A052913 -id:A052913 - cp's OEIS frontend

A007482 a(n) is the number of subsequences of [ 1, ..., 2n ] in which each odd number has an even neighbor.

1, 3, 11, 39, 139, 495, 1763, 6279, 22363, 79647, 283667, 1010295, 3598219, 12815247, 45642179, 162557031, 578955451, 2061980415, 7343852147, 26155517271, 93154256107, 331773802863, 1181629920803, 4208437368135

Offset: 0

N. J. A. Sloane

The even neighbor must differ from the odd number by exactly one.
If we defined this sequence by the recurrence (a(n) = 3*a(n-1) + 2*a(n-2)) that it satisfies, we could prefix it with an initial 0.
a(n) equals term (1,2) in M^n, M = the 3 X 3 matrix [1,1,2; 1,0,1; 2,1,1]. - Gary W. Adamson, Mar 12 2009
a(n) equals term (2,2) in M^n, M = the 3 X 3 matrix [0,1,0; 1,3,1; 0,1,0]. - Paul Barry, Sep 18 2009
From Gary W. Adamson, Aug 06 2010: (Start)
Starting with "1" = INVERT transform of A002605: (1, 2, 6, 16, 44, ...).
Example: a(3) = 39 = (16, 6, 2, 1) dot (1, 1, 3, 11) = (16 + 6 + 6 + 11). (End)
Pisano periods: 1, 1, 4, 1, 24, 4, 48, 2, 12, 24, 30, 4, 12, 48, 24, 4,272, 12, 18, 24, ... . - R. J. Mathar, Aug 10 2012
A007482 is also the number of ways of tiling a 3 X n rectangle with 1 X 1 squares, 2 X 2 squares and 2 X 1 (vertical) dominoes. - R. K. Guy, May 20 2015
With offset 1 (a(0) = 0, a(1) = 1) this is a divisibility sequence. - Michael Somos, Jun 03 2015
Number of elements of size 2^(-n) in a fractal generated by the second-order reversible cellular automaton, rule 150R (see the reference and the link). - Yuriy Sibirmovsky, Oct 04 2016
a(n) is the number of compositions (ordered partitions) of n into parts 1 (of three kinds) and 2 (of two kinds). - Joerg Arndt, Oct 05 2016
a(n) equals the number of words of length n over {0,1,2,3,4} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017
Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 2 X 2 cells and remove the cells that have two '1's in their modulo 3 coordinates. a(n) is the number of cells after n iterations. Cell configuration converges to a fractal with approximate dimension 1.833. - Peter Karpov, Apr 20 2017
This is the Lucas sequence U(P=3,Q=-2), and hence for n>=0, a(n+2)/a(n+1) equals the continued fraction 3 + 2/(3 + 2/(3 + 2/(3 + ... + 2/3))) with n 2's. - Greg Dresden, Oct 06 2019

			G.f. = 1 + 3*x + 11*x^2 + 39*x^3 + 139*x^4 + 495*x^5 + 1763*x^6 + ...
From _M. F. Hasler_, Jun 16 2019: (Start)
For n = 0, (1, ..., 2n) = () is the empty sequence, which is equal to its only subsequence, which satisfies the condition voidly, whence a(0) = 1.
For n = 1, (1, ..., 2n) = (1, 2); among the four subsequences {(), (1), (2), (1,2)} only (1) does not satisfy the condition, whence a(1) = 3.
For n = 2, (1, ..., 2n) = (1, 2, 3, 4); among the sixteen subsequences {(), ..., (1,2,3,4)}, the 5 subsequences (1), (3), (1,3), (2,3,4) and (1,2,3,4) do not satisfy the condition, whence a(2) = 16 - 5 = 11.
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 439.

T. D. Noe, Table of n, a(n) for n = 0..200
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 13, 29.
Alexander Burstein and Opel Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, arXiv:2002.12189 [math.CO], 2020.
R. K. Guy and William O. J. Moser, Numbers of subsequences without isolated odd members, Fibonacci Quarterly, 34, No. 2, 152-155 (1996). Math. Rev. 97d:11017.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 442
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..8)
Yuriy Sibirmovsky, A fractal with number of elements described by a(n)
Index entries for linear recurrences with constant coefficients, signature (3,2).

Row sums of triangle A073387.
Cf. A000045, A000129, A001045, A007455, A007481, A007483, A007484, A015518, A201000 (prime subsequence), A052913 (binomial transform), A026597 (inverse binomial transform).
Cf. A206776.

a007482 n = a007482_list !! (n-1)
a007482_list = 1 : 3 : zipWith (+)
               (map (* 3) $ tail a007482_list) (map (* 2) a007482_list)
-- Reinhard Zumkeller, Oct 21 2015

I:=[1,3]; [n le 2 select I[n] else 3*Self(n-1) + 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

a := n -> `if`(n=0, 1, 3^n*hypergeom([(1-n)/2,-n/2], [-n], -8/9)):
seq(simplify(a(n)), n = 0..23); # Peter Luschny, Jun 28 2017

a[n_]:=(MatrixPower[{{1,4},{1,2}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{3,2},{1,3},30] (* Harvey P. Dale, May 25 2013 *)
a[ n_] := Module[ {m = n + 1, s = 1}, If[ m < 0, {m, s} = -{m, (-2)^m}]; s SeriesCoefficient[ x / (1 - 3 x - 2 x^2), {x, 0, m}]]; (* Michael Somos, Jun 03 2015 *)
a[ n_] := With[{m = n + 1}, If[ m < 0, (-2)^m a[ -m], Expand[((3 + Sqrt[17])/2)^m - ((3 - Sqrt[17])/2)^m ] / Sqrt[17]]]; (* Michael Somos, Oct 13 2016 *)

a(n) := if n=0 then 1 elseif n=1 then 3 else 3*a(n-1)+2*a(n-2);
makelist(a(n),n,0,12); /* Emanuele Munarini, Jun 28 2017 */

{a(n) = 2*imag(( (3 + quadgen(68)) / 2)^(n+1))}; /* Michael Somos, Jun 03 2015 */

[lucas_number1(n,3,-2) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

G.f.: 1/(1-3*x-2*x^2).
a(n) = 3*a(n-1) + 2*a(n-2).
a(n) = (ap^(n+1)-am^(n+1))/(ap-am), where ap = (3+sqrt(17))/2 and am = (3-sqrt(17))/2.
Let b(0) = 1, b(k) = floor(b(k-1)) + 2/b(k-1); then, for n>0, b(n) = a(n)/a(n-1). - Benoit Cloitre, Sep 09 2002
The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)2^k*3^(n-2k). - Paul Barry, Apr 23 2005
a(n) = Sum_{k=0..n} A112906(n,k). - Philippe Deléham, Nov 21 2007
a(n) = - a(-2-n) * (-2)^(n+1) for all n in Z. - Michael Somos, Jun 03 2015
If c = (3 + sqrt(17))/2, then c^n = (A206776(n) + sqrt(17)*a(n-1)) / 2. - Michael Somos, Oct 13 2016
a(n) = 3^n*hypergeom([(1-n)/2,-n/2], [-n], -8/9) for n>=1. - Peter Luschny, Jun 28 2017
a(n) = round(((sqrt(17) + 3)/2)^(n+1)/sqrt(17)). The distance of the argument from the nearest integer is about 1/2^(n+3). - M. F. Hasler, Jun 16 2019
E.g.f.: (1/17)*exp(3*x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2)). - Stefano Spezia, Oct 07 2019
a(n) = (sqrt(2)*i)^n * ChebyshevU(n, -3*i/(2*sqrt(2))). - G. C. Greubel, Dec 24 2021
G.f.: 1/(1 - 3*x - 2*x^2) = Sum_{n >= 0} x^n * Product_{k = 1..n} (k + 2*x + 2)/(1 + k*x) (a telescoping series). Cf. A015518. - Peter Bala, May 08 2024

A078099 Array T(m,n) read by antidiagonals: T(m,n) = number of ways of 3-coloring an m X n grid (m >= 1, n >= 1).

Original entry on oeis.org

1, 2, 2, 4, 6, 4, 8, 18, 18, 8, 16, 54, 82, 54, 16, 32, 162, 374, 374, 162, 32, 64, 486, 1706, 2604, 1706, 486, 64, 128, 1458, 7782, 18150, 18150, 7782, 1458, 128, 256, 4374, 35498, 126534, 193662, 126534, 35498, 4374, 256, 512, 13122, 161926, 882180, 2068146, 2068146, 882180, 161926, 13122, 512

Offset: 1

N. J. A. Sloane, Dec 05 2002

We assume the top left point gets color 1 (or, in other words, take the total number of colorings and divide by 3). The rule for coloring is that horizontally or vertically adjacent points must have different colors. - N. J. A. Sloane, Feb 12 2013
Equals half the number of m X n binary matrices with no 2 X 2 circuit having the pattern 0011 in any orientation. - R. H. Hardin, Oct 06 2010
Also the number of Miura-ori foldings [Ginepro-Hull]. - N. J. A. Sloane, Aug 05 2015

			Array begins:
1       2       4       8       16      32      64      128     256     512 ...
2       6       18      54      162     486     1458    4374    13122 ...
4       18      82      374     1706    7782    35498   161926 ...
8       54      374     2604    18150   126534  882180 ...
16      162     1706    18150   193662 ...
32      486     7782    126534 ...
For the 1 X n case: the first point gets color 1, thereafter there are 2 choices for each color, so T(1,n) = 2^(n-1).
For the 2 X 2 case, the colorings are
12 12 12 13 13 13
21 23 31 31 32 21

Thomas C. Hull, Coloring Connections with Counting Mountain-Valley Assignments in (book) Origami^6: I. Mathematics, 2015, ed. Koryo Miura, Toshikazu Kawasaki, Tomohiro Tachi, Ryuhei Uehara, Robert J. Lang, Patsy Wang-Iverson, American Mathematical Soc., Dec 18, 2015, 368 pages
Michael S. Paterson (Warwick), personal communication.

Andrew Howroyd, Table of n, a(n) for n = 1..1128 (terms 1..120 from R. J. Mathar)
J. Ginepro, T. C. Hull, Counting Miura-ori Foldings, Journal of Integer Sequences, Vol. 17, 2014, #14.10.8
R. J. Mathar, Counting 2-way monotonic terrace forms..., vixra 1511.0225 (2015), Table T_{n x m}
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Vertex Coloring
Wikipedia, Graph Coloring

Cf. A207997, A020698, A078100. Main diagonal is A068253. Other diagonals produce A078101 and A078102.

Cf. A222444 (4 colorings), A222144 (5 colorings), A222281 (6 colorings), A222340 (7 colorings), A222462 (8 colorings).

with(linalg); t := transpose; M[1] := matrix(1,1,[1]); Z[1] := matrix(1,1,0); W[1] := evalm(M[1]+t(M[1])); v[1] := matrix(1,1,1);
for n from 2 to 6 do t1 := stackmatrix(M[n-1],Z[n-1]); t2 := stackmatrix(t(M[n-1]),M[n-1]); M[n] := t(stackmatrix(t(t1),t(t2))); Z[n] := matrix(2^(n-1),2^(n-1),0); W[n] := evalm(M[n]+t(M[n])); v[n] := matrix(1,2^(n-1),1); od:
T := proc(m,n) evalm( v[m] &* W[m]^(n-1) &* t(v[m]) ); end;

mmax = 10; M[1] = {{1}}; M[m_] := M[m] = {{M[m-1], Transpose[M[m-1]]}, {Array[0&, {2^(m-2), 2^(m-2)}], M[m-1]}} // ArrayFlatten; W[m_] := M[m] + Transpose[M[m]]; T[m_, 1] := 2^(m-1); T[1, n_] := 2^(n-1); T[m_, n_] := MatrixPower[W[m], n-1] // Flatten // Total; Table[T[m-n+1, n], {m, 1, mmax}, {n, 1, m}] // Flatten (* Jean-François Alcover, Feb 13 2016 *)

Let M[1] = [1], M[m+1] = the block matrix [ [ M[m], M[m]' ], [ 0, M[m] ] ], W[m] = M[m] + M[m]', then T(m, n) = sum of entries of W[m]^(n-1) (the prime denotes transpose).
T(3,n) = A052913(n). T(4,n) = 2*A078100(n).
T(n,m) = T(m,n). T(1,n)= A000079(n-1). T(2,n)=A025192(n). T(5,n) = 2*A207994(n). T(6,n) = 2*A207995(n). - R. J. Mathar, Nov 23 2015

More terms from Alois P. Heinz, Mar 23 2009

A060801 Invert transform of odd numbers: a(n) = Sum_{k=1..n} (2k+1)a(n-k), a(0)=1.

Original entry on oeis.org

1, 3, 14, 64, 292, 1332, 6076, 27716, 126428, 576708, 2630684, 12000004, 54738652, 249693252, 1138988956, 5195558276, 23699813468, 108107950788, 493140127004, 2249484733444, 10261143413212, 46806747599172, 213511451169436

Offset: 0

Vladeta Jovovic, Apr 27 2001

a(n) is the number of generalized compositions of n when there are 2*i+1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

Colin Barker, Table of n, a(n) for n = 0..1000
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017.
Stéphane Ouvry and Alexios P. Polychronakos, Signed area enumeration for lattice walks, Séminaire Lotharingien de Combinatoire (2023) Vol. 87B.
N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (5,-2).

Cf. A001906, A052530, A033453, A030017, A052913 (partial sums).

Join[{1}, LinearRecurrence[{5, -2}, {3, 14}, 22]] (* Jean-François Alcover, Aug 07 2018 *)

Vec((1 - x)^2 / (1 - 5*x + 2*x^2) + O(x^25)) \\ Colin Barker, Mar 19 2019

G.f.: (x^2-2*x+1)/(2*x^2-5*x+1).
G.f.: 1 / (1 - 3*x - 5*x^2 - 7*x^3 - 9*x^4 - 11*x^5 - ...). - Gary W. Adamson, Jul 27 2009
a(n) = 5*a(n-1) - 2*a(n-2) with a(1) = 3, a(2) = 14, for n >= 3. - Taras Goy, Mar 19 2019
a(n) = (2^(-2-n)*((5-sqrt(17))^n*(-7+sqrt(17)) + (5+sqrt(17))^n*(7+sqrt(17)))) / sqrt(17) for n > 0. - Colin Barker, Mar 19 2019
a(n) = A052913(n)-A052913(n-1). - R. J. Mathar, Sep 20 2020

A079162 a(n) = 5a(n-2) - 2a(n-4), with initial terms 0,1,2,4.

Original entry on oeis.org

0, 1, 2, 4, 10, 18, 46, 82, 210, 374, 958, 1706, 4370, 7782, 19934, 35498, 90930, 161926, 414782, 738634, 1892050, 3369318, 8630686, 15369322, 39369330, 70107974, 179585278, 319801226, 819187730, 1458790182, 3736768094, 6654348458, 17045465010, 30354161926, 77753788862

Offset: 0

Robert G. Wilson v, Dec 29 2002

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-2).

Cf. A005824, A052913.

a[0] = 0; a[1] = 1; a[n_] := a[n] = If[ OddQ[n], a[n - 1] + 2a[n - 2], 2a[n - 1] + a[n - 2]]; Table[a[n], {n, 0, 30}]
LinearRecurrence[{0,5,0,-2},{0,1,2,4},40] (* Harvey P. Dale, Jul 05 2022 *)

Also a(n) = a(n-1) + 2a(n-2) if n is odd, else a(n) = 2a(n-1) + a(n-2).
a(2n) = 2*A005824(2n), a(2n+1) = A052913(n) = A005824(2n) + A005824(2n+1).
G.f.: x*(1+2*x-x^2)/(1-5*x^2+2*x^4).

Corrected the g.f. and index in formula with A052913 R. J. Mathar, Apr 01 2009, May 02 2009

a(31) onwards from Andrew Howroyd, Mar 19 2025

A295515 The Euclid tree, read across levels.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 2, 3, 3, 1, 1, 4, 4, 3, 3, 5, 5, 2, 2, 5, 5, 3, 3, 4, 4, 1, 1, 5, 5, 4, 4, 7, 7, 3, 3, 8, 8, 5, 5, 7, 7, 2, 2, 7, 7, 5, 5, 8, 8, 3, 3, 7, 7, 4, 4, 5, 5, 1, 1, 6, 6, 5, 5, 9, 9, 4, 4, 11, 11, 7, 7, 10, 10, 3, 3, 11, 11, 8, 8

Offset: 1

Peter Luschny, Nov 25 2017

nonn

Set N(x) = 1 + floor(x) - frac(x) and let '"' denote the ditto operator, referring to the previously computed expression. Assume the first expression is '0'. Then [0, repeat(N("))] will generate the natural numbers 0, 1, 2, 3, ... and [0, repeat(1/N("))] will generate the rational numbers 0/1, 1/1, 1/2, 2/1, 1/3, 3/2, ... Every reduced nonnegative rational number r appears exactly once in this list as a relatively prime pair [n, d] = r = n/d. We list numerator and denominator one after the other in the sequence.
The apt name 'Euclid tree' is taken from the exposition of Malter, Schleicher and Don Zagier. It is sometimes called the Calkin-Wilf tree. The enumeration is based on Stern's diatomic series (which is a subsequence) and computed by a modification of Dijkstra's 'fusc' function.
The tree listed has root 0, the variant with root 1 is more widely used. Seen as sequences the difference between the two trees is trivial: it is enough to leave out the first two terms; but as trees they are markedly different (see the example section).

			The tree with root 0 starts:
                                      [0/1]
                  [1/1,                                    1/2]
        [2/1,                1/3,                3/2,                2/3]
   [3/1,      1/4,      4/3,      3/5,      5/2,      2/5,      5/3,      3/4]
[4/1, 1/5, 5/4, 4/7, 7/3, 3/8, 8/5, 5/7, 7/2, 2/7, 7/5, 5/8, 8/3, 3/7, 7/4, 4/5]
.
The tree with root 1 starts:
                                      [1/1]
                  [1/2,                                    2/1]
        [1/3,                3/2,                2/3,                3/1]
   [1/4,      4/3,      3/5,      5/2,      2/5,      5/3,      3/4,      4/1]
[1/5, 5/4, 4/7, 7/3, 3/8, 8/5, 5/7, 7/2, 2/7, 7/5, 5/8, 8/3, 3/7, 7/4, 4/5, 5/1]

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 3rd ed., 2004.

Peter Luschny, Table of n, row(n) for n = 1..12
N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.
Edsger Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232. EWD 578: More about the function 'fusc'.
Peter Luschny, Rational Trees and Binary Partitions.
A. Malter, D. Schleicher, and D. Zagier, New looks at old number theory, Amer. Math. Monthly, 120(3), 2013, pp. 243-264.
Moritz A. Stern, Über eine zahlentheoretische Funktion, J. Reine Angew. Math., 55 (1858), 193-220.
Index entries for sequences related to enumerating the rationals
Index entries for sequences related to trees

Cf. A002487, A174981, A294446 (Stern-Brocot tree), A294442 (Kepler's tree), A295511 (Schinzel-Sierpiński tree), A295512 (encoded by semiprimes).

# First implementation: use it only if you are not afraid of infinite loops.
a := x -> 1/(1+floor(x)-frac(x)): 0; do a(%) od;
# Second implementation:
lusc := proc(m) local a, b, n; a := 0; b := 1; n := m; while n > 0 do
if n mod 2 = 1 then b := a + b else a := a + b fi; n := iquo(n, 2) od; a end:
R := n -> 3*2^(n-1)-1 .. 2^n: # The range of level n.
EuclidTree_rat := n -> [seq(lusc(k+1)/lusc(k), k=R(n), -1)]:
EuclidTree_num := n -> [seq(lusc(k+1), k=R(n), -1)]:
EuclidTree_den := n -> [seq(lusc(k), k=R(n), -1)]:
EuclidTree_pair := n -> ListTools:-Flatten([seq([lusc(k+1), lusc(k)], k=R(n), -1)]):
seq(print(EuclidTree_pair(n)), n=1..5);

def A295515(n):
    if n == 1: return 0
    M = [0, 1]
    for b in (n//2 - 1).bits():
        M[b] = M[0] + M[1]
    return M[1]
print([A295515(n) for n in (1..85)])

Some characteristics in comparison to the tree with root 1, seen as a table with T(n,k) for n >= 1 and 1 <= k <= 2^(n-1). Here Tr(n,k), Tp(n,k), Tq(n,k) denotes the fraction r, the numerator of r and the denominator of r in row n and column k respectively.
                       With root 0:             With root 1:
Root Tr(1,1)           0/1                      1/1
Tp(n,1)                0,1,2,3,...              1,1,1,1,...
Tp(n,2^(n-1))          0,1,2,3,...              1,2,3,4,...
Tq(n,1)                1,1,1,1,...              1,2,3,4,...
Tq(n,2^(n-1))          1,2,3,4,...              1,1,1,1,...
Sum_k Tp(n,k)          0,2,8,26,...  A024023    1,3,9,27,...  A000244
Sum_k Tq(n,k)          1,3,9,27,...  A000244    1,3,9,27,...  A000244
Sum_k 2Tr(n,k)         0,3,9,21,...  A068156    2,5,11,23,... A083329
Sum_k Tp(n,k)Tq(n,k)   0,3,17,81,... A052913-1  1,4,18,82,... A052913
----
a(n) = A002487(floor(n/2)). - Georg Fischer, Nov 29 2022

cp's OEIS Frontend

A007482 a(n) is the number of subsequences of [ 1, ..., 2n ] in which each odd number has an even neighbor.

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A078099 Array T(m,n) read by antidiagonals: T(m,n) = number of ways of 3-coloring an m X n grid (m >= 1, n >= 1).

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A060801 Invert transform of odd numbers: a(n) = Sum_{k=1..n} (2*k+1)*a(n-k), a(0)=1.

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A079162 a(n) = 5*a(n-2) - 2*a(n-4), with initial terms 0,1,2,4.

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A295515 The Euclid tree, read across levels.

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A147840 a(n)=10*a(n-1)-8*a(n-2), a(0)=1, a(1)=8 .

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A060801 Invert transform of odd numbers: a(n) = Sum_{k=1..n} (2k+1)a(n-k), a(0)=1.

A079162 a(n) = 5a(n-2) - 2a(n-4), with initial terms 0,1,2,4.

A147840 a(n)=10a(n-1)-8a(n-2), a(0)=1, a(1)=8 .