(n)). If P is a sequence of polynomials [or triangular array having (row n)=(p(0),p(1),...,p(n))], then the Q-residues of the polynomials form a numerical sequence.

Following are examples in which Q is the triangle given by t(i,j)=1 for 0<=i<=j:

Q.....P...................Q-residue of P

1.....1...................A000079, 2^n

1....(x+1)^n..............A007051, (1+3^n)/2

1....(x+2)^n..............A034478, (1+5^n)/2

1....(x+3)^n..............A034494, (1+7^n)/2

1....(2x+1)^n.............A007582

1....(3x+1)^n.............A081186

1....(2x+3)^n.............A081342

1....(3x+2)^n.............A081336

1.....A040310.............A193649

1....(x+1)^n+(x-1)^n)/2...A122983

1....(x+2)(x+1)^(n-1).....A057198

1....(1,2,3,4,...,n)......A002064

1....(1,1,2,3,4,...,n)....A048495

1....(n,n+1,...,2n).......A087323

1....(n+1,n+2,...,2n+1)...A099035

1....p(n,k)=(2^(n-k))*3^k.A085350

1....p(n,k)=(3^(n-k))*2^k.A090040

1....A008288 (Delannoy)...A193653

1....A054142..............A101265

1....cyclotomic...........A193650

1....(x+1)(x+2)...(x+n)...A193651

1....A114525..............A193662

More examples:

Q...........P.............Q-residue of P

(x+1)^n...(x+1)^n.........A000110, Bell numbers

(x+1)^n...(x+2)^n.........A126390

(x+2)^n...(x+1)^n.........A028361

(x+2)^n...(x+2)^n.........A126443

(x+1)^n.....1.............A005001

(x+2)^n.....1.............A193660

A094727.....1.............A193657

(k+1).....(k+1)...........A001906 (even-ind. Fib. nos.)

(k+1).....(x+1)^n.........A112091

(x+1)^n...(k+1)...........A029761

(k+1)......A049310........A193663

(In these last four, (k+1) represents the triangle t(n,k)=k+1, 0<=k<=n.)

A051162...(x+1)^n.........A193658

A094727...(x+1)^n.........A193659

A049310...(x+1)^n.........A193664

A075362....A075362........A193665

Changing the notation slightly leads to the Mathematica program below and the following formulation for the Q-downstep of p: first, write t(n,k) as q(n,k). Define r(k)=Sum{q(k-1,i)*r(k-1-i) : i=0,1,...,k-1} Then row n of D(p) is given by v(n)=Sum{p(n,k)*r(n-k) : k=0,1,...,n}.

			First five rows of Q, coefficients of Fibonacci polynomials (A049310):
1
1...0
1...0...1
1...0...2...0
1...0...3...0...1
To obtain a(4)=15, downstep four times:
D(x^4+3*x^2+1)=(x^3+x^2+x+1)+3(x+1)+1: (1,1,4,5) [coefficients]
DD(x^4+3*x^2+1)=D(1,1,4,5)=(1,2,11)
DDD(x^4+3*x^2+1)=D(1,2,11)=(1,14)
DDDD(x^4+3*x^2+1)=D(1,14)=15.

Cf. A192872 (polynomial reduction), A193091 (polynomial augmentation), A193722 (the upstep operation and fusion of polynomial sequences or triangular arrays).

q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
f[n_, x_] := Fibonacci[n + 1, x];
p[n_, k_] := Coefficient[f[n, x], x, k]; (* A049310 *)
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 24}]    (* A193649 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]

Conjecture: G.f.: -(1+x)*(2*x-1) / ( (x-1)*(4*x^2+x-1) ). - R. J. Mathar, Feb 19 2015

A131056 A007318 * A131055.

Original entry on oeis.org

1, 3, 7, 17, 41, 97, 225, 513, 1153, 2561, 5633, 12289, 26625, 57345, 122881, 262145, 557057, 1179649, 2490369, 5242881, 11010049, 23068673, 48234497, 100663297, 209715201, 436207617, 905969665

Offset: 1

Gary W. Adamson, Jun 12 2007

			a(4) = 17 = (1, 3, 3, 1) dot (1, 2, 2, 4) = (1 + 6 + 6 + 4).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4)

Cf. A131055, A131054, A141222.

LinearRecurrence[{5,-8,4},{1,3,7,17},40] (* Harvey P. Dale, Apr 30 2022 *)

Vec(x*(1-2*x+2*x^3)/((1-x)*(1-2*x)^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2015

Binomial transform of A131055: (1, 2, 2, 4, 4, 6, 6, ...). A131056 = A131054 as an infinite lower triangular matrix * [1,2,3,...] as a vector.
G.f.: x*(1-2*x+2*x^3)/((1-x)*(1-2*x)^2); a(n)=-0^n/2+2^(n-1)*(n+1)+1. - Paul Barry, Jun 14 2008
a(n) = 2+A099035(n-1), n>1. - Juri-Stepan Gerasimov, Oct 02 2011

More terms from Paul Barry, Jun 14 2008

A087322 Triangle T read by rows: T(n, 1) = 2n + 1. For 1 < k <= n, T(n, k) = 2T(n,k-1) + 1.

Original entry on oeis.org

3, 5, 11, 7, 15, 31, 9, 19, 39, 79, 11, 23, 47, 95, 191, 13, 27, 55, 111, 223, 447, 15, 31, 63, 127, 255, 511, 1023, 17, 35, 71, 143, 287, 575, 1151, 2303, 19, 39, 79, 159, 319, 639, 1279, 2559, 5119, 21, 43, 87, 175, 351, 703, 1407, 2815, 5631, 11263, 23, 47, 95

Offset: 1

Amarnath Murthy, Sep 03 2003

With T(n,0) = n for n >= 0, this becomes J. M. Bergot's triangular array in the definition of A190730. - Petros Hadjicostas, Feb 15 2021

			Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
   3;
   5, 11;
   7, 15, 31;
   9, 19, 39,  79;
  11, 23, 47,  95, 191;
  13, 27, 55, 111, 223, 447;
  15, 31, 63, 127, 255, 511, 1023;
  17, 35, 71, 143, 287, 575, 1151, 2303;
  19, 39, 79, 159, 319, 639, 1279, 2559, 5119;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)

Cf. A087323, A099035, A190730.

A087322row[n_]:=NestList[2#+1&,2n+1,n-1];Array[A087322row,10] (* Paolo Xausa, Oct 17 2023 *)

T(n, k) = (n + 1)*2^k - 1 for n >= 1 and 1 <= k <= n.
From Petros Hadjicostas, Feb 15 2021: (Start)
Sum_{k=1..n} T(n,k) = A190730(n).
T(n,2) = 4*n + 3 for n >= 2.
T(n,n) = A087323(n).
T(n,n-1) = A099035(n) = (n+1)*2^(n-1) - 1 for n >= 2.
Recurrence: T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) for n >= 2 and 2 <= k <= n with initial conditions the values of T(n, 1) and T(n,2).
Bivariate o.g.f.: Sum_{n,k>=1} T(n,k)*x^n*y^k = (4*x^3*y^2 - 2*x^2*y - 4*x*y - x + 3)*x*y/((1 - 2*x*y)^2*(1 - x*y)*(1 - x)^2). (End)

Edited and extended by David Wasserman, May 06 2005

Name edited by Petros Hadjicostas, Feb 15 2021

A125026 Triangle read by rows: T(n,k) = k*binomial(n,k) + binomial(n-1,k) (1 <= k <= n).

Original entry on oeis.org

1, 3, 2, 5, 7, 3, 7, 15, 13, 4, 9, 26, 34, 21, 5, 11, 40, 70, 65, 31, 6, 13, 57, 125, 155, 111, 43, 7, 15, 77, 203, 315, 301, 175, 57, 8, 17, 100, 308, 574, 686, 532, 260, 73, 9, 19, 126, 444, 966, 1386, 1344, 876, 369, 91, 10, 21, 155, 615, 1530, 2562, 2982, 2430, 1365

Offset: 1

Gary W. Adamson, Nov 15 2006

Also A007318 * A127899 (unsigned) as a product of two infinite lower triangular matrices. - Gary W. Adamson, Feb 19 2007

			First few rows of the triangle are
   1;
   3,   2;
   5,   7,   3;
   7,  15,  13,   4;
   9,  26,  34,  21,   5;
  11,  40,  70,  65,  31,   6;
  13,  57, 125, 155, 111,  43,  7;
  ...

Cf. A099035 (row sums).

T:=(n,k)->k*binomial(n,k)+binomial(n-1,k): for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Edited by N. J. A. Sloane, Nov 29 2006

A135852 A007318 * A103516 as a lower triangular matrix.

Original entry on oeis.org

1, 3, 2, 8, 4, 3, 20, 6, 9, 4, 48, 8, 18, 16, 5, 112, 10, 30, 40, 25, 6, 256, 12, 45, 80, 75, 36, 7, 576, 14, 63, 140, 175, 126, 49, 8, 1280, 16, 84, 224, 350, 336, 196, 64, 9, 2816, 18, 108, 336, 630, 756, 588, 288, 81, 10

Offset: 0

Gary W. Adamson, Dec 01 2007

Binomial transform of triangle A103516.

			First few rows of the triangle are:
    1;
    3,  2;
    8,  4,  3;
   20,  6,  9,  4;
   48,  8, 18, 16,  5;
  112, 10, 30, 40, 25,  6;
  256, 12, 45, 80, 75, 36,  7;
  ...

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

Cf. A007318, A103516.
Cf. A001792 (1st column), A099035 (row sums).
Cf. A135853 (= A103516 * A007318).

T[n_, k_]:= If[n==0, 1, If[k==0, (n+2)*2^(n-1), (k+1)*Binomial[n, k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 07 2016 *)

def A135852(n,k):
    if (n==0): return 1
    elif (k==0): return (n+2)*2^(n-1)
    else: return (k+1)*binomial(n, k)
flatten([[A135852(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 07 2022

T(n, k) = (A007318 * A103516)(n, k).
T(n, 0) = A001792(n).
Sum_{k=0..n} T(n, k) = A099035(n+1).
T(n, k) = (k+1)*binomial(n, k), with T(n, 0) = (n+2)*2^(n-1), T(n, n) = n+1. - G. C. Greubel, Dec 07 2016

Offset changed to 0 by G. C. Greubel, Feb 07 2022

A130295 Erroneous duplicate of A125026.

Original entry on oeis.org

Offset: 1

Gary W. Adamson, May 20 2007

dead

This sequence was initially defined as "A051340 * A007318". However, the matrix product A051340 * A007318 is not well defined, because all elements of A051340 are strictly positive integers, as are all elements of the lower left of A007318. Therefore the matrix A051340 must be truncated to its lower left (setting A[i,j]=0 if j>i), which actually equals A130296. Then the product yields this sequence, which is identical to A125026.

Row sums = A099035 (not A083706 as stated initially): (1, 5, 15, 39, 95, 223, 511, ...).

			First few rows of the triangle A125026:
   1;
   3,  2;
   5,  7,   3;
   7, 15,  13,   4;
   9, 26,  34,  21,   5;
  11, 40,  70,  65,  31,  6;
  13, 57, 125, 155, 111, 43, 7;
  ...

Cf. A051340, A099035.

(A051340) * A007318 as infinite lower triangular matrices. [Here (A051340) is that matrix with the upper right triangle set to zero, which is actually A130296. - M. F. Hasler, Aug 15 2015]

Restored and edited by M. F. Hasler, Aug 15 2015

A128224 Duplicate of A125026.

Original entry on oeis.org

Offset: 1

Gary W. Adamson, Feb 19 2007

dead

Row sums = A099035: (1, 5, 15, 39, 95, 223, 511, 1151, ...).

			First few rows of the triangle:
   1;
   3,  2;
   5,  7,   3;
   7, 15,  13,   4;
   9, 26,  34,  21,   5;
  11, 40,  70,  65,  31,  6;
  13, 57, 125, 155, 111, 43, 7;
  ...

A007318 * A127899 (unsigned) as infinite lower triangular matrices.

Cf. A007318, A127899, A099035.

A187214 Number of gulls (or G-toothpicks) added at n-th stage in the first quadrant of the gullwing structure of A187212.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 5, 4, 2, 4, 6, 6, 8, 14, 15, 8, 2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 39, 16, 2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 40, 18, 8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 95

Offset: 1

Omar E. Pol, Mar 22 2011, Apr 06 2011

nonn

It appears that both a(2) and a(2^k - 1) are odd numbers, for k >= 2. Other terms are even numbers.

			At stage 1 we start in the first quadrant from a Q-toothpick centered at (1,0) with its endpoints at (0,0) and (1,1). There are no gulls in the structure, so a(1) = 0.
At stage 2 we place a gull (or G-toothpick) with its midpoint at (1,1) and its endpoints at (2,0) and (2,2), so a(2) = 1. There is only one exposed midpoint at (2,2).
At stage 3 we place a gull with its midpoint at (2,2), so a(3) = 1. There are two exposed endpoints.
At stage 4 we place two gulls, so a(4) = 2. There are two exposed endpoints.
At stage 5 we place two gulls, so a(5) = 2. There are four exposed endpoints.
And so on.
If written as a triangle begins:
0,
1,
1,2,
2,4,5,4,
2,4,6,6,8,14,15,8,
2,4,6,6,8,14,16,10,8,14,18,20,30,44,39,16,
2,4,6,6,8,14,16,10,8,14,18,20,30,44,40,18,8,14,18,20,30,44,42,28,...
It appears that rows converge to A151688.

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A099035, A151550, A151688, A187210, A187211, A187212, A187213, A187220.

read("transforms3") ;
L := BFILETOLIST("b187212.txt") ;
A187213 := DIFF(L) ;
seq( op(n,A187213)/2,n=2..nops(A187213)) ; # R. J. Mathar, Mar 30 2011

a(1)=0. a(n) = A187213(n)/2, for n >= 2.

It appears that a(2^k - 1) = A099035(k-1), for k >= 2.

Showing 1-8 of 8 results.

cp's OEIS Frontend

A193649 Q-residue of the (n+1)st Fibonacci polynomial, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

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A131056 A007318 * A131055.

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PARI

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A087322 Triangle T read by rows: T(n, 1) = 2*n + 1. For 1 < k <= n, T(n, k) = 2*T(n,k-1) + 1.

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A125026 Triangle read by rows: T(n,k) = k*binomial(n,k) + binomial(n-1,k) (1 <= k <= n).

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Maple

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A135852 A007318 * A103516 as a lower triangular matrix.

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Sage

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A130295 Erroneous duplicate of A125026.

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A128224 Duplicate of A125026.

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A187214 Number of gulls (or G-toothpicks) added at n-th stage in the first quadrant of the gullwing structure of A187212.

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A087322 Triangle T read by rows: T(n, 1) = 2n + 1. For 1 < k <= n, T(n, k) = 2T(n,k-1) + 1.