A104934 -id:A104934 - cp's OEIS frontend

A055099 Expansion of g.f.: (1 + x)/(1 - 3x - 2x^2).

1, 4, 14, 50, 178, 634, 2258, 8042, 28642, 102010, 363314, 1293962, 4608514, 16413466, 58457426, 208199210, 741512482, 2640935866, 9405832562, 33499369418, 119309773378, 424928058970, 1513403723666, 5390067288938, 19197009314146, 68371162520314, 243507506189234

Offset: 0

Wolfdieter Lang, Apr 26 2000

Row sums of triangle A054458.
a(n) = term (1,1) in M^n, M = the 3 X 3 matrix [1,1,1; 1,1,1; 2,2,1]. - Gary W. Adamson, Mar 12 2009
Equals the INVERT transform of A001333: (1, 3, 7, 17, 41, 99, ...). - Gary W. Adamson, Aug 14 2010
a(n) is the number of one sided n-step walks taking steps from {(0,1), (-1,0), (1,0), (1,1)}. - Shanzhen Gao, May 13 2011
Number of quaternary words of length n on {0,1,2,3} containing no subwords 03 or 30. - Philippe Deléham, Apr 27 2012
Pisano period lengths: 1, 1, 4, 1, 24, 4, 48, 1, 12, 24, 30, 4, 12, 48, 24, 2, 272, 12, 18, 24, ... - R. J. Mathar, Aug 10 2012
a(n) = A007481(2*n+1) - A007481(2*n) = A007481(2*(n+1)) - A007481(2*n+1). - Reinhard Zumkeller, Oct 25 2015
Number of length-n words on a,b,c,d avoiding aa and ab. For n >= 1, the number of such words ending with a or the number of those ending with b is A007482(n-1), and the number of those ending with c or the number of those ending with d is a(n-1). - Jianing Song, Jun 01 2022

			a(3) = 50 because among the 4^3 = 64 quaternary words of length 3 only 14 namely 003, 030, 031, 032, 033, 103, 130, 203, 230, 300, 301, 302, 303, 330 contain the subwords 03 or 30. - _Philippe Deléham_, Apr 27 2012

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (Problem 2.4.6).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
M. Abrate, S. Barbero, U. Cerruti, and N. Murru, Construction and composition of rooted trees via descent functions, Algebra, Volume 2013 (2013), Article ID 543913, 11 pages.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3 , example 17
A. S. Fraenkel, Heap games, numeration systems and sequences, arXiv:math/9809074 [math.CO], 1998; Annals of Combinatorics, 2 (1998), 197-210.
Shanzhen Gao and Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.
S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, 2010.
Sergey Kitaev and Jeffrey Remmel, (a,b)-rectangle patterns in permutations and words, arXiv:1304.4286 [math.CO], 2013.
Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv:1504.04404 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (3,2).

Column k=2 of A255256.

Cf. A001333, A002203, A007481, A007482, A054458, A104934, A202396.

a055099 n = a007481 (2 * n + 1) - a007481 (2 * n)
-- Reinhard Zumkeller, Oct 25 2015

I:=[1,4]; [n le 2 select I[n] else 3*Self(n-1) + 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jun 27 2021

a := proc(n) option remember; `if`(n < 2, [1, 4][n+1], (3*a(n-1) + 2*a(n-2))) end:
seq(a(n), n=0..23); # Peter Luschny, Jan 06 2019

max = 24; cv = ContinuedFraction[ Sqrt[2], max] // Convergents // Numerator; Series[ 1/(1 - cv.x^Range[max]), {x, 0, max}] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Jun 21 2013, after Gary W. Adamson *)
LinearRecurrence[{3, 2}, {1, 4}, 24] (* Jean-François Alcover, Sep 23 2017 *)

[(i*sqrt(2))^(n-1)*( i*sqrt(2)*chebyshev_U(n, -3*i/(2*sqrt(2))) + chebyshev_U(n-1, -3*i/(2*sqrt(2))) ) for n in (0..40)] # G. C. Greubel, Jun 27 2021

a(n) = a*c^n - b*d^n, a := (5 + sqrt(17))/(2*sqrt(17)), b := (5 - sqrt(17))/(2*sqrt(17)), c := (3 + sqrt(17))/2, d := (3 - sqrt(17))/2.
a(n) = Sum_{m=0..n} A054458(n, m).
a(n) = F32(n) + F32(n-1) with F32(n) = A007482(n), n >= 1, a(0) = 1.
a(n) = A007482(n) + A007482(n-1) = 2*A007482(n) - A104934(n). - R. J. Mathar, Jul 23 2010
a(n) = 3*a(n-1) + 2*a(n-2) with a(0) = 1, a(1) = 4. - Vincenzo Librandi, Dec 08 2010
a(n) = (Sum_{k = 0..n} A202396(n,k)*3^k)/2^n. - Philippe Deléham, Feb 05 2012
a(n) = (i*sqrt(2))^(n-1)*( i*sqrt(2)*ChebyshevU(n, -3*i/(2*sqrt(2))) + ChebyshevU(n-1, -3*i/(2*sqrt(2))) ). - G. C. Greubel, Jun 27 2021
a(n) = 2*a(n-1) + 2*A007482(n-1), n >= 1. - Jianing Song, Jun 01 2022
E.g.f.: exp(3*x/2)*(17*cosh(sqrt(17)*x/2) + 5*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, May 24 2024

Edited by N. J. A. Sloane, Jun 08 2010

A046717 a(n) = 2a(n-1) + 3a(n-2), a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 5, 13, 41, 121, 365, 1093, 3281, 9841, 29525, 88573, 265721, 797161, 2391485, 7174453, 21523361, 64570081, 193710245, 581130733, 1743392201, 5230176601, 15690529805, 47071589413, 141214768241, 423644304721, 1270932914165, 3812798742493, 11438396227481

Offset: 0

Gervais Deroo and M. Deroo

Form the digraph with matrix A = [0,1,1,1; 1,0,1,1; 1,1,0,1; 1,0,1,1]. Then the sequence 0,1,1,5,... or (3^(n-1)-(-1)^n)/2+0^n/3 with g.f. x(1-x)/(1-2x-3x^2) corresponds to the (1,2) term of A^n. - Paul Barry, Oct 02 2004
3*a(n+1) + a(n) = 4*A060925(n); a(n+1) = A015518(n) + A060925(n); a(n+1) - 6*A015518(n) = (-1)^n. - Creighton Dement, Nov 15 2004
The sequence corresponds to the (1,1) term of the matrix [1,2;2,1]^n. - Simone Severini, Dec 04 2004
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 4 times the bottom to get the new top. The limit of the sequence of fractions is 2. - Cino Hilliard, Sep 25 2005
a(n)^2 + (2*A015518(n))^2 = a(2n). E.g., a(3) = 13, 2*A015518(3) = 14, A046717(6) = 365. 13^2 + 14^2 = 365. - Gary W. Adamson, Jun 17 2006
Equals INVERTi transform of A104934: (1, 2, 8, 28, 100, 356, 1268, ...). - Gary W. Adamson, Jul 21 2010
a(n) is the number of compositions of n when there are 1 type of 1 and 4 types of other natural numbers. - Milan Janjic, Aug 13 2010
An elephant sequence, see A175655. For the central square just one A[5] vector, with decimal value 341, leads to this sequence (without the first leading 1). For the corner squares this vector leads to the companion sequence A015518 (without the leading 0). - Johannes W. Meijer, Aug 15 2010
Pisano period lengths: 1, 1, 2, 1, 4, 2, 6, 4, 2, 4, 10, 2, 6, 6, 4, 8, 16, 2, 18, 4, ... - R. J. Mathar, Aug 10 2012
a(n) is the number of words of length n over a ternary alphabet whose position in the lexicographic order is a multiple of two. - Alois P. Heinz, Apr 13 2022
a(n) is the sum, for k=0..3, of the number of walks of length n between two vertices at distance k of the cube graph. - Miquel A. Fiol, Mar 09 2024

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
P. D. Jarvis and J. G. Sumner, Matrix group structure and Markov invariants in the strand symmetric phylogenetic substitution model, arXiv preprint arXiv:1307.5574 [q-bio.PE], 2013.
Index entries for linear recurrences with constant coefficients, signature (2,3).

The first difference sequence of A015518.
Row sums of triangle A080928.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
Cf. A015518.
Cf. A104934. - Gary W. Adamson, Jul 21 2010

[n le 2 select 1 else 2*Self(n-1)+3*Self(n-2): n in [1..35]]; // Vincenzo Librandi, Jul 21 2013

[(3^n + (-1)^n)/2: n in [0..30]]; // G. C. Greubel, Jan 07 2018

a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2] od: seq(a[n], n=0..33); # Zerinvary Lajos, Dec 14 2008
seq(denom(((-2)^(2*n)+6^(2*n))/((-2)^n+6^n)),n=0..26)

Table[(3^n + (-1)^n)/2, {n, 0, 30}] (* Artur Jasinski, Dec 10 2006 *)
CoefficientList[ Series[(1 - x)/(1 - 2x - 3x^2), {x, 0, 30}], x]  (* Robert G. Wilson v, Apr 04 2011 *)
Table[ MatrixPower[{{1, 2}, {1, 1}}, n][[1, 1]], {n, 0, 30}] (* Robert G. Wilson v, Apr 04 2011 *)

{a(n) = (3^n+(-1)^n)/2};
for(n=0,30, print1(a(n), ", ")) /* modified by G. C. Greubel, Jan 07 2018 */

x='x+O('x^30); Vec((1-x)/((1+x)*(1-3*x))) \\ G. C. Greubel, Jan 07 2018

[lucas_number2(n,2,-3)/2 for n in range(0, 27)] # Zerinvary Lajos, Apr 30 2009

G.f.: (1-x)/((1+x)*(1-3*x)).
a(n) = (3^n + (-1)^n)/2.
a(n) = Sum_{k=0..n} binomial(n, 2k)2^(2k). - Paul Barry, Feb 26 2003
Binomial transform of A000302 (powers of 4) with interpolated zeros. Inverse binomial transform of A081294. - Paul Barry, Mar 17 2003
E.g.f.: exp(x)cosh(2x). - Paul Barry, Mar 17 2003
a(n) = ceiling(3^n/4) + floor(3^n/4) = ceiling(3^n/4)^2 - floor(3^n/4)^2. - Paul Barry, Jan 17 2005
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n,j)C(n-j,k)*(1+(-1)^(j-k))/2. - Paul Barry, May 21 2006
a(n) = Sum_{k=0..n} A098158(n,k)*4^(n-k). - Philippe Deléham, Dec 26 2007
a(n) = (3^n + (-1)^n)/2. - M. F. Hasler, Mar 20 2008
a(n) = 2 A015518(n) + (-1)^n; for n > 0, a(n) = A080925(n). - M. F. Hasler, Mar 20 2008
((1 + sqrt4)^n + (1 - sqrt4)^n)/2. The offset is 0. a(3)=13. - Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008
If p[1]=1 and p[i]=4 (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
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(4*k-1)/(x*(4*k+3) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
G.f.: G(0)/2, where G(k) = 1 + (-1)^k/(3^k - 3*9^k*x/(3*3^k*x + (-1)^k/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013

Description corrected by and more terms from Michael Somos

A033887 a(n) = Fibonacci(3*n + 1).

Original entry on oeis.org

1, 3, 13, 55, 233, 987, 4181, 17711, 75025, 317811, 1346269, 5702887, 24157817, 102334155, 433494437, 1836311903, 7778742049, 32951280099, 139583862445, 591286729879, 2504730781961, 10610209857723, 44945570212853, 190392490709135, 806515533049393, 3416454622906707

Offset: 0

N. J. A. Sloane

Binomial transform of A063727, and second binomial transform of (1,1,5,5,25,25,...), which is A074872 with offset 0. - Paul Barry, Jul 16 2003
Equals INVERT transform of A104934: (1, 2, 8, 28, 100, 356, ...) and INVERTi transform of A005054: (1, 4, 20, 100, 500, ...). - Gary W. Adamson, Jul 22 2010
a(n) is the number of compositions of n when there are 3 types of 1 and 4 types of other natural numbers. - Milan Janjic, Aug 13 2010
F(3*n+1) = 3^n*a(n;2/3), where a(n;d), n = 0, 1, ..., d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also the papers by Witula et al.). - Roman Witula, Jul 12 2012
We note that the remark above by Paul Barry can be easily obtained from the following scaling identity for delta-Fibonacci numbers y^n a(n;x/y) = Sum_{k=0..n} binomial(n,k) (y-1)^(n-k) a(k;x) and the fact that a(n;2)=5^floor(n/2). Indeed, for x=y=2 we get 2^n a(n;1) = Sum_{k=0..n} binomial(n,k) a(k;2) and, by A000045: Sum_{k=0..n} binomial(n,k) 2^k a(k;1) = Sum_{k=0..n} binomial(n,k) F(k+1) 2^k = 3^n a(n;2/3) = F(3n+1). - Roman Witula, Jul 12 2012
Except for the first term, this sequence can be generated by Corollary 1 (iv) of Azarian's paper in the references for this sequence. - Mohammad K. Azarian, Jul 02 2015
Number of 1’s in the substitution system {0 -> 110, 1 -> 11100} at step n from initial string "1" (1 -> 11100 -> 111001110011100110110 -> ...). - Ilya Gutkovskiy, Apr 10 2017
The o.g.f. of {F(m*n + 1)}A000045%20and%20L%20=%20A000032.%20-%20_Wolfdieter%20Lang">{n>=0}, for m = 1, 2, ..., is G(m,x) = (1 - F(m-1)*x) / (1 - L(m)*x + (-1)^m*x^2), with F = A000045 and L = A000032. - _Wolfdieter Lang, Feb 06 2023

			a(5) = Fibonacci(3*5 + 1) = Fibonacci(16) = 987. - _Indranil Ghosh_, Feb 04 2017

Indranil Ghosh, Table of n, a(n) for n = 0..1592
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, 7(38) (2012), 1871-1876.
Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010), #10.8.2, Example 13.
Gary Detlefs and Wolfdieter Lang, Improved Formula for the Multi-Section of the Linear Three-Term Recurrence Sequence, arXiv:2304.12937 [math.CO], 2023.
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013), #13.4.5.
Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, Binomials Transformation Formulae of Scaled Fibonacci Numbers, Open Mathematics, 15(1) (2017), 477-485.
Tanya Khovanova, Recursive Sequences.
Roman Witula, Binomials transformation formulae of scaled Lucas numbers, Demonstratio Mathematica, 46(1) (2013), 15-27.
Roman Witula and Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009), 310-329, MR2555042.
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A000032, A000045, A104934, A005054, A063727 (inverse binomial transform), A082761 (binomial transform), A001076, A001077.

Cf. A016777, A049310, A049651, A074872, A122070, A167808, A185384.

[Fibonacci(3*n+1): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Fibonacci[Range[1,5!,3]] (* Vladimir Joseph Stephan Orlovsky, May 18 2010 *)

a(n)=fibonacci(3*n+1) \\ Charles R Greathouse IV, Feb 03 2014

Vec((1-x)/(1-4*x-x^2) + O(x^100)) \\ Altug Alkan, Dec 10 2015

a(n) = A001076(n) + A001077(n) = A001076(n+1) - A001076(n).
a(n) = 2*A049651(n) + 1.
a(n) = 4*a(n-1) + a(n-2) for n>1, a(0)=1, a(1)=3;
G.f.: (1 - x)/(1 - 4*x - x^2).
a(n) = ((1 + sqrt(5))*(2 + sqrt(5))^n - (1 - sqrt(5))*(2 - sqrt(5))^n)/(2*sqrt(5)).
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n,j)*C(n-j,k)*F(n-j+1). - Paul Barry, May 19 2006
First differences of A001076. - Al Hakanson (hawkuu(AT)gmail.com), May 02 2009
a(n) = A167808(3*n+1). - Reinhard Zumkeller, Nov 12 2009
a(n) = Sum_{k=0..n} C(n,k)*F(n+k+1). - Paul Barry, Apr 19 2010
Let p[1]=3, p[i]=4, (i>1), and A be a 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) = Sum_{i=0..n} C(n,n-i)*A063727(i). - Seiichi Kirikami, Mar 06 2012
a(n) = Sum_{k=0..n} A122070(n,k) = Sum_{k=0..n} A185384(n,k). - Philippe Deléham, Mar 13 2012
a(n) = A000045(A016777(n)). - Michel Marcus, Dec 10 2015
a(n) = F(2*n)*L(n+1) + F(n-1)*(-1)^n for n > 0. - J. M. Bergot, Feb 09 2016
a(n) = Sum_{k=0..n} binomial(n,k)*5^floor(k/2)*2^(n-k). - Tony Foster III, Sep 03 2017
2*a(n) = Fibonacci(3*n) + Lucas(3*n). - Bruno Berselli, Oct 13 2017
a(n)^2 is the denominator of continued fraction [4,...,4, 2, 4,...,4], which has n 4's before, and n 4's after, the middle 2. - Greg Dresden and Hexuan Wang, Aug 30 2021
a(n) = i^n*(S(n, -4*i) + i*S(n-1, -4*i)), with i = sqrt(-1), and the Chebyshev S-polynomials (see A049310) with S(n, -1) = 0. From the simplified trisection formula. See the first entry above with A001076. - Gary Detlefs and Wolfdieter Lang, Mar 06 2023
E.g.f.: exp(2*x)*(5*cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x))/5. - Stefano Spezia, May 24 2024

A122542 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 4, 1, 0, 2, 8, 6, 1, 0, 2, 12, 18, 8, 1, 0, 2, 16, 38, 32, 10, 1, 0, 2, 20, 66, 88, 50, 12, 1, 0, 2, 24, 102, 192, 170, 72, 14, 1, 0, 2, 28, 146, 360, 450, 292, 98, 16, 1, 0, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1

Offset: 0

Philippe Deléham, Sep 19 2006, May 28 2007

Riordan array (1, x*(1+x)/(1-x)). Rising and falling diagonals are the tribonacci numbers A000213, A001590.

			Triangle begins:
  1;
  0, 1;
  0, 2,  1;
  0, 2,  4,   1;
  0, 2,  8,   6,   1;
  0, 2, 12,  18,   8,    1;
  0, 2, 16,  38,  32,   10,   1;
  0, 2, 20,  66,  88,   50,  12,   1;
  0, 2, 24, 102, 192,  170,  72,  14,   1;
  0, 2, 28, 146, 360,  450, 292,  98,  16,  1;
  0, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 1.

Other versions: A035607, A113413, A119800, A266213.
Diagonals: A000012, A005843, A001105, A035597 - A035606.
Columns: A000007, A040000, A008575, A005899, A008412-A008416, A008418, A008420, A035706 - A035745.
Sums include: A000007, A001333 (row), A001590 (diagonal), A007483, A057077 (signed row), A078016 (signed diagonal), A086901, A091928, A104934, A122558, A122690.
Cf. A155161, A059283.

a122542 n k = a122542_tabl !! n !! k
a122542_row n = a122542_tabl !! n
a122542_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [0, 1])
-- Reinhard Zumkeller, Jul 20 2013, Apr 17 2013

function T(n, k) // T = A122542
  if k eq 0 then return 0^n;
  elif k eq n then return 1;
  else return T(n-1,k) + T(n-1,k-1) + T(n-2,k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 27 2024

CoefficientList[#, y]& /@ CoefficientList[(1-x)/(1 - (1+y)x - y x^2) + O[x]^11, x] // Flatten (* Jean-François Alcover, Sep 09 2018 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, 0, T[n-1,k-1] +T[n-1,k] +T[n-2,k- 1] ]]; (* T = A122542 *)
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 27 2024 *)

def A122542_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)+2*sum(prec(n-i,k-1) for i in (2..n-k+1))
    return [prec(n, k) for k in (0..n)]
for n in (0..10): print(A122542_row(n)) # Peter Luschny, Mar 16 2016

Sum_{k=0..n} x^k*T(n,k) = A000007(n), A001333(n), A104934(n), A122558(n), A122690(n), A091928(n)  for x = 0, 1, 2, 3, 4, 5. - Philippe Deléham, Jan 25 2012
Sum_{k=0..n} 3^(n-k)*T(n,k) = A086901(n).
Sum_{k=0..n} 2^(n-k)*T(n,k) = A007483(n-1), n >= 1. - Philippe Deléham, Oct 08 2006
T(2*n, n) = A123164(n).
T(n, k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1), n > 1. - Philippe Deléham, Jan 25 2012
G.f.: (1-x)/(1-(1+y)*x-y*x^2). - Philippe Deléham, Mar 02 2012
From G. C. Greubel, Oct 27 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A057077(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A001590(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A078016(n). (End)

A208709 T(n,k)=Number of nXk 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than two of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 28, 32, 8, 16, 100, 196, 128, 16, 32, 356, 1268, 1372, 512, 32, 64, 1268, 8128, 16084, 9604, 2048, 64, 128, 4516, 52184, 185344, 204020, 67228, 8192, 128, 256, 16084, 334948, 2142580, 4226368, 2587924, 470596, 32768, 256, 512, 57284, 2149988

Offset: 1

R. H. Hardin Mar 01 2012

Table starts
...1.....2.......4.........8..........16............32..............64
...2.....8......28.......100.........356..........1268............4516
...4....32.....196......1268........8128.........52184..........334948
...8...128....1372.....16084......185344.......2142580........24754628
..16...512....9604....204020.....4226368......87985748......1830045552
..32..2048...67228...2587924....96373248....3613193828....135288700496
..64..8192..470596..32826932..2197585152..148378294612..10001404535216
.128.32768.3294172.416398420.50111214592.6093257064980.739367693888784

			Some solutions for n=4 k=3
..0..1..1....0..1..0....0..0..0....0..0..1....0..0..1....0..1..0....0..1..0
..1..0..0....0..0..0....0..1..1....0..0..0....0..1..1....0..0..0....1..0..0
..1..0..1....0..1..1....1..0..1....1..1..1....0..0..0....1..1..1....1..0..1
..1..1..1....0..1..1....0..1..0....0..0..0....0..1..0....1..0..0....1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..631

Column 2 is A004171(n-1)

Row 2 is A104934

A007484 a(n) = 3a(n-1) + 2a(n-2), with a(0)=2, a(1)=7.

Original entry on oeis.org

2, 7, 25, 89, 317, 1129, 4021, 14321, 51005, 181657, 646981, 2304257, 8206733, 29228713, 104099605, 370756241, 1320467933, 4702916281, 16749684709, 59654886689, 212464029485, 756701861833, 2695033644469, 9598504657073, 34185581260157, 121753753094617, 433632421804165

Offset: 0

N. J. A. Sloane

Number of subsequences of [1,...,2n+1] in which each even number has an odd neighbor.
Same as Pisot sequence E(2,7) (see A008776).
8*a(n) = A007482(n+2) + A007483(n+1) (conjectured, see A104934 for related formula). - Creighton Dement, Apr 15 2005
Conjecture verified using generating functions. - Robert Israel, Jul 12 2018
a(n) = sum of the elements of the matrix M^n, where M = {{1, 2}, {2, 2}}. - Griffin N. Macris, Mar 25 2016
a(3) = 25 is the only composite among the first 8 terms, but then the density of primes decreases, dropping below 50% at the 27th term. - M. F. Hasler, Jul 12 2018
a(n) is also the number of dominating sets in the (2n+1)-triangular snake graph for n > 0. - Eric W. Weisstein, Jun 09 2019

			G.f. = 2 + 7*x + 25*x^2 + 89*x^3 + 317*x^4 + 1129*x^5 + ... - _Michael Somos_, Jul 19 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
R. K. Guy and W. O. J. Moser, Numbers of subsequences without isolated odd members, Fibonacci Quarterly 34:2 (1996), pp. 152-155.
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Triangular Snake Graph
Index entries for linear recurrences with constant coefficients, signature (3,2).

Cf. A007455, A007481, A007483, A007484, A104934.

See A008776 for definitions of Pisot sequences.

a007484 n = a007484_list !! n
a007484_list = 2 : 7 : zipWith (+)
               (map (* 3) $ tail a007484_list) (map (* 2) a007484_list)
-- Reinhard Zumkeller, Nov 02 2015

A007484:=[2, 7]; [n le 2 select A007484[n] else 3*Self(n-1)+2*Self(n-2): n in [1..40]]; // Wesley Ivan Hurt, Jan 24 2017

A007484 := proc(n) option remember; if n=0 then 2; elif n=1 then 7; else 3*A007484(n-1)+2*A007484(n-2); fi; end;

LinearRecurrence[{3, 2}, {2, 7}, 40] (* Harvey P. Dale, Apr 24 2012 *)
Table[(2^-n ((3 - Sqrt[17])^n (-4 + Sqrt[17]) + (3 + Sqrt[17])^n (4 + Sqrt[17])))/Sqrt[17], {n, 0, 20}] // Expand (* Eric W. Weisstein, Jun 09 2019 *)
CoefficientList[Series[(2+x)/(1 -3x -2x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 09 2019 *)
a[ n_] := MatrixPower[{{1, 2}, {2, 2}}, n]//Flatten//Total; (* Michael Somos, Jul 19 2021 *)

a(n)=([0,1; 2,3]^n*[2;7])[1,1] \\ Charles R Greathouse IV, Mar 25 2016

A007484_vec(N)=Vec((2+x)/(1-3*x-2*x^2)+O(x^n)) \\ M. F. Hasler, Jul 12 2018

[(i*sqrt(2))^(n-1)*( i*2*sqrt(2)*chebyshev_U(n, -3*i/(2*sqrt(2))) + chebyshev_U(n-1, -3*i/(2*sqrt(2))) ) for n in (0..30)] # G. C. Greubel, Jul 18 2021

a(n) = nearest integer to (and converges rapidly to) (1+4/sqrt(17))*((3+sqrt(17))/2)^n. - N. J. A. Sloane, Jul 30 2016
If p[i] = Fibonacci(i+2) and if A is the 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-1)= det A. - Milan Janjic, May 08 2010
G.f.: (2 + x)/(1 - 3*x - 2*x^2). - M. F. Hasler, Jul 12 2018
From G. C. Greubel, Jul 18 2021: (Start)
a(n) = (i*sqrt(2))^(n-1)*( i*2*sqrt(2)*ChebyshevU(n, -3*i/(2*sqrt(2))) + ChebyshevU(n-1, -3*i/(2*sqrt(2))) ).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*((7*n-8*k)/(n-k))*2^k*3^(n-2*k-1) with a(0) = 2. (End)
If we extend the definition of A007483(m) to negative m by using the recurrence, then a(n) = A007483(-3-n)*(-2)^n holds for all n in Z. - Michael Somos, Jul 19 2021
E.g.f.: 2*exp(3*x/2)*(17*cosh(sqrt(17)*x/2) + 4*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, May 24 2024

Definition edited by N. J. A. Sloane, Jul 30 2016

A184761 T(n,k)=Half the number of nXk binary arrays with no 1 having an adjacent 1 both above and to its left.

Original entry on oeis.org

1, 2, 2, 4, 7, 4, 8, 25, 25, 8, 16, 89, 163, 89, 16, 32, 317, 1056, 1056, 317, 32, 64, 1129, 6847, 12397, 6847, 1129, 64, 128, 4021, 44391, 145778, 145778, 44391, 4021, 128, 256, 14321, 287802, 1713803, 3110914, 1713803, 287802, 14321, 256, 512, 51005, 1865917

Offset: 1

R. H. Hardin Jan 21 2011

Table starts
...1......2........4...........8.............16...............32
...2......7.......25..........89............317.............1129
...4.....25......163........1056...........6847............44391
...8.....89.....1056.......12397.........145778..........1713803
..16....317.....6847......145778........3110914.........66363023
..32...1129....44391.....1713803.......66363023.......2568513843
..64...4021...287802....20148584.....1415755252......99419347147
.128..14321..1865917...236878817....30202770902....3848174315295
.256..51005.12097367..2784890782...644326291402..148949599913987
.512.181657.78431296.32740859687.13745636657969.5765325542821919

			Some solutions for 4X3
..1..1..0....1..0..0....0..1..1....0..0..0....0..0..1....1..0..0....0..0..0
..0..0..0....0..0..1....0..0..1....0..0..1....1..0..0....0..1..0....1..0..1
..1..1..0....1..0..1....1..1..0....0..0..1....1..1..0....0..0..1....1..0..1
..0..0..0....0..0..0....0..0..1....1..1..0....0..1..0....1..0..0....0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..760

Column 2 is A007484(n-1) and 1/2 A055099 and 1/4 A104934(n+1)

A202206 a(n) = 3a(n-1)+3a(n-2) with a(0)=1 and a(1)=2.

Original entry on oeis.org

1, 2, 9, 33, 126, 477, 1809, 6858, 26001, 98577, 373734, 1416933, 5372001, 20366802, 77216409, 292749633, 1109898126, 4207943277, 15953524209, 60484402458, 229313780001, 869394547377, 3296124982134

Offset: 0

Philippe Deléham, Dec 14 2011

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

Cf. A104934, A112117.

LinearRecurrence[{3,3},{1,2},30] (* Harvey P. Dale, Aug 03 2020 *)

Vec((1-x)/(1-3*x-3*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 31 2012

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

A322940 a(n) = [x^n] (4x^2 + x - 1)/(2x^2 + 3*x - 1).

Original entry on oeis.org

1, 2, 4, 16, 56, 200, 712, 2536, 9032, 32168, 114568, 408040, 1453256, 5175848, 18434056, 65653864, 233829704, 832796840, 2966049928, 10563743464, 37623330248, 133997477672, 477239093512, 1699712235880, 6053614894664, 21560269155752, 76788037256584

Offset: 0

Peter Luschny, Jan 06 2019

nonn

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

Row sums of A322941.

Cf. A104934, A055099, A102865, A322939.

a := proc(n) option remember; `if`(n < 3, [1, 2, 4][n+1], 3*a(n-1) + 2*a(n-2)) end:
seq(a(n), n=0..26);

Join[{1}, LinearRecurrence[{3, 2}, {2, 4}, 26]] (* Jean-François Alcover, Jul 13 2019 *)

a(n) = 3*a(n-1) + 2*a(n-2) for n >= 3.
a(n) = 2*A104934(n-1) for n >= 1.
a(n) = 4*A055099(n-2) for n >= 2.
INVERT(a) = A102865.
INVERTi(a) = A322939. (See the link 'Transforms' at the bottom of the page.)

A202209 Triangle T(n,k), read by rows, given by (2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 13, 5, 0, 0, 34, 19, 1, 0, 0, 89, 65, 8, 0, 0, 0, 233, 210, 42, 1, 0, 0, 0, 610, 654, 183, 11, 0, 0, 0, 0, 1597, 1985, 717, 74, 1, 0, 0, 0, 0, 4181, 5911, 2622, 394, 14, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 14 2011

Riordan array ((1-x)/(1-3x+x^2), x^2/(1-3x+x^2)) .

			Triangle begins :
1
2, 0
5, 1, 0
13, 5, 0, 0
34, 19, 1, 0, 0
89, 65, 8, 0, 0, 0
233, 210, 42, 1, 0, 0, 0

Cf. A000045, A000079, A001519, A001870, A001906, A126124, A202207 (antidiagonal sums)

T(n,k) = 3*T(n-1,k) - T(n-2,k) + T(n-2,k-1).
G.f.: (1-x)/(1-3x+(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A057682(n+1), A000079(n), A122367(n), A025192(n), A052924(n), A104934(n), A202206(n), A122117(n), A197189(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6 respectively.
T(n,0) = A122367(n) = A000045(2n+1).

cp's OEIS Frontend

A055099 Expansion of g.f.: (1 + x)/(1 - 3*x - 2*x^2).

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A046717 a(n) = 2*a(n-1) + 3*a(n-2), a(0) = a(1) = 1.

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A033887 a(n) = Fibonacci(3*n + 1).

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A122542 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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A208709 T(n,k)=Number of nXk 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than two of its immediate leftward or upward or right-upward antidiagonal neighbors.

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A007484 a(n) = 3*a(n-1) + 2*a(n-2), with a(0)=2, a(1)=7.

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A055099 Expansion of g.f.: (1 + x)/(1 - 3x - 2x^2).

A046717 a(n) = 2a(n-1) + 3a(n-2), a(0) = a(1) = 1.

A007484 a(n) = 3a(n-1) + 2a(n-2), with a(0)=2, a(1)=7.

A202206 a(n) = 3a(n-1)+3a(n-2) with a(0)=1 and a(1)=2.

A322940 a(n) = [x^n] (4x^2 + x - 1)/(2x^2 + 3*x - 1).