A128209 -id:A128209 - cp's OEIS frontend

A005581 a(n) = (n-1)n(n+4)/6.

0, 0, 2, 7, 16, 30, 50, 77, 112, 156, 210, 275, 352, 442, 546, 665, 800, 952, 1122, 1311, 1520, 1750, 2002, 2277, 2576, 2900, 3250, 3627, 4032, 4466, 4930, 5425, 5952, 6512, 7106, 7735, 8400, 9102, 9842, 10621, 11440, 12300, 13202, 14147, 15136, 16170

Offset: 0

N. J. A. Sloane

A class of Boolean functions of n variables and rank 2.
Also, number of inscribable triangles within a (n+4)-gon sharing with them its vertices but not its sides. - Lekraj Beedassy, Nov 14 2003
a(n) = A111808(n,3) for n > 2. - Reinhard Zumkeller, Aug 17 2005
If X is an n-set and Y a fixed 2-subset of X then a(n-2) is equal to the number of (n-3)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
The sequence starting with offset 2 = binomial transform of [2, 5, 4, 1, 0, 0, 0, ...]. - Gary W. Adamson, Mar 20 2009
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 >= 4, a(n-4) is the number of (0,1) n X n matrices A <= P^(-1) + I + P having exactly two 1's in every row and column with perA=8. - Vladimir Shevelev, Apr 12 2010
Also arises as the number of triples of edges which can be chosen as the cut-points in the "three-opt" heuristic for a traveling salesman problem on (n+4) nodes. - James McDermott, Jul 10 2015
a(n) = risefac(n, 3)/3! - n is for n >= 1 also the number of independent components of a symmetric traceless tensor of rank 3 and dimension n. Here risefac is the rising factorial. - Wolfdieter Lang, Dec 10 2015
For n >= 2, a(n) is the number of characters in a word Q formed by concatenating all 'directed' ( left to right or vice versa), unrearranged subwords, from length 1 to (n-1), of a length (n-1) word q- allowing for the appearance of repeated subwords- and simply inserting an extra character for all subwords thus concatenated. - Christopher Hohl, May 30 2019

			In hexagon ABCDEF, the "interior" triangles are ACE and BDF, and a(6-4)=a(2)=2. - _Toby Gottfried_, Nov 12 2011
G.f. = 2*x^2 + 7*x^3 + 16*x^4 + 30*x^5 + 50*x^6 + 77*x^7 + 112*x^8 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
Joseph D. Konhauser, Dan Velleman and Stan Wagon,, Which Way Did the Bicycle Go?, MAA, 1996, p. 177.
V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, Vol. 3 (1992), pp. 15-19. - Vladimir Shevelev, Apr 12 2010
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #51 (the case k=3) (First published: San Francisco: Holden-Day, Inc., 1964).

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972. [Alternative scanned copy]
Armen G. Bagdasaryan and Ovidiu Bagdasar, On some results concerning generalized arithmetic triangles, Electronic Notes in Discrete Mathematics, Vol. 67 (2018), pp. 71-77.
Beáta Bényi, Miguel Méndez, José L. Ramírez and Tanay Wakhare, Restricted r-Stirling Numbers and their Combinatorial Applications, arXiv:1811.12897 [math.CO], 2018.
John Elias, Illustration: Triangular Staircasing
Richard K. Guy, Letter to N. J. A. Sloane, 1987.
Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart., Vol. 49, No. 3 (2011), pp. 231-243.
Milan Janjic, Two Enumerative Functions.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, Vol. 11, No. 4 (1999), pp. 127-138.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, Vol. 9, No. 6 (1999), pp. 593-605.
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Alice McLeod and William Moser, Counting cyclic binary strings, Math. Mag., Vol. 80, No. 1 (2007), pp. 29-37.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Dead link]
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Cached copy, May 15 2013]
Eric Weisstein's World of Mathematics, Trinomial Coefficient.
Index entries for sequences related to Boolean functions.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000217, A000292, A005582, A005586, A111808.
Cf. A176222, A000211, A052928, A128209.
Cf. A127672 (Chebyshev C).

[(n-1)*n*(n+4)/6 : n in [0..50]]; // Wesley Ivan Hurt, Jul 10 2015

A005581 := n->(n-1)*n*(n+4)/6: seq(A005581(n), n=0..50);
a:=n->sum ((j+3)*j/2,j=0..n): seq(a(n),n=-1..49); # Zerinvary Lajos, Dec 17 2006
seq((n+3)*binomial(n,3)/n, n=1..46); # Zerinvary Lajos, Feb 28 2007
A005581:=-(-2+z)/(z-1)**4; # Simon Plouffe in his 1992 dissertation
seq(sum(binomial(n,m), m=1..3)+n^2,n=-1..44); # Zerinvary Lajos, Jun 19 2008
A005581 := n -> GegenbauerC(`if`(3A005581(n)), n=0..50); # Peter Luschny, May 10 2016

Table[(n-1)*n*(n+4)/6, {n,0,50}] (* Stefan Steinerberger, Apr 10 2006 *)
LinearRecurrence[{4,-6,4,-1},{0,0,2,7},50] (* Harvey P. Dale, Sep 22 2012 *)

A005581(n):=(n-1)*n*(n+4)/6$ makelist(A005581(n),n,0,50); /* Martin Ettl, Dec 18 2012 */

{a(n) = n * (n+4) * (n-1) / 6}; /* Michael Somos, Apr 13 2007 */

concat([0, 0], Vec((x^2)*(2-x)/(1-x)^4 + O(x^50))) \\ Altug Alkan, Dec 10 2015

[(n-1)*n*(n+4)/6 for n in range(50)] # Danny Rorabaugh, Apr 20 2015

G.f.: (x^2)*(2-x)/(1-x)^4.
a(n) = binomial(n+1, n-2) + binomial(n, n-2).
a(n) = A027907(n, 3), n >= 0 (fourth column of trinomial coefficients). - N. J. A. Sloane, May 16 2003
Convolution of {1, 2, 3, ...} with {2, 3, 4, ...}. - Jon Perry, Jun 25 2003
a(n+2) = 2*te(n) - te(n-1), e.g., a(5) = 2*te(3) - te(2) = 2*20 - 10 = 30, where te(n) are the tetrahedral numbers A000292. - Jon Perry, Jul 23 2003
a(n) is the coefficient of x^3 in the expansion of (1+x+x^2)^n. For example, a(1)=0 since (1+x+x^2)^1=1+x+x^2. - Peter C. Heinig (algorithms(AT)gmx.de), Apr 09 2007
E.g.f.: (x^2 + x^3/6) * exp(x). - Michael Somos, Apr 13 2007
a(n) = - A005586(-4-n) for all n in Z. - Michael Somos, Apr 13 2007
a(n) = C(4+n,3)-(n+4)*(n+1), since C(4+n,3) = number of all triangles in (n+4)-gon, and (n+4)*(n+1)=number of triangles with at least one of the edges included. Example: n=0,in a square, all 4 possible triangles include some of the square's edges and C(4+n,3)-(n+4)*(n+1)=4-4*1=0 = number of other triangles = a(0). - Toby Gottfried, Nov 12 2011
a(n) = 2*binomial(n,2) + binomial(n,3). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(0)=0, a(1)=0, a(2)=2, a(3)=7, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Sep 22 2012
a(n) = A000292(n-1) + A000217(n-1) for all n in Z. - Michael Somos, Jul 29 2015
a(n+2) = -A127672(6+n, n), n >= 0, with A127672 giving the coefficients of Chebyshev's C polynomials. See the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 10 2015
a(n) = GegenbauerC(N, -n, -1/2) where N = 3 if 3Peter Luschny, May 10 2016
From Amiram Eldar, Jan 09 2022: (Start)
Sum_{n>=2} 1/a(n) = 163/200.
Sum_{n>=2} (-1)^n/a(n) = 12*log(2)/5 - 253/200. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000

A005582 a(n) = n(n+1)(n+2)*(n+7)/24.

Original entry on oeis.org

0, 2, 9, 25, 55, 105, 182, 294, 450, 660, 935, 1287, 1729, 2275, 2940, 3740, 4692, 5814, 7125, 8645, 10395, 12397, 14674, 17250, 20150, 23400, 27027, 31059, 35525, 40455, 45880, 51832, 58344, 65450, 73185, 81585, 90687, 100529, 111150, 122590, 134890

Offset: 0

N. J. A. Sloane

a(n) = number of Dyck (n+2)-paths with exactly 2 rows of peaks. A row of peaks is a maximal sequence of peaks all at the same height and 2 units apart. For example, UDUDUD ( = /\/\/\ ) contains exactly one row of peaks, as does UUUDDD, but UDUUDDUD has three and a(1)=2 counts UDUUDD, UUDDUD. - David Callan, Mar 02 2005
If X is an n-set and Y a fixed 2-subset of X then a(n-4) is equal to the number of (n-4)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
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>=7, a(n-7) is the number of (0,1) n X n matrices A<=P^(-1)+I+P having exactly two 1's in every row and column with perA=16. - Vladimir Shevelev, Apr 12 2010
Row 2 of the convolution array A213550. - Clark Kimberling, Jun 20 2012
a(n-1) = risefac(n, 4)/4! - risefac(n, 2)/2! is for n >= 1 also the number of independent components of a symmetric traceless tensor of rank 4 and dimension n. Here risefac is the rising factorial. - Wolfdieter Lang, Dec 10 2015
Consider the array formed by the second polygonal numbers of increasing rank:
  A000217(-1-n): 0,  1,  3,  6, 10, 15, ...
  A000270(-1-n): 1,  4,  9, 16, 25, 36, ...
  A000326(-1-n): 2,  7, 15, 26, 40, 57, ...
  A000384(-1-n): 3, 10, 21, 36, 55, 78, ...
Then the antidiagonal sums yield this sequence. - Michael Somos, Nov 23 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
Vladimir 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]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #51 (the case k=4) (First published: San Francisco: Holden-Day, Inc., 1964)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.
Milan Janjic, Two Enumerative Functions
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380. [See p. 301]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A005581.

Cf. A000211, A052928, A128209, A176222.

[seq(binomial(n,4)+2*binomial(n,3), n=2..43)]; # Zerinvary Lajos, Jul 26 2006
seq((n+4)*binomial(n,4)/n, n=3..43); # Zerinvary Lajos, Feb 28 2007
A005582:=(-2+z)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Table[n(n+1)(n+2)(n+7)/24,{n,0,40}] (* Harvey P. Dale, Jun 01 2012 *)

concat(0, Vec(x*(2-x)/(1-x)^5 + O(x^100))) \\ Altug Alkan, Dec 10 2015

a(n) = binomial(n+3, n-1) + binomial(n+2, n-1).
a(n) = binomial(n,4) + 2*binomial(n,3), n>=2. - Zerinvary Lajos, Jul 26 2006
From Colin Barker, Jan 28 2012: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(2-x)/(1-x)^5. (End)
a(n) = Sum_{k=1..n} ( Sum_{i=1..k} i(n-k+2) ). - Wesley Ivan Hurt, Sep 26 2013
a(n+1) = A127672(8+n, n), n >= 0, with the Chebyshev C-polynomial coefficients A127672(n, k). See the Abramowitz-Stegun reference.  - Wolfdieter Lang, Dec 10 2015
E.g.f.: (1/24)*x*(48 + 60*x + 16*x^2 + x^3)*exp(x). - G. C. Greubel, Jul 01 2017
Sum_{n>=1} 1/a(n) = 853/1225. - Amiram Eldar, Jan 02 2021
a(n) = A005587(-7-n) for all n in Z. - Michael Somos, Nov 23 2021

More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000

A176222 a(n) = (n^2 - 3*n + 1 + (-1)^n)/2.

Original entry on oeis.org

0, 3, 5, 10, 14, 21, 27, 36, 44, 55, 65, 78, 90, 105, 119, 136, 152, 171, 189, 210, 230, 253, 275, 300, 324, 351, 377, 406, 434, 465, 495, 528, 560, 595, 629, 666, 702, 741, 779, 820, 860, 903, 945, 990, 1034, 1081, 1127, 1176, 1224, 1275, 1325, 1378, 1430

Offset: 3

Vladimir Shevelev, Apr 12 2010

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).
Let T = P^(-1)+I+P.
  11000...01
  11100....0
  01110.....
  00111.....
  ..........
  00.....111
  10.....011
Then a(n) is the number of (0,1) n X n matrices A <= T (i.e., an element of A can be 1 only if T has a 1 at this place) having exactly two 1's in every row and column with per(A) = 4.
a(n) is the maximum number m such that m white kings and m black kings can coexist on an n+1 X n+1 chessboard without attacking each other. - Aaron Khan, Jul 05 2022

			For n=5 the reference matrix is:
  11001
  11100
  01110
  00111
  10011
There are 2^(3*n) = 32768 0-1 matrices obtained from removing one or more 1's in it.
There are 305 such matrices with permanent 4 and there are 13 such matrices with exactly two 1's in every column and every row.
There are 5 matrices having both properties. One of them is:
  10001
  01100
  01100
  00011
  10010
From _Aaron Khan_, Jul 05 2022: (Start)
Examples of the sequence when used for kings on a chessboard:
.
A solution illustrating a(2)=3:
  +-------+
  | B B B |
  | . . . |
  | W W W |
  +-------+
.
A solution illustrating a(3)=5:
  +---------+
  | B B B B |
  | B . . . |
  | . . . W |
  | W W W W |
  +---------+
(End)

V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3 (1992), 15-19.

G. C. Greubel, Table of n, a(n) for n = 3..1000
Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000211, A052928, A128209, A250000 (queens on a chessboard), A002620 (rooks on a chessboard), A355509 (knights on a chessboard).

[(n^2-3*n+1+(-1)^n)/2: n in [3..100]]; // Vincenzo Librandi, Mar 24 2011

A176222:=n->(n^2-3*n+1+(-1)^n)/2: seq(A176222(n), n=3..100); # Wesley Ivan Hurt, May 25 2015

Table[(n^2 - 3*n + 1 + (-1)^n)/2, {n, 3, 100}] (* or *) CoefficientList[Series[x (x - 3)/((1 + x)*(x - 1)^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, May 25 2015 *)
LinearRecurrence[{2,0,-2,1},{0,3,5,10},90] (* Harvey P. Dale, Jan 14 2024 *)

a(n)=(n^2-3*n+1+(-1)^n)/2 \\ Charles R Greathouse IV, Oct 16 2015

[n*(n-3)/2 + ((n+1)%2) for n in (3..60)] # G. C. Greubel, Mar 22 2022

a(n) = (n - t(n))*(n - 3 + t(n))/2, where t(n) = 1-(n mod 2).
G.f.: x^4*(3-x)/( (1+x)*(1-x)^3 ). - R. J. Mathar, Mar 06 2011
From Bruno Berselli, Sep 13 2011: (Start)
a(n) + a(n+1) = A005563(n-2).
a(-n) = A084265(n). (End)
a(n) = 1 -2*n +floor(n/2) +floor(n^2/2). - Wesley Ivan Hurt, Jun 14 2013
From Wesley Ivan Hurt, May 25 2015: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n>4.
a(n) = Sum_{i=(-1)^n..n-2} i. (End)
a(n) = A174239(n-2) * A174239(n-1). - Paul Curtz, Jul 17 2017
With offset 0, this is ceiling(n/2)*(2*floor(n/2)+3). - N. J. A. Sloane, Jan 16 2020
E.g.f.: (1/2)*((1-x)*exp(x/2) - exp(-x/2))^2. - G. C. Greubel, Mar 22 2022

Matrix class definition checked, edited and illustrated by Olivier Gérard, Mar 26 2011

A052953 Expansion of 2(1-x-x^2)/((1-x)(1+x)(1-2x)).

Original entry on oeis.org

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

Offset: 0

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

a(n) = sum of absolute values of terms in the (n+1)-th row of the triangle in A108561; - Reinhard Zumkeller, Jun 10 2005
a(n) = A078008(n+1) + 2*(1 + n mod 2). - Reinhard Zumkeller, Jun 10 2005
Essentially the same as A128209. - R. J. Mathar, Jun 14 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1024
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Apart from initial term, equals A026644(n+1) + 2.

Cf. A001045.

List([0..40], n-> (2^(n+1) +3 +(-1)^n)/3); # G. C. Greubel, Oct 21 2019

[(2^(n+1) +3 +(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Oct 21 2019

spec:= [S,{S=Union(Sequence(Union(Prod(Union(Z,Z),Z),Z)),Sequence(Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq((2^(n+1) +3 +(-1)^n)/3, n=0..40); # G. C. Greubel, Oct 21 2019

LinearRecurrence[{2,1,-2}, {2,2,4}, 40] (* G. C. Greubel, Oct 22 2019 *)
CoefficientList[Series[2(1-x-x^2)/((1-x)(1+x)(1-2x)),{x,0,40}],x] (* Harvey P. Dale, Aug 03 2025 *)

vector(41, n, (2^n +3 -(-1)^n)/3 ) \\ G. C. Greubel, Oct 21 2019

[(2^(n+1) +3 +(-1)^n)/3 for n in (0..40)] # G. C. Greubel, Oct 21 2019

G.f.: 2*(1-x-x^2)/((1-x^2)*(1-2*x)).
a(n) = a(n-1) + 2*a(n-2) - 2.
a(n) = 1 + Sum_{alpha=RootOf(-1+z+2*z^2)} (1 + 4*alpha)*alpha^(-1-n)/9.
a(2n) = 2*a(n-1)-2, a(2n+1) = 2*a(2n). - Lee Hae-hwang, Oct 11 2002
From Paul Barry, May 24 2004: (Start)
a(n) = A001045(n+1) + 1.
a(n) = (2^(n+1) - (-1)^(n+1))/3 + 1. (End)
E.g.f.: (2*exp(2*x) + 3*exp(x) + exp(-x))/3. - G. C. Greubel, Oct 21 2019

More terms from James Sellers, Jun 05 2000

A153643 Jacobsthal numbers A001045 incremented by 2.

Original entry on oeis.org

2, 3, 3, 5, 7, 13, 23, 45, 87, 173, 343, 685, 1367, 2733, 5463, 10925, 21847, 43693, 87383, 174765, 349527, 699053, 1398103, 2796205, 5592407, 11184813, 22369623, 44739245, 89478487, 178956973, 357913943, 715827885, 1431655767, 2863311533, 5726623063

Offset: 0

Paul Curtz, Dec 30 2008

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

Cf. A001045, A007395, A010684, A078008, A128209.

a:=[2,3,3];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Apr 02 2019

I:=[2,3,3]; [n le 3 select I[n] else 2*Self(n-1) +Self(n-2) -2*Self(n-3): n in [1..40]]; // G. C. Greubel, Apr 02 2019

LinearRecurrence[{1,2},{0,1}, 40] + 2 (* Harvey P. Dale, May 26 2014 *)
LinearRecurrence[{2,1,-2},{2,3,3}, 40] (* Georg Fischer, Apr 02 2019 *)

my(x='x+O('x^40)); Vec( (2-x-5*x^2)/((1-x^2)*(1-2*x)) ) \\ G. C. Greubel, Apr 02 2019

def A153643(n): return ((1<Chai Wah Wu, Apr 18 2025

((2-x-5*x^2)/((1-x^2)*(1-2*x))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 02 2019

a(n) = 2 + A001045(n) = A001045(n) + A007395(n) = 1 + A128209(n).
a(n) - A010684(n) = A078008(n), first differences of A001045. - Paul Curtz, Jan 17 2009
G.f.: (2 - x - 5*x^2)/((1+x)*(1-x)*(1-2*x)). - R. J. Mathar, Jan 23 2009
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n >= 3. - Andrew Howroyd, Feb 26 2018

Edited and extended by R. J. Mathar, Jan 23 2009

A153772 a(n) = (2^n + 2*(-1)^n - 6)/3.

Original entry on oeis.org

-1, -2, 0, 0, 4, 8, 20, 40, 84, 168, 340, 680, 1364, 2728, 5460, 10920, 21844, 43688, 87380, 174760, 349524, 699048, 1398100, 2796200, 5592404, 11184808, 22369620, 44739240, 89478484, 178956968, 357913940, 715827880

Offset: 0

Paul Curtz, Jan 01 2009

The array of T(n,k) with T(0,k) = A141325(k) and successive differences T(n,k) = T(n-1,k+1) - T(n-1,k) in further rows is
   1,  1,  1,  1,   3,  5, 9, 13, 21, 33, 55,..
   0,  0,  0,  2,   2,  4, 4,  8, 12, 22,..
   0,  0,  2,  0,   2,  0, 4,  4, 10,...
   0,  2, -2,  2,  -2,  4, 0,  6,..
   2, -4,  4, -4,   6, -4, 6,..
  -6,  8, -8, 10, -10, 10,...
with T(n,n) = A078008(n), T(n,n+1) = -A167030(n), T(n,n+2) = A128209(n), T(n,n+3) = -a(n). All these sequences along the diagonals obey the recurrences a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) and a(n) = 5*a(n-2) - 4*a(n-4).
Conjecture: For n >= 6, a(n) is the third largest natural number whose Collatz orbit has length n+2. - Markus Sigg, Sep 14 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000975, A005186, A033491.

[2^n/3 +2*(-1)^n/3-2: n in [0..40]]; // Vincenzo Librandi, Aug 07 2011

Table[(2^n + 2*(-1)^n - 6)/3, {n,0,25}] (* or *) LinearRecurrence[{2, 1, -2}, {-1, -2, 0}, 25] (* G. C. Greubel, Aug 27 2016 *)

a(n)=(2^n+2*(-1)^n-6)/3 \\ Charles R Greathouse IV, Aug 28 2016

a(n) = A078008(n) - 2.
a(n) = +2*a(n-1) +a(n-2) -2*a(n-3).
a(n) = a(n-1) + 2*a(n-2) + 4.
G.f.: (1 - 5*x^2) / ( (1-x)*(2*x-1)*(1+x) ).
E.g.f.: (1/3)*(2*exp(-x) - 6*exp(x) + exp(2*x)). - G. C. Greubel, Aug 27 2016
a(n) = 4*A000975(n-3) for n >= 3. - Markus Sigg, Sep 14 2020

A128208 Inverse of number triangle A128210.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 0, 5, 1, 0, 0, 0, 0, 11, 1, 0, 0, 0, 0, 0, 21, 1, 0, 0, 0, 0, 0, 0, 43, 1, 0, 0, 0, 0, 0, 0, 0, 85, 1, 0, 0, 0, 0, 0, 0, 0, 0, 171, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 341, 1

Offset: 0

Paul Barry, Feb 19 2007

Subdiagonal is A001045(n+1). Row sums are A128209.

			Triangle begins:
  1;
  1, 1;
  0, 1, 1;
  0, 0, 3, 1;
  0, 0, 0, 5,  1;
  0, 0, 0, 0, 11,  1;
  0, 0, 0, 0,  0, 21,  1;
  0, 0, 0, 0,  0,  0, 43,  1;
  0, 0, 0, 0,  0,  0,  0, 85,   1;
  0, 0, 0, 0,  0,  0,  0,  0, 171,   1;
  0, 0, 0, 0,  0,  0,  0,  0,   0, 341, 1;
  ...

A154890 Jacobsthal numbers A001045 alternatingly incremented by 3 and 5.

Original entry on oeis.org

3, 6, 4, 8, 8, 16, 24, 48, 88, 176, 344, 688, 1368, 2736, 5464, 10928, 21848, 43696, 87384, 174768, 349528, 699056, 1398104, 2796208, 5592408, 11184816, 22369624, 44739248, 89478488, 178956976, 357913944, 715827888, 1431655768, 2863311536, 5726623064

Offset: 0

Paul Curtz, Jan 17 2009

a(2n+1) = 2*a(2n).
a(n) = A153643(n)+A010684(n).
a(n+2) = 4*A128209(n).
a(n) = 2*a(n-1)+a(n-2)-2*a(n-3). G.f.: (3-11x^2)/((1-x)(1+x)(1-2x)). [R. J. Mathar, Jan 23 2009]

Edited and extended by R. J. Mathar, Jan 23 2009

A167167 A001045 with a(0) replaced by -1.

Original entry on oeis.org

-1, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, 178956971, 357913941, 715827883, 1431655765, 2863311531

Offset: 0

Paul Curtz, Oct 29 2009

Essentially the same as A001045, and perhaps also A152046.

Also the binomial transform of the sequence with terms (-1)^(n+1)*A128209(n).

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

Concatenation([-1], List([1..35], n-> (2^n -(-1)^n)/3) ); # G. C. Greubel, Dec 01 2019

[-1] cat [(2^n -(-1)^n)/3 : n in [1..35]]; // G. C. Greubel, Dec 01 2019

seq( `if`(n=0, -1, (2^n -(-1)^n)/3), n=0..35); # G. C. Greubel, Dec 01 2019

CoefficientList[Series[(2*x-1+2*x^2)/((1+x)*(1-2*x)), {x, 0, 35}], x] (* G. C. Greubel, Jun 04 2016 *)
Table[If[n==0, -1, (2^n -(-1)^n)/3], {n,0,35}] (* G. C. Greubel, Dec 01 2019 *)
LinearRecurrence[{1,2},{-1,1,1},40] (* Harvey P. Dale, Jul 23 2025 *)

vector(36, n, if(n==1, -1, (2^(n-1) +(-1)^n)/3 ) ) \\ G. C. Greubel, Dec 01 2019

[-1]+[lucas_number1(n, 1, -2) for n in (1..35)] # G. C. Greubel, Dec 01 2019

a(n) = A001045(n), n>0.
a(n) + a(n+1) = 2*A001782(n) = 2*A131577(n) = A155559(n) = A090129(n+2), n>0.
G.f.: (2*x^2 + 2*x - 1)/((1+x)*(1-2*x)).
E.g.f.: (exp(2*x) - exp(-x) - 3)/3. - G. C. Greubel, Dec 01 2019

Edited and extended by R. J. Mathar, Nov 01 2009

A364214 Numbers whose canonical representation as a sum of distinct Jacobsthal numbers (A280049) is palindromic.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 12, 15, 18, 21, 22, 30, 34, 42, 44, 49, 58, 63, 66, 71, 80, 85, 86, 102, 110, 126, 130, 146, 154, 170, 172, 183, 198, 209, 218, 229, 244, 255, 258, 269, 284, 295, 304, 315, 330, 341, 342, 374, 390, 422, 430, 462, 478, 510, 514, 546, 562, 594

Offset: 1

Amiram Eldar, Jul 14 2023

The even-indexed Jacobsthal numbers A001045(2*n) = A002450(n) = (4^n-1)/3, for n >= 1, are terms since their representation is 2*n-1 1's.
A001045(2*n+1) - 1 = A020988(n) = (2/3)*(4^n-1) is a term for n >= 1, since its representation is 2*n 1's.
A001045(n) + 1 = A128209(n) is a term for n >= 0, since its representation for n = 0 is 1 and its representation for n >= 1 is n-1 0's between 2 1's.
A160156(n) is a term for n >= 0 since its representation is n 0's interleaved with n+1 1's.

			The first 10 terms are:
   n  a(n)  A280049(a(n))
  --  ----  -------------
   1     1              1
   2     2             11
   3     4            101
   4     5            111
   5     6           1001
   6    10           1111
   7    12          10001
   8    15          10101
   9    18          11011
  10    21          11111

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001045, A002450, A003159, A020988, A280049.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341.

Position[Select[Range[1000], EvenQ[IntegerExponent[#, 2]] &], _?(PalindromeQ[IntegerDigits[#, 2]] &)] // Flatten

s(n) = if(n < 2, n > 0, n = s(n-1); until(valuation(n, 2)%2 == 0, n++); n); \\ A003159
is(n) = {my(d = binary(s(n))); d == Vecrev(d);}

cp's OEIS Frontend

A005581 a(n) = (n-1)*n*(n+4)/6.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

PARI

Sage

Formula

Extensions

A005582 a(n) = n*(n+1)*(n+2)*(n+7)/24.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A176222 a(n) = (n^2 - 3*n + 1 + (-1)^n)/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A052953 Expansion of 2*(1-x-x^2)/((1-x)*(1+x)*(1-2*x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A153643 Jacobsthal numbers A001045 incremented by 2.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Python

Sage

A005581 a(n) = (n-1)n(n+4)/6.

A005582 a(n) = n(n+1)(n+2)*(n+7)/24.

A052953 Expansion of 2(1-x-x^2)/((1-x)(1+x)(1-2x)).