A007051 -id:A007051 - 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

A191450 Dispersion of (3*n-1), read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 8, 4, 14, 23, 11, 6, 41, 68, 32, 17, 7, 122, 203, 95, 50, 20, 9, 365, 608, 284, 149, 59, 26, 10, 1094, 1823, 851, 446, 176, 77, 29, 12, 3281, 5468, 2552, 1337, 527, 230, 86, 35, 13, 9842, 16403, 7655, 4010, 1580, 689, 257, 104, 38, 15, 29525

Offset: 1

Clark Kimberling, Jun 05 2011

Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n) = {index of the row of D that contains n} is a fractal sequence. In this case s(n) = A016789(n-1), t(n) = A032766(n) [from term A032766(1) onward] and u(n) = A253887(n). [Author's original comment edited by Antti Karttunen, Jan 24 2015]

For other examples of such sequences, please see the Crossrefs section.

			The northwest corner of the square array:
  1,  2,  5,  14,  41,  122,  365,  1094,  3281,   9842,  29525,   88574, ...
  3,  8, 23,  68, 203,  608, 1823,  5468, 16403,  49208, 147623,  442868, ...
  4, 11, 32,  95, 284,  851, 2552,  7655, 22964,  68891, 206672,  620015, ...
  6, 17, 50, 149, 446, 1337, 4010, 12029, 36086, 108257, 324770,  974309, ...
  7, 20, 59, 176, 527, 1580, 4739, 14216, 42647, 127940, 383819, 1151456, ...
  9, 26, 77, 230, 689, 2066, 6197, 18590, 55769, 167306, 501917, 1505750, ...
  etc.
The leftmost column is A032766, and each successive column to the right of it is obtained by multiplying the left neighbor on that row by three and subtracting one, thus the second column is (3*1)-1, (3*3)-1, (3*4)-1, (3*6)-1, (3*7)-1, (3*9)-1, ... = 2, 8, 11, 17, 20, 26, ...

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Index entries for sequences that are permutations of the natural numbers

Inverse: A254047.
Transpose: A254051.
Column 1: A032766.
Cf. A007051, A057198, A199109, A199113 (rows 1-4).
Cf. A253887 (row index of n in this array) & A254046 (column index, see also A253786).
Cf. A000244, A016789, A038722, A048673, A254053, A254103, A254104.
Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191426-A191455.

A191450 := proc(r, c)
    option remember;
    if c = 1 then
        A032766(r) ;
    else
        A016789(procname(r, c-1)-1) ;
    end if;
end proc: # R. J. Mathar, Jan 25 2015

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=3n-1 (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191450 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191450 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

a(n,k)=3^(n-1)*(k*3\2*2-1)\2+1 \\ =3^(n-1)*(k*3\2-1/2)+1/2, but 30% faster. - M. F. Hasler, Jan 20 2015

(define (A191450 n) (A191450bi (A002260 n) (A004736 n)))
(define (A191450bi row col) (if (= 1 col) (A032766 row) (A016789 (- (A191450bi row (- col 1)) 1))))
(define (A191450bi row col) (/ (+ 3 (* (A000244 col) (- (* 2 (A032766 row)) 1))) 6)) ;; Another implementation based on L. Edson Jeffery's direct formula.
;; Antti Karttunen, Jan 21 2015

Conjecture: A(n,k) = (3 + (2*A032766(n) - 1)*A000244(k))/6. - L. Edson Jeffery, with slight changes by Antti Karttunen, Jan 21 2015

a(n) = A254051(A038722(n)). [When both this and transposed array A254051 are interpreted as one-dimensional sequences.] - Antti Karttunen, Jan 22 2015

Example corrected and description clarified by Antti Karttunen, Jan 24 2015

A198715 T(n,k)=Number of nXk 0..3 arrays with values 0..3 introduced in row major order and no element equal to any horizontal or vertical neighbor.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 5, 25, 25, 5, 14, 172, 401, 172, 14, 41, 1201, 6548, 6548, 1201, 41, 122, 8404, 107042, 250031, 107042, 8404, 122, 365, 58825, 1749965, 9548295, 9548295, 1749965, 58825, 365, 1094, 411772, 28609241, 364637102, 851787199, 364637102

Offset: 1

R. H. Hardin, Oct 29 2011

Number of colorings of the grid graph P_n X P_k using a maximum of 4 colors up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

			Table starts
....1........1............2...............5..................14
....1........4...........25.............172................1201
....2.......25..........401............6548..............107042
....5......172.........6548..........250031.............9548295
...14.....1201.......107042.........9548295...........851787199
...41.....8404......1749965.......364637102.........75987485516
..122....58825.....28609241.....13925032958.......6778819400772
..365...411772....467717288....531779578441.....604736581320925
.1094..2882401...7646461682..20307996787865...53948385378521909
.3281.20176804.125007943505.775536991678112.4812720805166620356
...
Some solutions with all values from 0 to 3 for n=6 k=4
..0..1..0..1....0..1..0..1....0..1..0..1....0..1..0..1....0..1..0..1
..1..0..1..0....1..0..1..0....1..0..1..0....1..0..1..0....1..0..1..0
..0..1..2..1....0..1..0..1....0..1..0..1....0..1..0..2....0..1..0..1
..1..2..0..3....2..0..3..0....2..0..1..0....1..2..1..3....1..2..3..0
..2..0..2..0....1..3..0..2....3..2..0..2....0..3..0..2....3..1..2..3
..3..2..0..1....3..2..1..0....0..3..2..1....3..1..3..0....1..3..1..0

Andrew Howroyd, Table of n, a(n) for n = 1..496 (terms 1..180 from R. H. Hardin)
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Vertex Coloring
Wikipedia, Graph Coloring

Columns 1-7 are A007051(n-2), A034494(n-1), A198710, A198711, A198712, A198713, A198714.
Main diagonal is A198709.
Cf. A207997 (3 colorings), A222444 (labeled 4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings), A207868 (unlimited).

A243505 Permutation of natural numbers, take the odd bisection of A122111 and divide the largest prime factor out: a(n) = A052126(A122111(2n-1)).

Original entry on oeis.org

1, 2, 4, 8, 3, 16, 32, 6, 64, 128, 12, 256, 9, 5, 512, 1024, 24, 18, 2048, 48, 4096, 8192, 10, 16384, 27, 96, 32768, 36, 192, 65536, 131072, 20, 72, 262144, 384, 524288, 1048576, 15, 54, 2097152, 7, 4194304, 144, 768, 8388608, 108, 1536, 288, 16777216, 40, 33554432, 67108864, 30

Offset: 1

Antti Karttunen, Jun 25 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Indranil Ghosh, Python program for computing the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse: A243506.
Cf. A000040, A000079, A006254, A007051, A052126, A105560.
Related or similar permutations: A122111, A064216, A241916, A243065-A243066, A244981-A244982, A244983-A244984, A244153-A244154.

A052126[n_] := n/FactorInteger[n][[-1, 1]];
A122111[n_] := Product[Prime[Sum[If[j < i, 0, 1], {j, #}]], {i, Max[#]}]&[ Flatten[Table[Table[PrimePi[f[[1]]], {f[[2]]}], {f, FactorInteger[n]}]]];
a[n_] := A052126[A122111[2n-1]];
Array[a, 60] (* Jean-François Alcover, Sep 23 2020 *)

(define (A243505 n) (A122111 (A064216 n)))

a(n) = A052126(A122111((2*n)-1)).
a(n) = A122111((2*n)-1) / A105560((2*n)-1).
As a composition of related permutations:
a(n) = A122111(A064216(n)).
a(n) = A241916(A243065(n)).
Other identities:
For all n >= 2, a(n) = A070003(A244984(n)-1) / A105560((2*n)-1).
For all n >= 1, a(A006254(n)) = A000079(n) and a(A007051(n)) = A000040(n).
For all n >= 1, A105560(2n-1) divides a(n).

A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 3, 2, 4, 8, 5, 6, 11, 23, 14, 7, 17, 32, 68, 41, 9, 20, 50, 95, 203, 122, 10, 26, 59, 149, 284, 608, 365, 12, 29, 77, 176, 446, 851, 1823, 1094, 13, 35, 86, 230, 527, 1337, 2552, 5468, 3281, 15, 38, 104, 257, 689, 1580, 4010, 7655, 16403, 9842, 16, 44, 113, 311, 770, 2066, 4739, 12029, 22964, 49208, 29525, 18, 47

Offset: 1

L. Edson Jeffery & Antti Karttunen, Jan 24 2015

This is transposed dispersion of (3n-1), starting from its complement A032766 as the first row of square array A(row,col). Please see the transposed array A191450 for references and background discussion about dispersions.

For any odd number x = A135765(row,col), the result after one combined Collatz step (3x+1)/2 -> x (A165355) is found in this array at A(row+1,col).

			The top left corner of the array:
   1,   3,   4,   6,   7,   9,  10,  12,   13,   15,   16,   18,   19,   21
   2,   8,  11,  17,  20,  26,  29,  35,   38,   44,   47,   53,   56,   62
   5,  23,  32,  50,  59,  77,  86, 104,  113,  131,  140,  158,  167,  185
  14,  68,  95, 149, 176, 230, 257, 311,  338,  392,  419,  473,  500,  554
  41, 203, 284, 446, 527, 689, 770, 932, 1013, 1175, 1256, 1418, 1499, 1661
...

Inverse: A254052.
Transpose: A191450.
Row 1: A032766.
Cf. A007051, A057198, A199109, A199113 (columns 1-4).
Cf. A254046 (row index of n in this array, see also A253786), A253887 (column index).
Array A135765(n,k) = 2*A(n,k) - 1.
Other related arrays: A254055, A254101, A254102.
Cf. also A000079, A000244, A003961, A007310, A032766, A002260, A004736, A038722, A165355, A254050.
Related permutations: A048673, A254053, A183209, A249745, A254103, A254104.

In A(n,k)-formulas below, n is the row, and k the column index, both starting from 1:
A(n,k) = (3 + ( A000244(n) * (2*A032766(k) - 1) )) / 6. - Antti Karttunen after L. Edson Jeffery's direct formula for A191450, Jan 24 2015
A(n,k) = A048673(A254053(n,k)). [Alternative formula.]
A(n,k) = (1/2) * (1 + A003961((2^(n-1)) * A254050(k))). [The above expands to this.]
A(n,k) = (1/2) * (1 + (A000244(n-1) * A007310(k))). [Which further reduces to this, equivalent to L. Edson Jeffery's original formula above.]
A(1,k) = A032766(k) and for n > 1: A(n,k) = (3 * A254051(n-1,k)) - 1. [The definition of transposed dispersion of (3n-1).]
A(n,k) = (1+A135765(n,k))/2, or when expressed one-dimensionally, a(n) = (1+A135765(n))/2.
A(n+1,k) = A165355(A135765(n,k)).
As a composition of related permutations. All sequences interpreted as one-dimensional:
a(n) = A048673(A254053(n)). [Proved above.]
a(n) = A191450(A038722(n)). [Transpose of array A191450.]

A047926 a(n) = (3^(n+1) + 2*n + 1)/4.

Original entry on oeis.org

1, 3, 8, 22, 63, 185, 550, 1644, 4925, 14767, 44292, 132866, 398587, 1195749, 3587234, 10761688, 32285049, 96855131, 290565376, 871696110, 2615088311, 7845264913, 23535794718, 70607384132, 211822152373, 635466457095, 1906399371260

Offset: 0

N. J. A. Sloane

Density of regular language L{0}* over {0,1,2,3} (i.e., number of strings of length n in L), where L is described by regular expression with c=3: Sum_{i=1..c} Product_{j=1..i} (j(1+...+j)*), where "Sum" stands for union and "Product" for concatenation. I.e., L = L((11*+11*2(1+2)*+11*2(1+2)*3(1+2+3)*)0*) - Nelma Moreira, Oct 10 2004

Conjecture: Number of representations of 3^(2n) as a sum a^2 + b^2 + c^2 with 0 < a <= b <= c. That is, a(1) = 3 because 3^2 = 1^2 + 2^2 + 2^2, a(2) = 3 because 3^4 = 1^2 + 4^2 + 8^2 = 3^2 + 6^2 + 6^2 = 4^2 + 4^2 + 7^2. - Zak Seidov, Mar 01 2012

M. Aigner, Combinatorial Search, Wiley, 1988, see Exercise 6.4.5.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC& LIACC, Universidade do Porto.
N. Moreira and R. Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A007051.

[(3^(n+1)+2*n+1)/4: n in [0..40]]; // Vincenzo Librandi, May 02 2011

Table[(3^(n+1)+2n+1)/4,{n,0,30}] (* or *) LinearRecurrence[{5,-7,3},{1,3,8},30] (* Harvey P. Dale, Apr 19 2019 *)

a(n)=(3^(n+1)+2*n+1)/4 \\ Charles R Greathouse IV, Mar 02 2012

[(gaussian_binomial(n,1,3)+n)/2 for n in range(1,28)] # Zerinvary Lajos, May 29 2009

From Paul Barry, Sep 03 2003: (Start)
a(n) = Sum_{k=0..n} (3^k + 1)/2. Partial sums of A007051.
G.f.: (1 - 2*x)/((1 - x)^2*(1 - 3*x)). (End)
For c = 3, a(c,n) = g(1,c)*n + Sum_{k=2..c} g(k,c)*k*(k^n - 1)/(k-1), where g(1,1) = 1, g(1,c) = g(1,c-1) + (-1)^(c-1)/(c-1)! for c > 1, and g(k,c) = g(k-1, c-1)/k, for c > 1 and 2 <= k <= c. - Nelma Moreira, Oct 10 2004
a(n+1) = 3*a(n) - n. - Franklin T. Adams-Watters, Jul 05 2014
E.g.f.: exp(x)*(1 + 2*x + 3*exp(2*x))/4. - Stefano Spezia, Sep 26 2023

A047848 Array A read by diagonals; n-th difference of (A(k,n), A(k,n-1),..., A(k,0)) is (k+2)^(n-1), for n=1,2,3,...; k=0,1,2,...

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 14, 6, 2, 1, 41, 22, 7, 2, 1, 122, 86, 32, 8, 2, 1, 365, 342, 157, 44, 9, 2, 1, 1094, 1366, 782, 260, 58, 10, 2, 1, 3281, 5462, 3907, 1556, 401, 74, 11, 2, 1, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1, 29525, 87382, 97657, 55988, 19609, 4682, 821, 112, 13, 2, 1

Offset: 0

Clark Kimberling

			Array, A(n, k), begins as:
  1, 2,  5,  14,   41, ... = A007051.
  1, 2,  6,  22,   86, ... = A047849.
  1, 2,  7,  32,  157, ... = A047850.
  1, 2,  8,  44,  260, ... = A047851.
  1, 2,  9,  58,  401, ... = A047852.
  1, 2, 10,  74,  586, ... = A047853.
  1, 2, 11,  92,  821, ... = A047854.
  1, 2, 12, 112, 1112, ... = A047855.
  1, 2, 13, 134, 1465, ... = A047856.
  1, 2, 14, 158, 1886, ... = A196791.
  1, 2, 15, 184, 2381, ... = A196792.
Downward antidiagonals, T(n, k), begins as:
      1;
      2,     1;
      5,     2,     1;
     14,     6,     2,     1;
     41,    22,     7,     2,     1;
    122,    86,    32,     8,     2,    1;
    365,   342,   157,    44,     9,    2,   1;
   1094,  1366,   782,   260,    58,   10,   2,   1;
   3281,  5462,  3907,  1556,   401,   74,  11,   2,  1;
   9842, 21846, 19532,  9332,  2802,  586,  92,  12,  2, 1;
  29525, 87382, 97657, 55988, 19609, 4682, 821, 112, 13, 2, 1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A047857 (row sums), A196793 (main diagonal).

A:= func< n,k | ((n+3)^k +n+1)/(n+2) >; // array A047848
[A(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2025

A[n_, k_]:= ((n+3)^k +n+1)/(n+2);
A047848[n_, k_]:= A[k,n-k];
Table[A047848[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 11 2025 *)

def A(n,k): return (pow(n+3,k) +n+1)//(n+2) # array A047848
print(flatten([[A(k,n-k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 11 2025

A(n, k) = ((n+3)^k + n + 1)/(n+2). - Ralf Stephan, Feb 14 2004
From G. C. Greubel, Jan 11 2025: (Start)
T(n, k) = ((k+3)^(n-k) + k + 1)/(k+2) (antidiagonal triangle).
T(n, n) = A196793(n).
Sum_{k=0..n} T(n, k) = A047857(n). (End)

A100774 a(n) = 2*(3^n - 1).

Original entry on oeis.org

0, 4, 16, 52, 160, 484, 1456, 4372, 13120, 39364, 118096, 354292, 1062880, 3188644, 9565936, 28697812, 86093440, 258280324, 774840976, 2324522932, 6973568800, 20920706404, 62762119216, 188286357652, 564859072960, 1694577218884

Offset: 0

Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Apr 06 2005

a(n) is the number of steps which are made when generating all n-step nonreversing random walks that begin in a fixed point P on a two-dimensional square lattice. To make one step means to move along one edge on the lattice.
These are also the first local maxima reached in the Collatz trajectories of 2^n - 1. - David Rabahy, Oct 30 2017
Also the graph diameter of the n-Sierpinski carpet graph. - Eric W. Weisstein, Mar 13 2018
a(n) is the number of edge covers of F_{n,2}, which has adjacent vertices u and w, and n vertices each adjacent to both u and w. An edge cover is a subset of the edges where each vertex is adjacent to at least one vertex. To cover each of the n vertices v_i, we need to have at least the edge uv_i or wv_i or both, giving us three choices for each. We can then add the edge uw or not, which is 2*3^n choices. But we need to remove the case where all uv_i's were chosen and uw not chosen, and all ww_i's were chosen and uw not chosen. - Feryal Alayont, Jun 17 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..196
Eric Weisstein's World of Mathematics, Graph Diameter
Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A003462, A007051, A024023, A029858, A034472, A048473, A058481, A067771, A079004, A115099, A134931.

[2*(3^n - 1): n in [0..25] ]; // Vincenzo Librandi, Apr 30 2011

Table[2 (3^n - 1), {n, 0, 24}] (* Alonso del Arte, Nov 08 2012 *)
2 (3^Range[0, 20] - 1) (* Eric W. Weisstein, Mar 13 2018 *)
LinearRecurrence[{4, -3}, {4, 16}, {0, 20}] (* Eric W. Weisstein, Mar 13 2018 *)
CoefficientList[Series[4 x/(1 - 4 x + 3 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 13 2018 *)

A100774(n):=2*(3^n - 1)$
makelist(A100774(n),n,0,30); /* Martin Ettl, Nov 09 2012 */

a(n)=2*3^n-2 \\ Charles R Greathouse IV, Nov 09 2012

a(n) = 2*(3^n - 1).
a(n) = 4*Sum_{i=0..n-1} 3^i.
a(n) = 4*A003462(n).
a(n) = A048473(n) - 1. - Paul Curtz, Jan 19 2009
G.f.: 4*x/((1-x)*(1-3*x)). - Eric W. Weisstein, Mar 13 2018
a(n) = 4*a(n-1) - 3*a(n-2). - Eric W. Weisstein, Mar 13 2018
From Elmo R. Oliveira, Dec 06 2023: (Start)
a(n) = 2*A024023(n).
a(n) = 3*a(n-1) + 4 for n>0.
E.g.f.: 2*(exp(3*x) - exp(x)). (End)

A134931 a(n) = (5*3^n-3)/2.

Original entry on oeis.org

1, 6, 21, 66, 201, 606, 1821, 5466, 16401, 49206, 147621, 442866, 1328601, 3985806, 11957421, 35872266, 107616801, 322850406, 968551221, 2905653666, 8716961001, 26150883006, 78452649021, 235357947066, 706073841201, 2118221523606

Offset: 0

Rolf Pleisch, Jan 29 2008

Numbers n where the recurrence s(0)=1, if s(n-1) >= n then s(n) = s(n-1) - n else s(n) = s(n-1) + n produces s(n)=0. - Hugo Pfoertner, Jan 05 2012
A046901(a(n)) = 1. - Reinhard Zumkeller, Jan 31 2013
Binomial transform of A146523: (1, 5, 10, 20, 40, ...) and double binomial transform of A010685: (1, 4, 1, 4, 1, 4, ...). - Gary W. Adamson, Aug 25 2016
Also the number of maximal cliques in the (n+1)-Hanoi graph. - Eric W. Weisstein, Dec 01 2017
a(n) is the least k such that f(a(n-1)+1) + ... + f(k) > f(a(n-2)+1) + ... + f(a(n-1)) for n > 1, where f(n) = 1/(n+1). Because Sum_{k=1..5*3^(n-1)} 1/(a(n)+3*k-1) + 1/(a(n)+3*k) + 1/(a(n)+3*k+1) - 1/((a(n)+1+5*3^n)*5*3^(n-1)) < Sum_{k=1..5*3^(n-1)} 1/(a(n-1)+k+1) < Sum_{k=1..5*3^(n-1)} 1/(a(n)+3*k-1) + 1/(a(n)+3*k) + 1/(a(n)+3*k+1), we have 1 < 1/3 + 1/4 + ... + 1/7 < 1/8 + 1/9 + ... + 1/22 < ... . - Jinyuan Wang, Jun 15 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Hanoi Graph
Eric Weisstein's World of Mathematics, Maximal Clique
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A003462, A007051, A034472, A024023, A067771, A029858.

Cf. A005030, A010685, A046901, A060816, A116952, A146523, A225918.

[(5*3^n-3)/2: n in [0..30]]; // Vincenzo Librandi, Jun 05 2011

seq((5*3^n-3)/2, n= 0..25); # Gary Detlefs, Jun 22 2010

a=1; lst={a}; Do[a=a*3+3; AppendTo[lst,a], {n,0,100}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)
Table[(5 3^n - 9)/6, {n, 20}] (* Eric W. Weisstein, Dec 01 2017 *)
(5 3^Range[20] - 9)/6 (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{4, -3}, {1, 6}, 20] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[(1 + 2 x)/(1 - 4 x + 3 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

a(n) = (5*3^n-3)/2; /* Joerg Arndt, Apr 14 2013 */

a(n) = 3*(a(n-1) + 1), with a(0)=1.
From R. J. Mathar, Jan 31 2008: (Start)
O.g.f.: (5/2)/(1-3*x) - (3/2)/(1-x).
a(n) = (A005030(n) - 3)/2. (End)
a(n) = A060816(n+1) - 1. - Philippe Deléham, Apr 14 2013
E.g.f.: exp(x)*(5*exp(2*x) - 3)/2. - Stefano Spezia, Aug 28 2023

More terms from Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Previous Showing 31-40 of 204 results. Next

cp's OEIS Frontend

A060816 a(0) = 1; a(n) = (5*3^(n-1) - 1)/2 for n > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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.)

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A191450 Dispersion of (3*n-1), read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

Formula

Extensions

A198715 T(n,k)=Number of nXk 0..3 arrays with values 0..3 introduced in row major order and no element equal to any horizontal or vertical neighbor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A243505 Permutation of natural numbers, take the odd bisection of A122111 and divide the largest prime factor out: a(n) = A052126(A122111(2n-1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n*(2*floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A047926 a(n) = (3^(n+1) + 2*n + 1)/4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A047848 Array A read by diagonals; n-th difference of (A(k,n), A(k,n-1),..., A(k,0)) is (k+2)^(n-1), for n=1,2,3,...; k=0,1,2,...

A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...