cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A181435 First column in matrix inverse of a mixed convolution of A052906.

Original entry on oeis.org

1, -4, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, -1, -1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 0, 1, -1, -1, 0, -1, 1, 0, 0, 1
Offset: 1

Views

Author

Mats Granvik, Oct 20 2010

Keywords

Comments

It appears that except for the second term, the sequence is identical to the Möbius function.

Links

Crossrefs

Cf. A003688, A006190, A013946, A008683, A178536, A181434.

Programs

  • Maple
    b := proc(n)
    option remember;
    local c;
    c := 3;
    if n <= 3 then
        op(n,[1,c,c^2]) ;
    else
        c*procname(n-1)+procname(n-2) ;
    end if;
end proc:
A := proc(n,k)
    if n >= k then
        b(n-k+1) ;
    else
        0 ;
    end if;
end proc:
B := proc(n,k)
    if modp(n,k) = 0 then
        1;
    else
        0;
    end if;
end proc:
AB := proc(n,k)
    option remember;
    add( A(n,j)*B(j,k),j=1..n) ;
end proc:
ABinv := proc(n,k)
    option remember;
    if k > n then
        0;
    elif k = n then
        1;
    else
        -add( AB(n,j)*procname(j,k),j=k..n-1) ;
    end if;
end proc:
A181435 := proc(n)
    ABinv(n,1) ;
end proc:
for n from 1 do
    printf("%d %d\n",n,ABinv(n,1)) ;
end do: # R. J. Mathar, Oct 06 2017
  • Mathematica
    Clear[t, n, k, nn, b, A, c]; nn = 77; c = 3; b[0] = 1; b[1] = 1; b[n_] := b[n] = c*b[n - 1] + b[n - 2]; t[n_, 1] = If[n >= 1, b[n], 0]; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}] - Sum[t[n - i, k], {i, 1, k - 1}], 0]; MatrixForm[A = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]; Inverse[A][[All, 1]] (* Mats Granvik, Sep 16 2017 *)

Formula

From Mats Granvik, Sep 16 2017: (Start)
a(n) as the matrix inverse of a mixed convolution: Let c = 3 and let the sequence b be defined by the recurrence: b(1) = 1, b(2) = c, b(3) = c^2; for n >= 4, b(n) = c*b(n-1) + b(n-2), so b(n) = A052906(n-1), and let the lower triangular matrix A be: If n >= k then A(n,k) = b(n - k + 1) else A(n,k) = 0, and let B be the lower triangular matrix A051731. Then the matrix inverse (A.B)^-1 will have a(n) as its first column.
The matrix product T = A.B can be defined as follows: Let c = 3 and the sequence b be defined by the recurrence b(0) = 1, b(1) = 1; for b >= 2, b(n) = c*b(n - 1) + b(n - 2); and let T be the lower triangular matrix defined by the recurrence: T(n, 1) = If n >= 1 then T(n, 1) = b(n) else T(n, 1) = 0; for k >= 2, T(n, k) = If n >= k then (Sum_{i=1..k-1} T(n - i, k - 1) - T(n - i, k)) else 0. (Then the matrix inverse of T will have a(n) as its first column.)
(End)

A006190 a(n) = 3*a(n-1) + a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, 141481, 467280, 1543321, 5097243, 16835050, 55602393, 183642229, 606529080, 2003229469, 6616217487, 21851881930, 72171863277, 238367471761, 787274278560, 2600190307441, 8587845200883, 28363725910090
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Denominators of continued fraction convergents to (3+sqrt(13))/2. - Benoit Cloitre, Jun 14 2003
a(n) and A006497(n) occur in pairs: (a,b): (1,3), (3,11), (10,36), (33,119), (109,393), ... such that b^2 - 13a^2 = 4(-1)^n. - Gary W. Adamson, Jun 15 2003
Form the 4-node graph with matrix A = [1,1,1,1; 1,1,1,0; 1,1,0,1; 1,0,1,1]. Then this sequence counts the walks of length n from the vertex with degree 5 to one (any) of the other vertices. - Paul Barry, Oct 02 2004
a(n+1) is the binomial transform of A006138. - Paul Barry, May 21 2006
a(n+1) is the diagonal sum of the exponential Riordan array (exp(3x),x). - Paul Barry, Jun 03 2006
Number of paths in the right half-plane from (0,0) to the line x=n-1, consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(3)=10 because we have hh, H, UD, DU, hU, Uh, UU, hD, Dh and DD. - Emeric Deutsch, Sep 03 2007
Equals INVERT transform of A000129. Example: a(5) = 109 = (29, 12, 5, 2, 1) dot (1, 1, 3, 10, 33) = (29 + 12 + 15 + 20 + 33). - Gary W. Adamson, Aug 06 2010
For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 3's along the main diagonal, 1's along the superdiagonal and subdiagonal, and 0's everywhere else. - John M. Campbell, Jul 08 2011
These numbers could also be called "bronze Fibonacci numbers". Indeed, for n >= 1, F(n+1) = ceiling(phi*F(n)), if n is even and F(n+1) = floor(phi*F(n)), if n is odd, where phi is the golden ratio; analogously, for Pell numbers (A000129), or "silver Fibonacci numbers", P(n+1) = ceiling(delta*a(n)), if n is even and P(n+1) = floor(delta*a(n)), if n is odd, where delta = delta_S = 1 + sqrt(2) is the silver ratio. Here, for n >= 1, we have a(n+1) = ceiling(c*a(n)), if n is even and a(n+1) = floor(c*a(n)), if n is odd, where c = (3 + sqrt(13))/2 is the bronze ratio (cf. comment in A098316). - Vladimir Shevelev, Feb 23 2013
Let p(n,x) denote the Fibonacci polynomial, defined by p(1,x) = 1, p(2,x) = x, p(n,x) = x*p(n-1,x) + p(n-2,x). Let q(n,x) be the numerator polynomial of the rational function p(n, x + 1 + 1/x). Then q(n,1) = a(n). - Clark Kimberling, Nov 04 2013
The (1,1)-entry of the matrix A^n where A = [0,1,0; 1,2,1; 1,1,2]. - David Neil McGrath, Jul 18 2014
a(n+1) counts closed walks on K2, containing three loops on the other vertex. Equivalently the (1,1)-entry of A^(n+1) where the adjacency matrix of digraph is A = (0,1; 1,3). - David Neil McGrath, Oct 29 2014
For n >= 1, a(n) equals the number of words of length n-1 on alphabet {0,1,2,3} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
With offset 1 is the INVERTi transform of A001076. - Gary W. Adamson, Jul 24 2015
Apart from the initial 0, this is the p-INVERT transform of (1,0,1,0,1,0,...) for p(S) = 1 - 3 S. See A291219. - Clark Kimberling, Sep 02 2017
From Rogério Serôdio, Mar 30 2018: (Start)
This is a divisibility sequence (i.e., if n|m then a(n)|a(m)).
gcd(a(n),a(n+k)) = a(gcd(n, k)) for all positive integers n and k. (End)
Numbers of straight-chain fatty acids involving oxo and/or hydroxy groups, if cis-/trans isomerism and stereoisomerism are neglected. - Stefan Schuster, Apr 04 2018
Number of 3-compositions of n restricted to odd parts (and allowed zeros); see Hopkins & Ouvry reference. - Brian Hopkins, Aug 17 2020
From Michael A. Allen, Jan 25 2023: (Start)
Also called the 3-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 3 kinds of squares available. (End)
a(n) is the number of compositions of n when there are P(k) sorts of parts k, with k,n >= 1, P(k) = A000129(k) is the k-th Pell number (see example below). - Enrique Navarrete, Dec 15 2023

Examples

			From _Enrique Navarrete_, Dec 15 2023: (Start)
From the comment on compositions with Pell number of parts, A000129(k), there are A000129(1)=1 type of 1, A000129(2)=2 types of 2, A000129(3)=5 types of 3, A000129(4)=12 types of 4, A000129(5)=29 types of 5 and A000129(6)=70 types of 6.
The following table gives the number of compositions of n=6:
Composition, number of such compositions, number of compositions of this type:
  6,              1,     70;
  5+1,            2,     58;
  4+2,            2,     48;
  3+3,            1,     25;
  4+1+1,          3,     36;
  3+2+1,          6,     60;
  2+2+2,          1,      8;
  3+1+1+1,        4,     20;
  2+2+1+1,        6,     24;
  2+1+1+1+1,      5,     10;
  1+1+1+1+1+1,    1,      1;
for a total of a(6)=360 compositions of n=6. (End).

References

  • H. L. Abbott and D. Hanson, A lattice path problem, Ars Combin., 6 (1978), 163-178.
  • A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 128.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • L.-N. Machaut, Query 3436, L'Intermédiaire des Mathématiciens, 16 (1909), 62-63. - N. J. A. Sloane, Mar 08 2022

Links

Crossrefs

Row sums of Pascal's rhombus (A059317). Also row sums of triangle A054456(n, m).
Sequences with g.f. 1/(1-k*x-x^2) or x/(1-k*x-x^2): A000045 (k=1), A000129 (k=2), this sequence (k=3), A001076 (k=4), A052918 (k=5), A005668 (k=6), A054413 (k=7), A041025 (k=8), A099371 (k=9), A041041 (k=10), A049666 (k=11), A041061 (k=12), A140455 (k=13), A041085 (k=14), A154597 (k=15), A041113 (k=16), A178765 (k=17), A041145 (k=18), A243399 (k=19), A041181 (k=20).
Cf. A006497, A052906, A175182 (Pisano periods), A201001 (prime subsequence), A092936 (squares).
Cf. A243399.

Programs

  • GAP
    a:=[0,1];; for n in [3..30] do a[n]:=3*a[n-1]+a[n-2]; od; a; # Muniru A Asiru, Mar 31 2018
  • Haskell
    a006190 n = a006190_list !! n
a006190_list = 0 : 1 : zipWith (+) (map (* 3) $ tail a006190_list) a006190_list
-- Reinhard Zumkeller, Feb 19 2011
  • Magma
    [ n eq 1 select 0 else n eq 2 select 1 else 3*Self(n-1)+Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 19 2011
  • Maple
    a[0]:=0: a[1]:=1: for n from 2 to 35 do a[n]:= 3*a[n-1]+a[n-2] end do: seq(a[n],n=0..30); # Emeric Deutsch, Sep 03 2007
A006190:=-1/(-1+3*z+z**2); # Simon Plouffe in his 1992 dissertation, without the leading 0
seq(combinat[fibonacci](n,3),n=0..30); # R. J. Mathar, Dec 07 2011
  • Mathematica
    a[n_] := (MatrixPower[{{1, 3}, {1, 2}}, n].{{1}, {1}})[[2, 1]]; Table[ a[n], {n, -1, 24}] (* Robert G. Wilson v, Jan 13 2005 *)
LinearRecurrence[{3,1},{0,1},30] (* or *) CoefficientList[Series[x/ (1-3x-x^2), {x,0,30}], x] (* Harvey P. Dale, Apr 20 2011 *)
Table[If[n==0, a1=1; a0=0, a2=a1; a1=a0; a0=3*a1+a2], {n, 0, 30}] (* Jean-François Alcover, Apr 30 2013 *)
Table[Fibonacci[n, 3], {n, 0, 30}] (* Vladimir Reshetnikov, May 08 2016 *)
  • PARI
    a(n)=if(n<1,0,contfracpnqn(vector(n,i,2+(i>1)))[2,1])
  • PARI
    a(n)=([1,3;1,2]^n)[2,1] \\ Charles R Greathouse IV, Mar 06 2014
  • PARI
    concat([0],Vec(x/(1-3*x-x^2)+O(x^30))) \\ Joerg Arndt, Apr 30 2013
  • Sage
    [lucas_number1(n,3,-1) for n in range(0, 30)] # Zerinvary Lajos, Apr 22 2009

Formula

G.f.: x/(1 - 3*x - x^2).
From Benoit Cloitre, Jun 14 2003: (Start)
a(3*n) = 2*A041019(5*n-1), a(3*n+1) = A041019(5*n), a(3*n+2) = A041019(5*n+3).
a(2*n) = 3*A004190(n-1); a(3*n) = 10*A041613(n-1) for n >= 1. (End)
From Gary W. Adamson, Jun 15 2003: (Start)
a(n-1) + a(n+1) = A006497(n).
A006497(n)^2 - 13*a(n)^2 = 4(-1)^n. (End)
a(n) = U(n-1, (3/2)i)(-i)^(n-1), i^2 = -1. - Paul Barry, Nov 19 2003
a(n) = Sum_{k=0..n-1} binomial(n-k-1,k)*3^(n-2*k-1). - Paul Barry, Oct 02 2004
a(n) = F(n, 3), the n-th Fibonacci polynomial evaluated at x=3.
Let M = {{0, 1}, {1, 3}}, v[1] = {0, 1}, v[n] = M.v[n - 1]; then a(n) = Abs[v[n][[1]]]. - Roger L. Bagula, May 29 2005 [Or a(n) = [M^(n+1)]{1,1}. - _L. Edson Jeffery, Aug 27 2013]
From Paul Barry, May 21 2006: (Start)
a(n+1) = Sum_{k=0..n} Sum_{j=0..n-k} C(k,j)*C(n-j,k)*2^(k-j).
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(k,j)*C(n-j,k)*2^(n-j-k).
a(n+1) = Sum_{k=0..floor(n/2)} C(n-k,k)*3^(n-2*k).
a(n) = Sum_{k=0..n} C(k,n-k)*3^(2*k-n). (End)
E.g.f.: exp(3*x/2)*sinh(sqrt(13)*x/2)/(sqrt(13)/2). - Paul Barry, Jun 03 2006
a(n) = (ap^n - am^n)/(ap - am), with ap = (3 + sqrt(13))/2, am = (3 - sqrt(13))/2.
Let C = (3 + sqrt(13))/2 = exp arcsinh(3/2) = 3.3027756377... Then C^n, n > 0 = a(n)*(1/C) + a(n+1). Let X = the 2 X 2 matrix [0, 1; 1, 3]. Then X^n = [a(n-1), a(n); a(n), a(n+1)]. - Gary W. Adamson, Dec 21 2007
1/3 = 3/(1*10) + 3/(3*33) + 3/(10*109) + 3/(33*360) + 3/(109*1189) + ... . - Gary W. Adamson, Mar 16 2008
a(n) = ((3 + sqrt(13))^n - (3 - sqrt(13))^n)/(2^n*sqrt(13)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009
a(p) == 13^((p-1)/2) mod p, for odd primes p. - Gary W. Adamson, Feb 22 2009
From Johannes W. Meijer, Jun 12 2010: (Start)
Limit_{k->oo} a(n+k)/a(k) = (A006497(n) + a(n)*sqrt(13))/2.
Limit_{n->oo} A006497(n)/a(n) = sqrt(13). (End)
Sum_{k>=1} (-1)^(k-1)/(a(k)*a(k+1)) = (sqrt(13)-3)/2. - Vladimir Shevelev, Feb 23 2013
From Vladimir Shevelev, Feb 24 2013: (Start)
(1) Expression a(n+1) via a(n): a(n+1) = (3*a(n) + sqrt(13*a(n)^2 + 4*(-1)^n))/2;
(2) a^2(n+1) - a(n)*a(n+2) = (-1)^n;
(3) Sum_{k=1..n} (-1)^(k-1)/(a(k)*a(k+1)) = a(n)/a(n+1);
(4) a(n)/a(n+1) = (sqrt(13)-3)/2 + r(n), where |r(n)| < 1/(a(n+1)*a(n+2)). (End)
a(n) = sqrt(13*(A006497(n))^2 + (-1)^(n-1)*52)/13. - Vladimir Shevelev, Mar 13 2013
Sum_{n >= 1} 1/( a(2*n) + 1/a(2*n) ) = 1/3; Sum_{n >= 1} 1/( a(2*n + 1) - 1/a(2*n + 1) ) = 1/9. - Peter Bala, Mar 26 2015
From Rogério Serôdio, Mar 30 2018: (Start)
Some properties:
(1) a(n)*a(n+1) = 3*Sum_{k=1..n} a(k)^2;
(2) a(n)^2 + a(n+1)^2 = a(2*n+1);
(3) a(n)^2 - a(n-2)^2 = 3*a(n-1)*(a(n) + a(n-2));
(4) a(m*(p+1)) = a(m*p)*a(m+1) + a(m*p-1)*a(m);
(5) a(n-k)*a(n+k) = a(n)^2 + (-1)^(n+k+1)*a(k)^2;
(6) a(2*n) = a(n)*(3*a(n) + 2*a(n-1));
(7) 3*Sum_{k=2..n+1} a(k)*a(k-1) is equal to a(n+1)^2 if n odd, and is equal to a(n+1)^2 - 1 if n is even;
(8) a(n) = alpha(k)*a(n-2*k+1) + a(n-4*k+2), where alpha(k) = ap^(2*k-1) + am^(2*k-1), with ap = (3 + sqrt(13))/2, am = (3 - sqrt(13))/2;
(9) 131|Sum_{k=n..n+9} a(k), for all positive n. (End)
From Kai Wang, Feb 10 2020: (Start)
a(n)^2 - a(n+r)*a(n-r) = (-1)^(n-r)*a(r)^2 - Catalan's identity.
arctan(1/a(2n)) - arctan(1/a(2n+2)) = arctan(a(2)/a(2n+1)).
arctan(1/a(2n)) = Sum_{m>=n} arctan(a(2)/a(2m+1)).
The same formula holds for Fibonacci numbers and Pell numbers. (End)
a(n+2) = 3^(n+1) + Sum_{k=0..n} a(k)*3^(n-k). - Greg Dresden and Gavron Campbell, Feb 22 2022
G.f. = 1/(1-Sum_{k>=1} P(k)*x^k), P(k)=A000129(k) (with a(0)=1). - Enrique Navarrete, Dec 17 2023
G.f.: x/(1 - 3*x - x^2) = Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (m*k + 3 - m + x)/(1 + m*k*x) ) for arbitrary m (a telescoping series). - Peter Bala, May 08 2024
Sum_{n>=0} a(n)/phi^(3*n) = 1, where phi = A001622 is the golden ratio. - Diego Rattaggi, Apr 07 2025

Extensions

Second formula corrected by Johannes W. Meijer, Jun 02 2010

A052542 a(n) = 2*a(n-1) + a(n-2), with a(0) = 1, a(1) = 2, a(2) = 4.

Original entry on oeis.org

1, 2, 4, 10, 24, 58, 140, 338, 816, 1970, 4756, 11482, 27720, 66922, 161564, 390050, 941664, 2273378, 5488420, 13250218, 31988856, 77227930, 186444716, 450117362, 1086679440, 2623476242, 6333631924, 15290740090, 36915112104, 89120964298, 215157040700
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Apart from the initial 1, this sequence is simply twice the Pell numbers, A000129. - Antonio Alberto Olivares, Dec 31 2003
Image of 1/(1-2x) under the mapping g(x) -> g(x/(1+x^2)). - Paul Barry, Jan 16 2005
The intermediate convergents to 2^(1/2) begin with 4/3, 10/7, 24/17, 58/41; essentially, numerators = A052542 and denominators = A001333. - Clark Kimberling, Aug 26 2008
a(n) is the number of generalized compositions of n+1 when there are 2*i-2 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010
Apart from the initial 1, this is the p-INVERT transform of (1,0,1,0,1,0,...) for p(S) = 1 - 2 S. See A291219. - Clark Kimberling, Sep 02 2017
Conjecture: Apart from the initial 1, a(n) is the number of compositions of two types of n having no even parts. - Gregory L. Simay, Feb 17 2018
For n>0, a(n+1) is the length of tau^n(10) where tau is the morphism: 1 -> 101, 0 -> 1. See Song and Wu. - Michel Marcus, Jul 21 2020
The above conjecture is true, as the g.f. can be written as 1/(1 - (2*x)/(1 - x^2)). - John Tyler Rascoe, Jun 01 2024

Links

Crossrefs

Cf. A052906. Essentially first differences of A001333.

Programs

  • GAP
    a:=[2,4];; for n in [3..40] do a[n]:=2*a[n-1]+a[n-2]; od; a; # G. C. Greubel, May 09 2019
  • Haskell
    a052542 n = a052542_list !! n
a052542_list = 1 : 2 : 4 : tail (zipWith (+)
               (map (* 2) $ tail a052542_list) a052542_list)
-- Reinhard Zumkeller, Feb 24 2015
  • Magma
    I:=[2,4]; [n le 2 select I[n] else 2*Self(n-1) +Self(n-2): n in [1..40]]; // G. C. Greubel, May 09 2019
  • Maple
    spec := [S,{S=Sequence(Prod(Union(Z,Z),Sequence(Prod(Z,Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
A052542 := proc(n)
    option remember;
    if n <=2 then
        2^n;
    else
        2*procname(n-1)+procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Sep 23 2016
A052542List := proc(m) local A, P, n; A := [1,2]; P := [1,1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(A), P[-2]]);
A := [op(A), P[-1]] od; A end: A052542List(31); # Peter Luschny, Mar 26 2022
  • Mathematica
    Join[{1}, LinearRecurrence[{2, 1}, {2, 4}, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2012 *)
  • PARI
    Vec((1-x^2)/(1-2*x-x^2) +O(x^40)) \\ Charles R Greathouse IV, Nov 20 2011
  • Sage
    ((1-x^2)/(1-2*x-x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

Formula

G.f.: (1 - x^2)/(1 - 2*x - x^2).
Recurrence: a(0)=1, a(2)=4, a(1)=2, a(n) + 2*a(n+1) - a(n+2) = 0;
a(n) = Sum_{alpha = RootOf(-1+2*x+x^2)} (1/2)*(1-alpha)*alpha^(-n-1).
a(n) = 2*A001333(n-1) + a(n-1), n > 1. A001333(n)/a(n) converges to sqrt(1/2). - Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003
Binomial transform of A094024. a(n) = 0^n + ((1 + sqrt(2))^n - (1 - sqrt(2))^n)/sqrt(2). - Paul Barry, Apr 22 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1, k)2^(n-2k). - Paul Barry, Jan 16 2005
If p[i] = 2*(i mod 2) and if A is Hessenberg matrix of order n defined by A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i=j+1), and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = det A. - Milan Janjic, May 02 2010
a(n) = round(sqrt(Pell(2n) + Pell(2n-1))). - Richard R. Forberg, Jun 22 2014
a(n) = 2*A000129(n) + A000007(n) - Iain Fox, Nov 30 2017
a(n) = A000129(n) - A000129(n-2). - Gregory L. Simay, Feb 17 2018

A003688 a(n) = 3*a(n-1) + a(n-2), with a(1)=1 and a(2)=4.

Original entry on oeis.org

1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564, 21932293, 72437443, 239244622, 790171309, 2609758549, 8619446956, 28468099417, 94023745207, 310539335038, 1025641750321, 3387464586001, 11188035508324, 36951571110973
Offset: 1

Views

Author

Frans J. Faase

Keywords

Comments

Number of 2-factors in K_3 X P_n.
Form the graph with matrix [1,1,1,1;1,1,1,0;1,1,0,1;1,0,1,1]. The sequence 1,1,4,13,... with g.f. (1-2*x)/(1-3*x-x^2) counts closed walks of length n at the vertex of degree 5. - Paul Barry, Oct 02 2004
a(n) is term (1,1) in M^n, where M is the 3x3 matrix [1,1,2; 1,1,1; 1,1,1]. - Gary W. Adamson, Mar 12 2009
Starting with 1, INVERT transform of A003945: (1, 3, 6, 12, 24, ...). - Gary W. Adamson, Aug 05 2010
Row sums of triangle
m/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....3
.2..|..1.....3.....9
.3..|..1.....6.....9.....27
.4..|..1.....6....27.....27...81
.5..|..1.....9....27....108...81...243
.6..|..1.....9....54....108..405...243...729
.7..|..1....12....54....270..405..1458...729..2187
which is the triangle for numbers 3^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 12 2012
Pisano period lengths: 1, 3, 1, 6, 12, 3, 16, 12, 6, 12, 8, 6, 52, 48, 12, 24, 16, 6, 40, 12, ... - R. J. Mathar, Aug 10 2012
a(n-1) is the number of length-n strings of 4 letters {0,1,2,3} with no two adjacent nonzero letters identical. The general case (strings of L letters) is the sequence with g.f. (1+x)/(1-(L-1)*x-x^2). - Joerg Arndt, Oct 11 2012

Examples

			G.f. = x + 4*x^2 + 13*x^3 + 43*x^4 + 142*x^5 + 469*x^6 + 1549*x^7 + 5116*x^8 + ...

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Crossrefs

Partial sums of A052906. Pairwise sums of A006190.
Cf. A003945, A006190, A049310, A055830, A136159, A290948.
Cf. A374439.

Programs

  • Magma
    [n le 2 select 4^(n-1) else 3*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 19 2011
  • Maple
    with(combinat): a:=n->fibonacci(n,3)-2*fibonacci(n-1,3): seq(a(n), n=2..25); # Zerinvary Lajos, Apr 04 2008
  • Mathematica
    a[n_] := (MatrixPower[{{1, 3}, {1, 2}}, n].{{1}, {1}})[[1, 1]]; Table[ a[n], {n, 0, 23}] (* Robert G. Wilson v, Jan 13 2005 *)
LinearRecurrence[{3,1},{1,4},30] (* Harvey P. Dale, Mar 15 2015 *)
  • PARI
    a(n)=([0,1; 1,3]^(n-1)*[1;4])[1,1] \\ Charles R Greathouse IV, Aug 14 2017
  • SageMath
    @CachedFunction
def a(n): # a = A003688
    if (n<3): return 4^(n-1)
    else: return 3*a(n-1) + a(n-2)
[a(n) for n in range(1,41)] # G. C. Greubel, Dec 26 2023

Formula

a(n) = ((13 - sqrt(13))/26)*((3 + sqrt(13))/2)^n + ((13 + sqrt(13))/26)*((3 - sqrt(13))/2)^n. - Paul Barry, Oct 02 2004
a(n) = Sum_{k=0..n} 2^k*A055830(n,k). - Philippe Deléham, Oct 18 2006
Starting (1, 1, 4, 13, 43, 142, 469, ...), row sums (unsigned) of triangle A136159. - Gary W. Adamson, Dec 16 2007
G.f.: x*(1+x)/(1-3*x-x^2). - Philippe Deléham, Nov 03 2008
a(n) = A006190(n) + A006190(n-1). - Sergio Falcon, Nov 26 2009
For n>=2, a(n) = F_n(3) + F_(n+1)(3), where F_n(x) is Fibonacci polynomial (cf. A049310): F_n(x) = Sum_{i=0..floor((n-1)/2)} binomial(n-i-1,i) * x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012
G.f.: G(0)*(1+x)/(2-3*x), where G(k) = 1 + 1/(1 - (x*(13*k-9))/( x*(13*k+4) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013
a(n)^2 is the denominator of continued fraction [3,3,...,3, 5, 3,3,...,3], which has n-1 3's before, and n-1 3's after, the middle 5. - Greg Dresden, Sep 18 2019
a(n) = Sum_{k=0..n} A046854(n-1,k)*3^k. - R. J. Mathar, Feb 10 2024
a(n) = 3^n*Sum_{k=0..n} A374439(n, k)*(-1/3)^k. - Peter Luschny, Jul 26 2024

Extensions

Formula added by Olivier Gérard, Aug 15 1997
Name clarified by Michel Marcus, Oct 16 2016

A180148 a(n) = 3*a(n-1) + a(n-2) with a(0)=2 and a(1)=5.

Original entry on oeis.org

2, 5, 17, 56, 185, 611, 2018, 6665, 22013, 72704, 240125, 793079, 2619362, 8651165, 28572857, 94369736, 311682065, 1029415931, 3399929858, 11229205505, 37087546373, 122491844624, 404563080245, 1336181085359, 4413106336322, 14575500094325, 48139606619297
Offset: 0

Views

Author

Johannes W. Meijer, Aug 13 2010

Keywords

Comments

Inverse binomial transform of A052961 (without the leading 1).
For n >= 1, also the number of matchings in the n-alkane graph. - Eric W. Weisstein, Jul 14 2021

Links

Crossrefs

Cf. A003688, A052906.
Appears in A180142.
Cf. A000602 (more information on n-alkanes).

Programs

  • Maple
    a:= n-> (<<0|1>, <1|3>>^n. <<2, 5>>)[1,1]:
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 14 2021
  • Mathematica
    LinearRecurrence[{3, 1}, {5, 7}, 20] (* Eric W. Weisstein, Jul 14 2021 *)
CoefficientList[Series[(2 - x)/(1 - 3 x - x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jul 14 2021 *)
  • PARI
    a(n)=([0,1;1,3]^n*[2;5])[1,1] \\ Charles R Greathouse IV, Oct 13 2016

Formula

G.f.: (2-x)/(1-3*x-x^2).
a(n) = 3*a(n-1) + a(n-2) with a(0)=2 and a(1)=5.
a(n) = ((4+7*A)*A^(-n-1) + (4+7*B)*B^(-n-1))/13 with A = (-3+sqrt(13))/2 and B = (-3-sqrt(13))/2.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^n*2/(A006497(n) - A006190(n)*sqrt(13)).
a(n) = 2 * Sum_{k=0..n-2} A168561(n-2,k)*3^k + 5 * Sum_{k=0..n-1} A168561(n-1,k)*3^k, n>0. - R. J. Mathar, Feb 14 2024
a(n) = 2*A006190(n+1) - A006190(n). - R. J. Mathar, Feb 14 2024

A249580 List of quadruples (r,s,t,u): the matrix M = [[4,12,9][2,5,3][1,2,1]] is raised to successive negative powers, then (r,s,t,u) are the square roots of M[1,3], M[1,1], M[3,3], M[3,1] respectively.

Original entry on oeis.org

3, -1, -2, 1, -9, 4, 7, -3, 30, -13, -23, 10, -99, 43, 76, -33, 327, -142, -251, 109, -1080, 469, 829, -360, 3567, -1549, -2738, 1189, -11781, 5116, 9043, -3927, 38910, -16897, -29867, 12970, -128511, 55807, 98644, -42837
Offset: 1

Views

Author

Russell Walsmith, Nov 02 2014

Keywords

Comments

The sequence, in reverse order, comprises numbers to the left of a(0) in A249579, where the terms would be labeled a(-1), a(-2), a(-3), ... .
This sequence 'factors' into four sequences with alternating signs. Ignoring signage, they are A052906, A003688, A052924 and A006190 (listed as crossrefs below).

Examples

			M^-1 = [[1,-6,9][-1,5,-6][1,-4,4]]. sqrt(M[1,3]) = 3, sqrt(M[1,1]) = -1, sqrt(M[3,3]) = -2, sqrt(M[3,1]) = 1. Then r = 3; s = -1; t = -2; ; u = 1.
M^-2 = [[16,-72,81][-12,55,-63][9,-42,49]]. sqrt(M[1,3]) = -9, sqrt(M[1,1]) = 4, sqrt(M[3,3]) = 7, sqrt(M[3,1]) = -3. Then r = -9; s = 4; t = 7; ; u = -3.

Links

Crossrefs

Cf. A249579. Disregarding signage, a(4n) = A052906; a(4n+1) = A003688; a(4n+2) = A052924; a(4n+3) = A006190.

Programs

  • Mathematica
    m[e_] := MatrixPower[{{4, 12, 9}, {2, 5, 3}, {1, 2, 1}}, -e]; g[e_, x_, y_] := (-1)^If[x == y, e, e + 1] Sqrt@ m[e][[x, y]]; f[e_] := {g[e, 1, 3], g[e, 1, 1], g[e, 3, 3], g[e, 3, 1]}; Array[f, 10] // Flatten (* Robert G. Wilson v, Dec 19 2014 *)
  • PARI
    Vec(-x*(x^6+x^5+x^3-2*x^2-x+3)/(x^8-3*x^4-1) + O(x^100)) \\ Colin Barker, Nov 06 2014

Formula

a(n) = -3*a(n-4)+a(n-8). - Colin Barker, Nov 06 2014
G.f.: -x*(x^6+x^5+x^3-2*x^2-x+3) / (x^8-3*x^4-1). - Colin Barker, Nov 06 2014
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