A052953 -id:A052953 - cp's OEIS frontend

A078008 Expansion of (1 - x)/((1 + x)(1 - 2x)).

1, 0, 2, 2, 6, 10, 22, 42, 86, 170, 342, 682, 1366, 2730, 5462, 10922, 21846, 43690, 87382, 174762, 349526, 699050, 1398102, 2796202, 5592406, 11184810, 22369622, 44739242, 89478486, 178956970, 357913942, 715827882, 1431655766, 2863311530, 5726623062

Offset: 0

N. J. A. Sloane, Nov 17 2002

Conjecture: a(n) = the number of fractions in the infinite Farey row of 2^n terms with even denominators. Compare the Salamin & Gosper item in the Beeler et al. link. - Gary W. Adamson, Oct 27 2003
Counts closed walks starting and ending at the same vertex of a triangle. 3*a(n) = P(C_n, 3) chromatic polynomial for 3 colors on cyclic graph C_n. A078008(n) + 2*A001045(n) = 2^n provides decomposition of Pascal's triangle. - Paul Barry, Nov 17 2003
Permutations with one fixed point avoiding 123 and 132.
General form: iterate k -> 2^n - k. See also A001045. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
The inverse g.f. generates sequence 1, 0, -2, -2, -2, -2, ...
a(n) gives the number of oriented (i.e., unreduced for symmetry) meanders on an (n+2) X 3 rectangular grid; see A201145. - Jon Wild, Nov 22 2011
Pisano period lengths: 1, 1, 6, 1, 4, 6, 6, 2, 18, 4, 10, 6, 12, 6, 12, 2, 8, 18, 18, 4, ... - R. J. Mathar, Aug 10 2012
a(n) is the number of length n binary words that end in an odd length run of 0's if we do not include the first letter of the word in our run length count. a(4) =6 because we have 0000, 0010, 0110, 1000, 1010, 1110. - Geoffrey Critzer, Dec 16 2013
a(n) is the top left entry of the n-th power of any of the six 3 X 3 matrices [0, 1, 1; 1, 1, 1; 1, 0, 0], [0, 1, 1; 1, 1, 0; 1, 1, 0], [0, 1, 1; 1, 0, 1; 1, 1, 0], [0, 1, 1; 1, 1, 0; 1, 0, 1], [0, 1, 1; 1, 0, 1; 1, 0, 1] or [0, 1, 1; 1, 0, 0; 1, 1, 1]. - R. J. Mathar, Feb 04 2014
a(n) is the number of compositions of n into parts of two kinds without part 1. - Gregory L. Simay, Jun 04 2018
a(n) is the number of words of length n over a binary alphabet whose position in the lexicographic order is a multiple of three.  a(3) = 2: aba, bab. - Alois P. Heinz, Apr 13 2022
a(n) is the number of words of length n over a ternary alphabet starting with a fixed letter (say, 'a') and ending in a different letter, such that no two adjacent letters are the same. a(4) = 6: abab, abac, abcb, acab, acac, acbc. - Ignat Soroko, Jul 19 2023

			G.f. = 1 + 2*x^2 + 2*x^3 + 6*x^4 + 10*x^5 + 22*x^6 + ... - _Michael Somos_, Mar 18 2022

Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 1.1.10a.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms n=0..300 from T. D. Noe)
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.]
Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015. See Section 4.6.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Paul Barry and A. Hennessy, Notes on a Family of Riordan Arrays and Associated Integer Hankel Transforms , JIS 12 (2009) 09.5.3.
M. Beeler, R. W. Gosper, and R. C. Schroeppel, R. HAKMEM. MIT AI Memo 239, Feb 29 1972. (Item #54 by Salamin & Gosper)
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, arXiv:1907.06517 [math.CO], 2019.
Ernesto Estevez-Rams, Cristy Azanza Ricardo and Beatriz Aragón Fernández, An alternative expression for counting the number of close-packaged polytypes, Z. Krist. 220 (2005) 592-595, eq (4)
Leonhard Euler, Introductio in analysin infinitorum, (1748), section 216.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 13.
T. Mansour and A. Robertson, Refined Restricted Permutations Avoiding Subsets of Patterns of Length Three, arXiv:math/0204005 [math.CO], 2002.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for linear recurrences with constant coefficients, signature (1,2).
Index entries for sequences related to Chebyshev polynomials.

First differences of A001045.
See A151575 for a signed version.
Bisections: A047849, A020988.
Cf. A151548, A139250, A151555, A153006.

List([0..40], n-> (2^n + 2*(-1)^n)/3); # G. C. Greubel, Jun 28 2019

[(2^n + 2*(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Jun 28 2019

Table[(2^n + 2*(-1)^n)/3, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008; modified by G. C. Greubel, Jun 28 2019 *)
CoefficientList[Series[(1-x)/(1-x-2x^2),{x,0,40}],x]  (* Harvey P. Dale, Mar 30 2011 *)
LinearRecurrence[{1, 2}, {1, 0}, 40] (* Jean-François Alcover, Sep 23 2017 *)

a(n)=(1<Charles R Greathouse IV, Jun 10 2011

def A078008(n): return ((1<Chai Wah Wu, Apr 18 2025

[(2^n + 2*(-1)^n)/3 for n in (0..40)] # G. C. Greubel, Jun 28 2019

Euler expands(1-x)/(1 - x - 2*x^2) into an infinite series and finds that the coefficient of the n-th term is (2^n + (-1)^n 2)/3. Section 226 shows that Euler could have easily found the recursion relation: a(n) = a(n-1) + 2a(n-2) with a(0) = 1 and a(1) = 0. - V. Frederick Rickey (fred-rickey(AT)usma.edu), Feb 10 2006. [Typos corrected by Jaume Oliver Lafont, Jun 01 2009]
a(n) = Sum_{k=0..floor(n/3)} binomial(n, f(n)+3*k) where f(n) = (0, 2, 1, 0, 2, 1, ...) = A080424(n). - Paul Barry, Feb 20 2003
E.g.f. (exp(2*x) + 2*exp(-x))/3. - Paul Barry, Apr 20 2003
a(n) = A001045(n) + (-1)^n = A000079(n) - 2*A001045(n). - Paul Barry, Feb 20 2003
a(n) = (2^n + 2*(-1)^n)/3. - Mario Catalani (mario.catalani(AT)unito.it), Aug 29 2003
a(n) = T(n, i/(2*sqrt(2)))*(-i*sqrt(2))^n - U(n-1, i/(2*sqrt(2)))*(-i*sqrt(2))^(n-1)/2. - Paul Barry, Nov 17 2003
From Paul Barry, Jul 30 2004: (Start)
a(n) = 2*a(n-1) + 2*(-1)^n for n > 0, with a(0)=1.
a(n) = Sum_{k=0..n} (-1)^k*(2^(n-k-1) + 0^(n-k)/2). (End)
a(n) = A014113(n-1) for n > 0; a(n) = A052953(n-1) - 2*(n mod 2) = sum of n-th row of the triangle in A108561. - Reinhard Zumkeller, Jun 10 2005
A137208(n+1) - 2*A137208(n) = a(n) signed. - Paul Curtz, Aug 03 2008
a(n) = A001045(n+1) - A001045(n) - Paul Curtz, Feb 09 2009
If p[1] =0, and p[i]=2, (i>1), 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, Apr 29 2010
a(n) = 2*(a(n-2) + a(n-3) + a(n-4) ... + a(0)), that is, twice the sum of all the previous terms except the last; with a(0) = 1 and a(1) = 0. - Benoit Jubin, Nov 21 2011
a(n+1) = 2*A001045(n). - Benoit Jubin, Nov 22 2011
G.f.: 1 - x + x*Q(0), where Q(k) = 1 + 2*x^2 + (2*k+3)*x - x*(2*k+1 + 2*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013
G.f.: 1+ x^2*Q(0), where Q(k) = 1 + 1/(1 - x*(4*k+1+2*x)/(x*(4*k+3+2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 01 2014
a(n) = 3*a(n-2) + 2*a(n-3). - David Neil McGrath, Sep 10 2014
a(n) = (-1)^n * A151575(n). - G. C. Greubel, Jun 28 2019
a(n)+a(n+1) = 2^n. - R. J. Mathar, Feb 24 2021
a(n) = -a(2-n) * (-2)^(n-1) = (3/2)*(a(n-1) + a(n-2)) - a(n-3) for all n in Z. - Michael Somos, Mar 18 2022

A112468 Riordan array (1/(1-x), x/(1+x)).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, -1, 1, 1, 0, 2, -2, 1, 1, 1, -2, 4, -3, 1, 1, 0, 3, -6, 7, -4, 1, 1, 1, -3, 9, -13, 11, -5, 1, 1, 0, 4, -12, 22, -24, 16, -6, 1, 1, 1, -4, 16, -34, 46, -40, 22, -7, 1, 1, 0, 5, -20, 50, -80, 86, -62, 29, -8, 1, 1, 1, -5, 25, -70, 130, -166, 148, -91, 37, -9, 1, 1, 0, 6, -30, 95, -200, 296, -314, 239, -128, 46, -10, 1

Offset: 0

Paul Barry, Sep 06 2005

Row sums are A040000. Diagonal sums are A112469. Inverse is A112467. Row sums of k-th power are 1, k+1, k+1, k+1, .... Note that C(n,k) = Sum_{j=0..n-k} C(n-j-1, n-k-j).
Equals row reversal of triangle A112555 up to sign, where log(A112555) = A112555 - I. Unsigned row sums equals A052953 (Jacobsthal numbers + 1). Central terms of even-indexed rows are a signed version of A072547. Sums of squared terms in rows yields A112556, which equals the first differences of the unsigned central terms. - Paul D. Hanna, Jan 20 2006
Sum_{k=0..n} T(n,k)*x^k = A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively (see the square array in A112739). - Philippe Deléham, Feb 22 2014

			Triangle starts
  1;
  1,  1;
  1,  0,  1;
  1,  1, -1,  1;
  1,  0,  2, -2,  1;
  1,  1, -2,  4, -3,  1;
  1,  0,  3, -6,  7, -4,  1;
Matrix log begins:
  0;
  1,  0;
  1,  0,  0;
  1,  1, -1,  0;
  1,  1,  1, -2,  0;
  1,  1,  1,  1, -3,  0; ...
Production matrix begins
  1,  1,
  0, -1,  1,
  0,  0, -1,  1,
  0,  0,  0, -1,  1,
  0,  0,  0,  0, -1,  1,
  0,  0,  0,  0,  0, -1,  1,
  0,  0,  0,  0,  0,  0, -1,  1.
- _Paul Barry_, Apr 08 2011

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
H. Belbachir and F. Bencherif, On some properties of bivariate Fibonacci and Lucas Polynomials, JIS 11 (2008) 08.2.6.
Hacene Belbachir and Athmane Benmezai, Expansion of Fibonacci and Lucas Polynomials: An Answer to Prodinger's Question, Journal of Integer Sequences, Vol. 15 (2012), #12.7.6.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
H. Prodinger, On the expansion of Fibonacci and Lucas Polynomials, JIS 12 (2009) 09.1.6.

Cf. A174294, A174295, A174296, A174297. - Mats Granvik, Mar 15 2010
Cf. A072547 (central terms), A112555 (reversed rows), A112465, A052953, A112556, A112739, A119258.
See A279006 for another version.

T:= function(n,k)
    if k=0 or k=n then return 1;
    else return T(n-1,k-1) - T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 13 2019

a112468 n k = a112468_tabl !! n !! k
a112468_row n = a112468_tabl !! n
a112468_tabl = iterate (\xs -> zipWith (-) ([2] ++ xs) (xs ++ [0])) [1]
-- Reinhard Zumkeller, Jan 03 2014

function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1,k-1) - T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 13 2019

T := (n,k,m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k,k+1)*hypergeom( [1,n+1],[k+2],m)/(k+1)!; A112468 := (n,k) -> T(n,n-k,-1);
seq(print(seq(simplify(A112468(n,k)),k=0..n)),n=0..10); # Peter Luschny, Jul 25 2014

T[n_, 0] = 1; T[n_, n_] = 1; T[n_, k_ ]:= T[n, k] = T[n-1, k-1] - T[n-1, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Jean-François Alcover, Mar 06 2013 *)

{T(n,k)=local(m=1,x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff(polcoeff((1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x),n,X),k,Y)} \\ Paul D. Hanna, Jan 20 2006

T(n,k) = if(k==0 || k==n, 1, T(n-1, k-1) - T(n-1, k)); \\ G. C. Greubel, Nov 13 2019

@CachedFunction
def T(n, k):
    if (k<0 or n<0): return 0
    elif (k==0 or k==n): return 1
    else: return T(n-1, k-1) - T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 13 2019

Triangle T(n,k) read by rows: T(n,0)=1, T(n,k) = T(n-1,k-1) - T(n-1,k). - Mats Granvik, Mar 15 2010
Number triangle T(n, k)= Sum_{j=0..n-k} C(n-j-1, n-k-j)*(-1)^(n-k-j).
G.f. of matrix power T^m: (1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x). G.f. of matrix log: x*(1-2*x*y+x^2*y)/(1-x*y)^2/(1-x). - Paul D. Hanna, Jan 20 2006
T(n, k) = R(n,n-k,-1) where R(n,k,m) = (1-m)^(-n+k)-m^(k+1)*Pochhammer(n-k,k+1)*hyper2F1([1,n+1],[k+2],m)/(k+1)!. - Peter Luschny, Jul 25 2014

A026644 a(n) = a(n-1) + 2*a(n-2) + 2, for n>=3, where a(0)= 1, a(1)= 2, a(2)= 4.

Original entry on oeis.org

1, 2, 4, 10, 20, 42, 84, 170, 340, 682, 1364, 2730, 5460, 10922, 21844, 43690, 87380, 174762, 349524, 699050, 1398100, 2796202, 5592404, 11184810, 22369620, 44739242, 89478484, 178956970, 357913940, 715827882, 1431655764, 2863311530, 5726623060

Offset: 0

Clark Kimberling

Number of moves to solve Chinese rings puzzle.
a(n-1) (with a(0):=0) enumerates all sequences of length m=1,2,...,floor(n/2) with nonzero integer entries n_i satisfying sum |n_i| <= n-m. Rephrasing K. A. Meissner's example p. 6. Example n=4: from length m=1: [1], [2], [3], each in 2 signed versions; from m=2: [1,1] in 2^2 = 4 signed versions. Hence a(3) = a(4-1) = 3*2 + 1*4 = 10.
Also the number of different 3-colorings (out of 4 colors) for the vertices of all triangulated planar polygons on a base with n+1 vertices if the colors of the two base vertices are fixed. - Patrick Labarque, Mar 23 2010
For n > 0, also the total distance that the disks travel from the leftmost peg to the rightmost peg in the Tower of Hanoi puzzle, in the unique solution with 2^n-1 moves (see links). - Sela Fried, Dec 17 2023

Richard I. Hess, Compendium of Over 7000 Wire Puzzles, privately printed, 1991.
Richard I. Hess, Analysis of Ring Puzzles, booklet distributed at 13th International Puzzle Party, Amsterdam, Aug 20 1993.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Thomas Baruchel, Properties of the cumulated deficient binary digit sum, arXiv:1908.02250 [math.NT], 2019.
Sela Fried, Economically solving the Tower of Hanoi puzzle.
Nicolas Gastineau and O. Togni, On S-packing edge-colorings of cubic graphs, arXiv preprint arXiv:1711.10906 [cs.DM], 2017.
Lee Hae-hwang, Illustration of initial terms in terms of rosemary plants
Krzysztof A. Meissner, Black hole entropy in Loop Quantum Gravity, arXiv:gr-qc/0407052, 2004.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Row sums of A026637.
For n >= 1, equals twice A000975, also A001045 - 1.
A167030 is an essentially identical sequence.

[n eq 0 select 1 else (2^(n+2) -3-(-1)^n)/3 : n in [0..40]]; // G. C. Greubel, Jun 28 2024

f:=n-> if n mod 2 = 0 then (2^(n+2)-4)/3 else (2^(n+2)-2)/3; fi;

Join[{1}, Floor[(2^Range[3, 40] - 2)/3]] (* or *) LinearRecurrence[{2,1,-2},{1,2,4,10},40] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
CoefficientList[Series[(1-x^2+2x^3)/((1-x)(1-x-2x^2)),{x,0,1001}],x] (* Vincenzo Librandi, Apr 04 2012 *)

Vec((1-x^2+2*x^3)/(1-x)/(1-x-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Apr 04 2012

def A026644(n): return ((4<Chai Wah Wu, Apr 17 2025

[(2^(n+2)-3-(-1)^n)/3 + int(n==0) for n in range(41)] # G. C. Greubel, Jun 28 2024

a(2*k) = 2*a(2*k-1), a(2*k+1) = 2*a(2*k) + 2. - Peter Shor, Apr 11 2002
For n>0: a(n+1) = a(n) + 2*b(n+1) + 4*b(n), where b(k) = A001045(k). - N. J. A. Sloane, May 16 2003
For n>0: if n mod 2 = 0 then (2^(n+2)-4)/3 else (2^(n+2)-2)/3. - Richard Hess
a(2*n) = 2*n-1 + Sum_{k=0..2*n-1} a(k), n>0; a(2*n+1) = 2*n+1 + Sum_{k=0..n} a(k). - Lee Hae-hwang, Sep 17 2002; corrected by R. J. Mathar, Oct 21 2008
a(n) = 2*n + 2*Sum_{k=1..n-2} a(k), n>0. - Lee Hae-hwang, Sep 19 2002; corrected by R. J. Mathar, Oct 21 2008
From Paul Barry, Oct 24 2007: (Start)
G.f.: (1 - x^2 + 2*x^3)/((1 - x)*(1 - x - 2*x^2)).
a(n) = J(n+2) - 1 + 0^n, where J(n) = A001045(n) (Jacobsthal numbers).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = 0^n + Sum_{k=0..n} (2 - 2*0^(n-k))*J(k+1). (End)
a(n) = A052953(n+1) - 2, n>0. [Moved from A020988, R. J. Mathar, Oct 21 2008]
a(n) = floor(A097074(n+1)/2), n>0. - Gary Detlefs, Dec 19 2010
a(n) = A169969(2*n-1) - 1, n>=2; a(n) = 3*2^(n-1) - 1 - A169969(2*n-7), n>=5 . - Yosu Yurramendi, Jul 05 2016
a(n+3) = 3*2^(n+2) - 2 - a(n), n>=1, a(1)=2, a(2)=4, a(3)=10 . - Yosu Yurramendi, Jul 05 2016
a(n) + A084170(n) = 3*2^n - 2, n>=1. - Yosu Yurramendi, Jul 05 2016
E.g.f: (3 - 4*cosh(x) + 4*cosh(2*x) - 2*sinh(x) + 4*sinh(2*x))/3. - Ilya Gutkovskiy, Jul 05 2016
a(n+3) = 9*2^n + A084170(n), n>=0. - Yosu Yurramendi, Jul 07 2016
a(n) = A000975(n+1) - A000035(n+1), n>0, a(0)=1. - Yuchun Ji, Aug 05 2020

Recurrence in definition line found by Lee Hae-hwang, Apr 03 2002

A108561 Triangle read by rows: T(n,0)=1, T(n,n)=(-1)^n, T(n+1,k)=T(n,k-1)+T(n,k) for 0 < k < n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 1, -1, 1, 2, 2, 0, 1, 1, 3, 4, 2, 1, -1, 1, 4, 7, 6, 3, 0, 1, 1, 5, 11, 13, 9, 3, 1, -1, 1, 6, 16, 24, 22, 12, 4, 0, 1, 1, 7, 22, 40, 46, 34, 16, 4, 1, -1, 1, 8, 29, 62, 86, 80, 50, 20, 5, 0, 1, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, -1, 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 0, 1, 1, 11, 56, 174, 367

Offset: 0

Reinhard Zumkeller, Jun 10 2005

Sum_{k=0..n} T(n,k) = A078008(n);
Sum_{k=0..n} abs(T(n,k)) = A052953(n-1) for n > 0;
T(n,1) = n - 2 for n > 1;
T(n,2) = A000124(n-3) for n > 2;
T(n,3) = A003600(n-4) for n > 4;
T(n,n-6) = A001753(n-6) for n > 6;
T(n,n-5) = A001752(n-5) for n > 5;
T(n,n-4) = A002623(n-4) for n > 4;
T(n,n-3) = A002620(n-1) for n > 3;
T(n,n-2) = A008619(n-2) for n > 2;
T(n,n-1) = n mod 2 for n > 0;
T(2*n,n) = A072547(n+1).
Sum_{k=0..n} T(n,k)*x^k = A232015(n), A078008(n), A000012(n), A040000(n), A001045(n+2), A140725(n+1) for x = 2, 1, 0, -1, -2, -3 respectively. - Philippe Deléham, Nov 17 2013, Nov 19 2013
(1,a^n) Pascal triangle with a = -1. - Philippe Deléham, Dec 27 2013
T(n,k) = A112465(n,n-k). - Reinhard Zumkeller, Jan 03 2014

			From _Philippe Deléham_, Nov 17 2013: (Start)
Triangle begins:
  1;
  1, -1;
  1,  0,  1;
  1,  1,  1, -1;
  1,  2,  2,  0,  1;
  1,  3,  4,  2,  1, -1;
  1,  4,  7,  6,  3,  0,  1; (End)

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
C. Merino, S. D. Noble, M. Ramirez-Ibanez, R. Villarroel-Flores, On the structure of the h-vector of a paving matroid, Eur. J. Comb. 33, No. 8, 1787-1799 (2012).
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318 (a=1), A008949(a=2), A164844(a=10).
Similar to the triangles A035317, A059259, A080242, A112555.
Cf. A228196
Cf. A072547 (central terms).

Flat(List([0..13],n->List([0..n],k->Sum([0..k],i->Binomial(n,i)*(-2)^(k-i))))); # Muniru A Asiru, Feb 19 2018

a108561 n k = a108561_tabl !! n !! k
a108561_row n = a108561_tabl !! n
a108561_tabl = map reverse a112465_tabl
-- Reinhard Zumkeller, Jan 03 2014

A108561 := (n, k) -> add(binomial(n, i)*(-2)^(k-i), i = 0..k):
seq(seq(A108561(n,k), k = 0..n), n = 0..12); # Peter Bala, Feb 18 2018

Clear[t]; t[n_, 0] = 1; t[n_, n_] := t[n, n] = (-1)^Mod[n, 2]; t[n_, k_] := t[n, k] = t[n-1, k] + t[n-1, k-1]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013 *)

def A108561_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^k*prec(n, k) for k in (1..n-1)]+[(-1)^(n+1)]
for n in (1..12): print(A108561_row(n)) # Peter Luschny, Mar 16 2016

G.f.: (1-y*x)/(1-x-(y+y^2)*x). - Philippe Deléham, Nov 17 2013
T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 17 2013
From Peter Bala, Feb 18 2018: (Start)
T(n,k) = Sum_{i = 0..k} binomial(n,i)*(-2)^(k-i), 0 <= k <= n.
The n-th row polynomial is the n-th degree Taylor polynomial of the rational function (1 + x)^n/(1 + 2*x) about 0. For example, for n = 4, (1 + x)^4/(1 + 2*x) = 1 + 2*x + 2*x^2 + x^4 + O(x^5). (End)

Definition corrected by Philippe Deléham, Dec 26 2013

A128209 Jacobsthal numbers(A001045) + 1.

Original entry on oeis.org

1, 2, 2, 4, 6, 12, 22, 44, 86, 172, 342, 684, 1366, 2732, 5462, 10924, 21846, 43692, 87382, 174764, 349526, 699052, 1398102, 2796204, 5592406, 11184812, 22369622, 44739244, 89478486, 178956972, 357913942, 715827884, 1431655766, 2863311532, 5726623062

Offset: 0

Paul Barry, Feb 19 2007

Row sums of A128208.
Essentially the same as A052953. - R. J. Mathar, Jun 14 2008
Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 1, a(n+1) is the number of different representations of matrix P^(-1)+I+P by sum of permutation matrices. - Vladimir Shevelev, Apr 12 2010
a(n) is the rank of Fibonacci(n+2) in row n of A049456 (regarded as an irregular triangle read by rows). - N. J. A. Sloane, Nov 23 2016

V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19. [From Vladimir Shevelev, Apr 12 2010]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67. [Annotated and corrected scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A167030, A153643, A049456.

[1+2^n/3-(-1)^n/3: n in [0..40]]; // Vincenzo Librandi, Aug 11 2011

CoefficientList[Series[(1-3*x^2)/(1-2*x-x^2+2*x^3),{x,0,40}],x] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)

a(n)=2^n\/3+1 \\ Charles R Greathouse IV, Jan 31 2012

def A128209(n): return ((1<Chai Wah Wu, Apr 17 2025

a(n) = 1 + 2^n/3 - (-1)^n/3.

G.f.: (1-3*x^2)/(1 - 2*x - x^2 + 2*x^3).

A103196 a(n) = (1/9)(2^(n+3)-(-1)^n(3n-1)).

Original entry on oeis.org

1, 2, 3, 8, 13, 30, 55, 116, 225, 458, 907, 1824, 3637, 7286, 14559, 29132, 58249, 116514, 233011, 466040, 932061, 1864142, 3728263, 7456548, 14913073, 29826170, 59652315, 119304656, 238609285, 477218598, 954437167

Offset: 0

Creighton Dement, Mar 18 2005

A floretion-generated sequence relating to the Jacobsthal sequence A001045 as well as to A095342 (Number of elements in n-th string generated by a Kolakoski(5,1) rule starting with a(1)=1). (a(n)) may be seen as the result of a certain transform of the natural numbers (see program code).

Floretion Algebra Multiplication Program, FAMP Code: 4jesleftforseq[A*B] with A = + 'i + 'j + i' + j' + 'ii' + 'jj' + 'ij' + 'ji' + e and B = - .25'i + .25'j + .25'k + .25i' - .25j' + .25k' - .25'ii' + .25'jj' + .25'kk' + .25'ij' + .25'ik' + .25'ji' + .25'jk' - .25'ki' - .25'kj' - .25e; 1vesforseq[A*B](n) = n, ForType: 1A.

Index entries for linear recurrences with constant coefficients, signature (0,3,2).

Cf. A001045, A095342, A053088, A034299, A052953, A026644.

Table[(2^(n+3)-(-1)^n (3n-1))/9,{n,0,30}] (* or *) LinearRecurrence[ {0,3,2},{1,2,3},40] (* Harvey P. Dale, Jul 09 2018 *)

G.f. (2x+1)/((1-2x)(x+1)^2); Superseeker results: a(n) + a(n+1) = A001045(n+3); a(n+1) - a(n) = A095342(n+1); a(n+2) - a(n+1) - a(n) = A053088(n+1) = A034299(n+1) - A034299(n); a(n) + 2a(n+1) + a(n+2) = 2^(n+3); a(n+2) - 2a(n+1) + a(n) = A053088(n+1) - A053088(n); a(n+2) - a(n) = A001045(n+4) - A001045(n+3) = A052953(n+3) - A052953(n+2) = A026644(n+2) - A026644(n+1);

a(n)=sum{k=0..n+2, (-1)^(n-k)*C(n+2, k)phi(phi(3^k))}; a(n)=sum{k=0..n+2, (-1)^(n-k)*C(n+2, k)(2*3^k/9+C(1, k)/3+4*C(0, k)/9)}; a(n)=sum{k=0..n+2, J(n-k+3)((-1)^(k+1)-2C(1, k)+4C(0, k))} where J(n)=A001045(n); a(n)=A113954(n+2). - Paul Barry, Nov 09 2005

cp's OEIS Frontend

A078008 Expansion of (1 - x)/((1 + x)*(1 - 2*x)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Python

Sage

Formula

A112468 Riordan array (1/(1-x), x/(1+x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A026644 a(n) = a(n-1) + 2*a(n-2) + 2, for n>=3, where a(0)= 1, a(1)= 2, a(2)= 4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

SageMath

Formula

Extensions

A108561 Triangle read by rows: T(n,0)=1, T(n,n)=(-1)^n, T(n+1,k)=T(n,k-1)+T(n,k) for 0 < k < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Haskell

Maple

Mathematica

Sage

Formula

Extensions

A128209 Jacobsthal numbers(A001045) + 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

A078008 Expansion of (1 - x)/((1 + x)(1 - 2x)).