A133566 -id:A133566 - cp's OEIS frontend

A133567 A007318 * A133566.

1, 1, 1, 1, 3, 1, 1, 6, 3, 1, 1, 10, 6, 5, 1, 1, 15, 10, 15, 5, 1, 1, 21, 15, 35, 15, 7, 1, 1, 28, 21, 70, 35, 28, 7, 1, 1, 36, 28, 126, 70, 84, 28, 9, 1, 1, 45, 36, 210, 126, 210, 84, 45, 9, 1

Offset: 1

Gary W. Adamson, Sep 16 2007

Row sums = A083329: (1, 2, 5, 11, 23, 47, 95, ...).
From Clark Kimberling, Feb 28 2012: (Start)
A133567 is jointly generated with A133084 as an array of coefficients of polynomials v(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = u(n-1,x) + (x+1)*v(n-1)x and v(n,x) = x*u(n-1,x) + v(n-1,x) + 1. See the Mathematica section. (End)

			First few rows of the triangle:
  1;
  1,  1;
  1,  3,  1;
  1,  6,  3,  1;
  1, 10,  6,  5,  1;
  1, 15, 10, 15,  5,  1;
  1, 21, 15, 35, 15,  7,  1;
  ...

Cf. A133566, A083329, A133084.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A133567 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A133084 *)
(* Clark Kimberling, Feb 28 2012 *)

Binomial transform of triangle A133566.

A133603 The matrix-vector product A133566 * A000108.

Original entry on oeis.org

1, 1, 3, 5, 19, 42, 174, 429, 1859, 4862, 21658, 58786, 266798, 742900, 3417340, 9694845, 45052515, 129644790, 607283490, 1767263190, 8331383610, 24466267020, 115948830660, 343059613650, 1632963760974, 4861946401452

Offset: 0

Gary W. Adamson, Sep 18 2007

A133602 is a companion sequence.

			a(5) = C(5) = 42.
a(6) = 174 = C(6) + C(5) = 132 + 42.

Cf. A133566, A000108, A133602, A024492 (bisection).

A133603 := proc(n)
    add(A133566(n+1,k+1)*A000108(k),k=0..n) ;
end proc: # R. J. Mathar, Jun 20 2015

A133566 * A000108 where A133566 = an infinite lower triangular matrix and A000108 = the Catalan sequence. For odd n, a(n) = C(n). For even n, a(n) = C(n) + C(n-1) = A005807(n-1).
Conjecture: n*(n-2)*(3*n-1)*(n+1)*a(n) -8*n*(2*n-3)*a(n-1) -4*(n-1)*(3*n+2)*(
2*n-3)*(2*n-5)*a(n-2)=0. - R. J. Mathar, Jun 20 2015

A133647 A133566 * A000244.

Original entry on oeis.org

1, 3, 12, 27, 108, 243, 972, 2187, 8748, 19683, 78732, 177147, 708588, 1594323, 6377292, 14348907, 57395628, 129140163, 516560652, 1162261467, 4649045868, 10460353203, 41841412812, 94143178827, 376572715308, 847288609443, 3389154437772, 7625597484987, 30502389939948

Offset: 0

Gary W. Adamson, Sep 19 2007

A133646 is a companion case.

			a(3) = 27 = 3^3.
a(4) = 108 = 4 * 3^3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,9).

Cf. A000244, A133566, A133646.

Join[{1},Flatten[NestList[9#&,{3,12},20]]] (* or *) LinearRecurrence[{0,9},{1,3,12},40] (* Harvey P. Dale, Aug 01 2019 *)

A133566 * A000244, where A133566 = an infinite lower triangular matrix and A000244 = [3^0, 3^1, 3^2, ...]. For odd n, a(n) = 3^n. For even n, a(n) = 4 * 3^(n-1).
From Philippe Deléham, Apr 06 2012: (Start)
G.f.: (1+3*x+3*x^2)/(1-9*x^2).
a(n+2) = 9*a(n), a(0) = 1, a(1) = 3, a(2) = 12. (End)
From Amiram Eldar, Jun 02 2025: (Start)
Sum_{n>=0} 1/a(n) = 47/32.
Sum_{n>=0} (-1)^n/a(n) = 23/32. (End)

More terms from Philippe Deléham, Apr 06 2012

A133805 Triangle read by rows: A007318 * A133566 * A133080.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 7, 6, 4, 1, 11, 10, 11, 5, 1, 16, 15, 25, 15, 6, 1, 22, 21, 50, 35, 22, 7, 1, 29, 28, 91, 70, 63, 28, 8, 1, 37, 36, 154, 126, 154, 84, 37, 9, 1, 46, 45, 246, 210, 336, 210, 129, 45, 10, 1, 56, 55, 375, 330, 672, 462, 375, 165, 56, 11, 1, 67, 66, 550, 495, 1254, 924, 957, 495, 231, 66, 12, 1

Offset: 1

Gary W. Adamson, Sep 23 2007

Row sums = A133806: (1, 3, 8, 18, 38, 78, 318, ...).

Left column = A000124.

			First few rows of the triangle:
   1;
   2,  1;
   4,  3,  1;
   7,  6,  4,  1;
  11, 10, 11,  5,  1;
  16, 15, 25, 15,  6,  1;
  22, 21, 50, 35, 22,  7,  1;
  29, 28, 91, 70, 63, 28,  8,  1;
  ...

Cf. A000124, A133080, A133566, A133804, A133806.

Binomial transform of (A133566 * A133080) where (A133566 * A133080) = an infinite lower triangular matrix with (1,1,1,...) in the main and subdiagonals and (1,0,1,0,1,...) in the subsubdiagonal.

a(21) = 1 inserted and more terms from Georg Fischer, Jun 08 2023

A133569 A133566 * A007318 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 1, 3, 3, 1, 2, 7, 9, 5, 1, 1, 5, 10, 10, 5, 1, 2, 11, 25, 30, 20, 7, 1, 1, 7, 21, 35, 35, 21, 7, 1, 2, 15, 49, 91, 105, 77, 35, 9, 1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1

Offset: 1

Gary W. Adamson, Sep 16 2007

Row sums = A122756: (1, 2, 6, 8, 24, 32, 96, ...).

With offset (0,4); triangle T(n,k), 0 <= k <= n, given by [1,1,-3,1,0,0,0,0,0,0,0,...] DELTA [1,0,-2,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 15 2007

			First few rows of the triangle:
  1;
  1,  1;
  2,  3,  1;
  1,  3,  3,  1;
  2,  7,  9,  5,  1;
  1,  5, 10, 10,  5,  1;
  2, 11, 25, 30, 20,  7,  1;
  1,  7, 21, 35, 35, 21,  7,  1;
  ...

Cf. A133566, A122756.

T(n,k) = T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(2,2) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Sep 17 2014

G.f.: (-1-x-x*y-x^2-x^2*y)/((x*y+1+x)*(-1+x+x*y)). - R. J. Mathar, Aug 12 2015

A133804 Triangle read by rows: A007318 * A133080 * A133566.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 14, 10, 5, 1, 6, 25, 20, 16, 6, 1, 7, 41, 35, 41, 21, 7, 1, 8, 63, 56, 91, 56, 29, 8, 1, 9, 92, 84, 182, 126, 92, 36, 9, 1, 10, 129, 120, 336, 252, 246, 120, 46, 10, 1, 11, 175, 165, 582, 462, 582, 330, 175, 55, 11, 1, 12, 231, 220, 957, 792, 1254, 792, 550, 220, 67, 12, 1

Offset: 1

Gary W. Adamson, Sep 23 2007

Row sums = A133124: (1, 3, 7, 16, 35, 74, 153, ...).

A133805 = binomial transform of (A133566 * A133080).

			First few rows of the triangle:
  1;
  2,  1;
  3,  3,  1;
  4,  7,  4,  1;
  5, 14, 10,  5,  1;
  6, 25, 20, 16,  6,  1;
  7, 41, 35, 41, 21,  7,  1;
  ...

Cf. A133080, A133124, A133566.

Binomial transform of (A133080 * A133566), where A133080 * A133566 = an infinite lower triangular matrix with (1,1,1,...) in the main and subdiagonals and (0,1,0,1,0,...) in the subsubdiagonal.

a(21) = 1 inserted and more terms from Georg Fischer, Jun 08 2023

A133807 A007318 * (A097806 + A133566 - I), where I is the identity matrix.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 4, 1, 5, 16, 10, 6, 1, 6, 25, 20, 20, 6, 1, 7, 36, 35, 50, 21, 8, 1, 8, 49, 56, 105, 56, 35, 8, 1, 9, 64, 84, 196, 126, 112, 36, 10, 1, 10, 81, 120, 336, 252, 294, 120, 54, 10, 1

Offset: 1

Gary W. Adamson, Sep 23 2007

Row sums = A133806: (1, 3, 8, 18, 38, 78, 158, ...).

			First few rows of the triangle:
  1;
  2,  1;
  3,  4,  1;
  4,  9,  4,  1;
  5, 16, 10,  6,  1;
  6, 25, 20, 20,  6,  1;
  7, 36, 35, 50, 21,  8,  1;
  ...

Cf. A133566, A007318 , A097806, A133806.

Binomial transform of matrix M, where M = (A097806 + A133566 - I) = triangle with (1,1,1,...) in the main diagonal, (1,2,1,2,1,...) in the subdiagonal and the rest zeros. I = Identity matrix.

A040001 1 followed by {1, 2} repeated.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 0

N. J. A. Sloane

Continued fraction for sqrt(3).
Also coefficient of the highest power of q in the expansion of the polynomial nu(n) defined by: nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(1,1), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
nu(0)=1 nu(1)=1; nu(2)=2; nu(3)=3+q; nu(4)=5+3q+2q^2; nu(5)=8+7q+6q^2+4q^3+q^4; nu(6)=13+15q+16q^2+14q^3+11q^4+5q^5+2q^6.
From Jaroslav Krizek, May 28 2010: (Start)
a(n) = denominators of arithmetic means of the first n positive integers for n >= 1.
See A026741(n+1) or A145051(n) - denominators of arithmetic means of the first n positive integers. (End)
From R. J. Mathar, Feb 16 2011: (Start)
This is a prototype of multiplicative sequences defined by a(p^e)=1 for odd primes p, and a(2^e)=c with some constant c, here c=2. They have Dirichlet generating functions (1+(c-1)/2^s)*zeta(s).
Examples are A153284, A176040 (c=3), A040005 (c=4), A021070, A176260 (c=5), A040011, A176355 (c=6), A176415 (c=7), A040019, A021059 (c=8), A040029 (c=10), A040041 (c=12). (End)
a(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A000325(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
For n > 0: denominators of row sums of the triangular enumeration of rational numbers A226314(n,k) / A054531(n,k), 1 <= k <= n; see A226555 for numerators. - Reinhard Zumkeller, Jun 10 2013
From Jianing Song, Nov 01 2022: (Start)
For n > 0, a(n) is the minimal gap of distinct numbers coprime to n. Proof: denote the minimal gap by b(n). For odd n we have A058026(n) > 0, hence b(n) = 1. For even n, since 1 and -1 are both coprime to n we have b(n) <= 2, and that b(n) >= 2 is obvious.
The maximal gap is given by A048669. (End)

			1.732050807568877293527446341... = 1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + ...))))
G.f. = 1 + x + 2*x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + ...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 144.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Ashok Kumar Gupta and Ashok Kumar Mittal, Bifurcating continued fractions, arXiv:math/0002227 [math.GM] (2000).
Michael Somos, Rational Function Multiplicative Coefficients.
Eric Weisstein's World of Mathematics, Square Root.
Eric Weisstein's World of Mathematics, Theodorus's Constant.
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000034, A002194, A133566, A083329 (binomial Transf).

Apart from a(0) the same as A134451.

a040001 0 = 1; a040001 n = 2 - mod n 2
a040001_list = 1 : cycle [1, 2]  -- Reinhard Zumkeller, Apr 16 2015

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[3],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
PadRight[{1},120,{2,1}] (* Harvey P. Dale, Nov 26 2015 *)

{a(n) = 2 - (n==0) - (n%2)} /* Michael Somos, Jun 11 2003 */

{ allocatemem(932245000); default(realprecision, 12000); x=contfrac(sqrt(3)); for (n=0, 20000, write("b040001.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

Multiplicative with a(p^e) = 2 if p even; 1 if p odd. - David W. Wilson, Aug 01 2001
G.f.: (1 + x + x^2) / (1 - x^2). E.g.f.: (3*exp(x)-2*exp(0)+exp(-x))/2. - Paul Barry, Apr 27 2003
a(n) = (3-2*0^n +(-1)^n)/2. a(-n)=a(n). a(2n+1)=1, a(2n)=2, n nonzero.
a(n) = sum{k=0..n, F(n-k+1)*(-2+(1+(-1)^k)/2+C(2, k)+0^k)}. - Paul Barry, Jun 22 2007
Row sums of triangle A133566. - Gary W. Adamson, Sep 16 2007
Euler transform of length 3 sequence [ 1, 1, -1]. - Michael Somos, Aug 04 2009
Moebius transform is length 2 sequence [ 1, 1]. - Michael Somos, Aug 04 2009
a(n) = sign(n) + ((n+1) mod 2) = 1 + sign(n) - (n mod 2). - Wesley Ivan Hurt, Dec 13 2013

A084068 a(1) = 1, a(2) = 2; a(2k) = 2a(2k-1) - a(2k-2), a(2k+1) = 4a(2k) - a(2k-1).

Original entry on oeis.org

1, 2, 7, 12, 41, 70, 239, 408, 1393, 2378, 8119, 13860, 47321, 80782, 275807, 470832, 1607521, 2744210, 9369319, 15994428, 54608393, 93222358, 318281039, 543339720, 1855077841, 3166815962, 10812186007, 18457556052, 63018038201, 107578520350

Offset: 1

Benoit Cloitre, May 10 2003

The upper principal and intermediate convergents to 2^(1/2), beginning with 2/1, 3/2, 10/7, 17/12, 58/41, form a strictly decreasing sequence; essentially, numerators=A143609 and denominators=A084068. - Clark Kimberling, Aug 27 2008
From Peter Bala, Mar 23 2018: (Start)
Define a binary operation o on the real numbers by x o y = x*sqrt(1 + y^2) + y*sqrt(1 + x^2). The operation o is commutative and associative with identity 0. We have
  a(2*n + 1) = 1 o 1 o ... o 1 (2*n + 1 terms) and
  a(2*n) = (1/sqrt(2))*(1 o 1 o ... o 1) (2*n terms). Cf. A049629, A108412 and A143608.
This is a fourth-order divisibility sequence. Indeed, a(2*n) = U(2*n)/sqrt(2) and a(2*n+1) = U(2*n+1), where U(n) is the Lehmer sequence [Lehmer, 1930] defined by the recurrence U(n) = 2*sqrt(2)*U(n-1) - U(n-2) with U(0) = 0 and U(1) = 1. The solution to the recurrence is U(n) = (1/2)*( (sqrt(2) + 1)^n - (sqrt(2) - 1)^n ).
It appears that this sequence consists of those numbers m such that 2*m^2 = floor( m*sqrt(2) * ceiling(m*sqrt(2)) ). Cf. A084069. (End)
Conjecture: a(n) is the earliest occurrence of n in A348295, which is to say, a(n) is the least m such that Sum_{k=1..m} (-1)^(floor(k*(sqrt(2)-1))) = Sum_{k=1..m} (-1)^A097508(k) = n. This has been confirmed for the first 32 terms by Chai Wah Wu, Oct 21 2021. - Jianing Song, Jul 16 2022

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

Indranil Ghosh, Table of n, a(n) for n = 1..2608
Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.
D. H. Lehmer, An extended theory of Lucas' functions, Annals of Mathematics, Second Series, Vol. 31, No. 3 (Jul., 1930), pp. 419-448.
Eric Weisstein's World of Mathematics, Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Bisections are A001542 and A002315.

Cf. A084069, A084070, A133566, A079496, A001541, A001653, A008844, A055792, A049629, A108412, A143608.

a := proc (n) if `mod`(n, 2) = 1 then (1/2)*(sqrt(2) + 1)^n - (1/2)*(sqrt(2) - 1)^n else (1/2)*((sqrt(2) + 1)^n - (sqrt(2) - 1)^n)/sqrt(2) end if;
end proc:
seq(simplify(a(n)), n = 1..30); # Peter Bala, Mar 25 2018

a[n_] := ((Sqrt[2]+1)^n - (Sqrt[2]-1)^n) ((-1)^n(Sqrt[2]-2) + (Sqrt[2]+2))/8;
Table[Simplify[a[n]], {n, 30}] (* after Paul Barry, Peter Luschny, Mar 29 2018 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,6,0]^(n-1)*[1;2;7;12])[1,1] \\ Charles R Greathouse IV, Jun 20 2015

"A Diofloortin equation": n such that 2*n^2=floor(n*sqrt(2)*ceiling(n*sqrt(2))).
a(n)*a(n+3) = -2 + a(n+1)*a(n+2).
From Paul Barry, Jun 06 2006: (Start)
G.f.: x*(1+x)^2/(1-6*x^2+x^4);
a(n) = ((sqrt(2)+1)^n-(sqrt(2)-1)^n)*((sqrt(2)/8-1/4)*(-1)^n+sqrt(2)/8+1/4);
a(n) = Sum_{k=0..floor(n/2)} 2^k*(C(n,2*k)-C(n-1,2*k+1)*(1+(-1)^n)/2). (End)
A000129(n+1) = A079496(n) + a(n). - Gary W. Adamson, Sep 18 2007
Equals A133566 * A000129, where A000129 = the Pell sequence. - Gary W. Adamson, Sep 18 2007
From Peter Bala, Mar 23 2018: (Start)
a(2*n + 2) = a(2*n + 1) + sqrt( (1 + a(2*n + 1)^2)/2 ).
a(2*n + 1) = 2*a(2*n) + sqrt( (1 + 2*a(2*n)^2) ).
More generally,
a(2*n+2*m+1) = sqrt(2)*a(2*n) o a(2*m+1), where o is the binary operation defined above, that is,
a(2*n+2*m+1) = sqrt(2)*a(2*n)*sqrt(1 + a(2*m+1)^2) + a(2*m+1)*sqrt(1 + 2*a(2*n)^2).
sqrt(2)*a(2*(n + m)) = (sqrt(2)*a(2*n)) o (sqrt(2)*a(2*m)), that is,
a(2*n+2*m) = a(2*n)*sqrt(1 + 2*a(2*m)^2) + a(2*m)*sqrt(1 + 2*a(2*n)^2).
sqrt(1 + 2*a(2*n)^2) = A001541(n).
1 + 2*a(2*n)^2 = A055792(n+1).
a(2*n) - a(2*n-1) = A001653(n).
(1 + a(2*n+1)^2)/2 = A008844(n). (End)
a(n) = A000129(n) for even n and A001333(n) for odd n. - R. J. Mathar, Oct 15 2021

A114752 a(2n)=2n, a(2n+1)=4n+1.

Original entry on oeis.org

1, 2, 5, 4, 9, 6, 13, 8, 17, 10, 21, 12, 25, 14, 29, 16, 33, 18, 37, 20, 41, 22, 45, 24, 49, 26, 53, 28, 57, 30, 61, 32, 65, 34, 69, 36, 73, 38, 77, 40, 81, 42, 85, 44, 89, 46, 93, 48, 97, 50, 101, 52, 105, 54, 109, 56, 113, 58, 117, 60, 121, 62, 125, 64, 129, 66, 133, 68, 137

Offset: 1

Amarnath Murthy, Nov 15 2005

Original definition (typos corrected):  The following triangle contains n consecutive numbers beginning from n in ascending order if n is odd else in descending order. 1 3 2 3 4 5 7 6 5 4 5 6 7 8 9 11 10 9 8 7 6 ... Sequence contains the leading diagonal.
Equals A133566 * [1,2,3,...]. - Gary W. Adamson, Sep 16 2007
The sequence satisfies a divisibility property described by E. Angelini on the SeqFan list, cf. link. - M. F. Hasler, Mar 22 2013
First difference of A014255 (shown easily from the Nurikabe property of that sequence, or by manipulating the linear recurrence representations). - Allan C. Wechsler, Oct 20 2022

			Contribution by _M. F. Hasler_, Mar 22 2013: (Start)
The triangle described in the original definition starts
   1
   3  2
   3  4  5
   7  6  5  4
   5  6  7  8  9
  11 10  9  8  7  6. (End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Angelini, k-chunks sum and division by k, post to the SeqFan list, Mar 22 2013.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A114751, A114753, A133566, A014255

With[{nn=40},Riffle[Range[1,4nn,4],Range[2,2nn,2]]] (* Harvey P. Dale, Apr 14 2017 *)

a(n)=n+if(n%2,n\2*2) \\ Charles R Greathouse IV, Oct 16 2015

a(2n) = 2n, a(2n+1) = 4n+1. - Joshua Zucker, May 05 2006
G.f.: x*(1+2*x+3*x^2)/(1-x^2)^2. - Philippe Deléham, Mar 02 2012
a(n) = (3n-(n-1)*(-1)^n-1)/2. - Bruno Berselli, Mar 02 2012

More terms from Joshua Zucker, May 05 2006

Simpler definition from M. F. Hasler, Mar 22 2013

cp's OEIS Frontend

A133567 A007318 * A133566.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A133603 The matrix-vector product A133566 * A000108.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A133647 A133566 * A000244.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A133805 Triangle read by rows: A007318 * A133566 * A133080.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A133569 A133566 * A007318 as infinite lower triangular matrices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A133804 Triangle read by rows: A007318 * A133080 * A133566.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A133807 A007318 * (A097806 + A133566 - I), where I is the identity matrix.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A040001 1 followed by {1, 2} repeated.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

A084068 a(1) = 1, a(2) = 2; a(2k) = 2a(2k-1) - a(2k-2), a(2k+1) = 4a(2k) - a(2k-1).