A006012 -id:A006012 - cp's OEIS frontend

A084130 a(n) = 8a(n-1) - 8a(n-2), a(0)=1, a(1)=4.

1, 4, 24, 160, 1088, 7424, 50688, 346112, 2363392, 16138240, 110198784, 752484352, 5138284544, 35086401536, 239584935936, 1635988275200, 11171226714112, 76281907511296, 520885446377472, 3556828310929408, 24287542916415488

Offset: 0

Paul Barry, May 16 2003

Binomial transform of A001541.
Let A be the unit-primitive matrix (see [Jeffery]) A = A_(8,3) = [0,0,0,1; 0,0,2,0; 0,2,0,1; 2,0,2,0]. Then A084130(n) = (1/4)*Trace(A^(2*n)). (Cf. A006012, A001333.) - L. Edson Jeffery, Apr 04 2011
a(n) is also the rational part of the Q(sqrt(2)) integer giving the length L(n) of a variant of the Lévy C-curve, given by Kival Ngaokrajang, at iteration step n. See A057084. - Wolfdieter Lang, Dec 18 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
L. E. Jeffery, Unit-primitive matrices
Index entries for linear recurrences with constant coefficients, signature (8,-8).

Cf. A000129, A001333, A001541, A002203, A006012, A057084, A084130, A084131.

[n le 2 select 4^(n-1) else 8*(Self(n-1) -Self(n-2)): n in [1..41]]; // G. C. Greubel, Oct 13 2022

LinearRecurrence[{8,-8},{1,4},30] (* Harvey P. Dale, Sep 25 2014 *)

{a(n)= if(n<0, 0, real((4+ 2*quadgen(8))^n))}

A084130=BinaryRecurrenceSequence(8,-8,1,4)
[A084130(n) for n in range(41)] # G. C. Greubel, Oct 13 2022

a(n) = (4+sqrt(8))^n/2 + (4-sqrt(8))^n/2.
G.f.: (1-4*x)/(1-8*x+8*x^2).
E.g.f.: exp(4*x)*cosh(sqrt(8)*x).
a(n) = A057084(n) - 4*A057084(n-1). - R. J. Mathar, Nov 10 2013
From G. C. Greubel, Oct 13 2022: (Start)
a(2*n) = 2^(3*n-1)*A002203(2*n).
a(2*n+1) = 2^(3*n+2)*A000129(2*n+1). (End)

A094803 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 1, s(2n) = 3.

Original entry on oeis.org

1, 3, 9, 28, 90, 296, 988, 3328, 11272, 38304, 130416, 444544, 1516320, 5174144, 17659840, 60282880, 205795456, 702583296, 2398676736, 8189409280, 27960021504, 95460743168, 325921881088, 1112763940864, 3799207806976, 12971294957568, 44286747439104, 151204366286848

Offset: 1

Herbert Kociemba, Jun 11 2004

In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n)) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = j, s(2n) = k.

Counts all paths of length (2*n+1), n >= 0, starting and ending at the initial node and ending at the nodes 1, 2, 3, 4 and 5 on the path graph P_7, see the Maple program. - Johannes W. Meijer, May 29 2010

Michael De Vlieger, Table of n, a(n) for n = 1..1876
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (6,-10,4).

Cf. A006012, A024175, A030436.

with(GraphTheory): G:=PathGraph(7): A:= AdjacencyMatrix(G): nmax:=25; n2:=2*nmax: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..5); od: seq(a(2*n+1),n=0..nmax-1); # Johannes W. Meijer, May 29 2010

f[n_] := FullSimplify[ TrigToExp[(1/4)Sum[ Sin[Pi*k/8]Sin[3Pi*k/8](2Cos[Pi*k/8])^(2n), {k, 1, 7}]]]; Table[ f[n], {n, 25}] (* Robert G. Wilson v, Jun 18 2004 *)
Rest@ CoefficientList[Series[-x (1 - 3 x + x^2)/((2 x - 1)*(2 x^2 - 4 x + 1)), {x, 0, 25}], x] (* Michael De Vlieger, Aug 04 2021 *)

a(n) = (1/4)*Sum_{k=1..7} sin(Pi*k/8)*sin(3*Pi*k/8)*(2*cos(Pi*k/8))^(2n).
a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3).
G.f.: -x*(1 - 3*x + x^2)/((2*x - 1)*(2*x^2 - 4*x + 1)).
E.g.f.: (2*sinh(x)^2 + sinh(2*x) + sqrt(2)*exp(2*x)*sinh(sqrt(2)*x))/4. - Stefano Spezia, Jun 14 2023

A111567 Binomial transform of A048654: generalized Pellian with second term equal to 4.

Original entry on oeis.org

1, 5, 18, 62, 212, 724, 2472, 8440, 28816, 98384, 335904, 1146848, 3915584, 13368640, 45643392, 155836288, 532058368, 1816560896, 6202126848, 21175385600, 72297288704, 246838383616, 842758957056, 2877359060992

Offset: 0

Creighton Dement, Aug 06 2005

Dropping the leading 1, this becomes the 4th row of the 2-shuffle Phi_2(W(sqrt(2))) of the Fraenkel-Kimberling publication. - R. J. Mathar, Aug 17 2009

Floretion Algebra Multiplication Program, FAMP Code: 1lesseq[K*J] with K = + .5'i + .5'j + .5k' + .5'kk' and J = + .5i' + .5j' + 2'kk' + .5'ki' + .5'kj'.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
A. S. Fraenkel and Clark Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discr. Math. 126 (1-3) (1994) 137-149. [From _R. J. Mathar_, Aug 17 2009]
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A007052, A006012, A111566.

LinearRecurrence[{4,-2},{1,5},30] (* Harvey P. Dale, Jul 01 2016 *)

a[0]:1$
a[1]:5$
a[n]:=4*a[n-1]-2*a[n-2]$
A111567(n):=a[n]$
makelist(A111567(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n) = 4*a(n-1) - 2*a(n-2), a(0) = 1, a(1) = 5. Program "FAMP" returns: A111566(n) = A007052(n) - A006012(n) + a(n).
From R. J. Mathar, Apr 02 2008: (Start)
O.g.f.: (1+x)/(1-4*x+2*x^2).
a(n) = A007070(n) + A007070(n-1). (End)
a(n) = ((2+sqrt(18))*(2+sqrt(2))^n + (2-sqrt(18))*(2-sqrt(2))^n)/4, offset 0. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
a(n) = ((5+sqrt(32))(2+sqrt(2))^n+(5-sqrt(32))(2-sqrt(2))^n)/2 offset 0. - Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Typo in definition corrected by Klaus Brockhaus, Aug 09 2009

A081704 Let f(0)=1, f(1)=t, f(n+1) = (f(n)^2+t^n)/f(n-1). f(t) is a polynomial with integer coefficients. Then a(n) = f(n) when t=3.

Original entry on oeis.org

1, 3, 12, 51, 219, 942, 4053, 17439, 75036, 322863, 1389207, 5977446, 25719609, 110665707, 476169708, 2048851419, 8815747971, 37932185598, 163213684077, 702271863591, 3021718265724, 13001775737847, 55943723892063, 240713292246774, 1035735289557681

Offset: 0

Victor Ufnarovski (ufn(AT)maths.lth.se), Apr 02 2003

f satisfies the linear recursion f(n+1) = (t+2)*f(n)-t*f(n-1). For t=3 this gives a(n+1) = 5*a(n)-3*a(n-1).

Given the 3 X 3 matrix [1,1,1; 1,1,2; 1,1,3] = M, a(n) = term (1,1) in M^(n+1). - Gary W. Adamson, Aug 06 2010

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-3).

Cf. A006012, A001519.

Equals 3*A018902(n-1) for n>0.

f := proc(n) if n=0 then 1 elif n=1 then t else sort(simplify((f(n-1)^2+t^(n-1))/f(n-2)),t) fi end; a := i->subs(t=3,f(i));

a[0]=1; a[1]=3; a[n_] := a[n]=5a[n-1]-3a[n-2]; Array[a,25,0]
LinearRecurrence[{5,-3},{1,3},30] (* Harvey P. Dale, Jul 28 2013 *)

Vec((1-2*x)/(1-5*x+3*x^2) + O(x^30)) \\ Colin Barker, Nov 26 2016

a(n+1) = (a(n)^2 + 3^n) / a(n-1).
From Philippe Deléham, Nov 14 2008: (Start)
G.f.: (1-2*x)/(1-5*x+3*x^2).
a(n) = Sum_{k, 0<=k<=n} A147703(n,k)*2^k. (End)
a(n) = (2^(-1-n)*((5-sqrt(13))^n*(-1+sqrt(13)) + (1+sqrt(13))*(5+sqrt(13))^n))/sqrt(13). - Colin Barker, Nov 26 2016
E.g.f.: exp(5*x/2)*(sqrt(13)*cosh(sqrt(13)*x/2) + sinh(sqrt(13)*x/2))/sqrt(13). - Stefano Spezia, Jul 09 2022

A111566 a(n) = ((1+sqrt(8))(2+sqrt(2))^n + (1-sqrt(8))(2-sqrt(2))^n)/2.

Original entry on oeis.org

1, 6, 22, 76, 260, 888, 3032, 10352, 35344, 120672, 412000, 1406656, 4802624, 16397184, 55983488, 191139584, 652591360, 2228086272, 7607162368, 25972476928, 88675582976, 302757378048, 1033678346240, 3529198628864, 12049437822976, 41139354034176, 140458540490752

Offset: 0

Creighton Dement, Aug 06 2005

Binomial transform of A048655: generalized Pellian with second term equal to 5.

Floretion Algebra Multiplication Program, FAMP Code: 1vesseq[K*J] with K = + .5'i + .5'j + .5k' + .5'kk' and J = + .5i' + .5j' + 2'kk' + .5'ki' + .5'kj'.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A007052, A006012, A111567, A007070.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+2*r)*(2+r)^n+(1-2*r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 27 2009

LinearRecurrence[{4,-2},{1,6},30] (* Harvey P. Dale, Jan 31 2015 *)

x='x+O('x^30); Vec((1+2*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jan 27 2018

a(n) = 4*a(n-1) - 2*a(n-2), a(0) = 1, a(1) = 6.
Program "FAMP" returns: a(n) = A007052(n) - A006012(n) + A111567(n).
From R. J. Mathar, Apr 02 2008: (Start)
O.g.f.: (1+2*x)/(1-4*x+2*x^2).
a(n) = A007070(n) + 2*A007070(n-1). (End)
a(n) = Sum_{k=0..n} A207543(n,k)*2^k. - Philippe Deléham, Feb 25 2012
a(n) = 4*A007070(n) - A007052(n+1). - Yuriy Sibirmovsky, Sep 13 2016
E.g.f.: exp(2*x)*(cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, May 26 2024

Edited by N. J. A. Sloane, Jul 27 2009 using new definition from Al Hakanson (hawkuu(AT)gmail.com)

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 1, 0, 0, 1, 10, 5, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (1/(1-x), x^2/(1-x)^2).
A skewed version of triangular array A085478.
Mirror image of triangle in A098158.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
From Gus Wiseman, Jul 08 2025: (Start)
After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:
  {}  {1}        {1,3}    .  .  .
      {2}        {1,4}
      {3}        {2,4}
      {4}        {1,2,4}
      {1,2}      {1,3,4}
      {2,3}
      {3,4}
      {1,2,3}
      {2,3,4}
      {1,2,3,4}
Requiring n-1 gives A202064.
For anti-runs instead of runs we have A384893.
(End)

			Triangle begins :
1
1, 0
1, 1, 0
1, 3, 0, 0
1, 6, 1, 0, 0
1, 10, 5, 0, 0, 0
1, 15, 15, 1, 0, 0, 0
1, 21, 35, 7, 0, 0, 0, 0
1, 28, 70, 28, 1, 0, 0, 0, 0

Column k = 1 is A000217.
Column k = 2 is A000332.
Row sums are A011782 (or A000079 shifted right).
Removing all zeros gives A034839 (requiring n-1 A034867).
Last nonzero term in each row appears to be A093178, requiring n-1 A124625.
Reversing rows gives A098158, without zeros A109446.
Without the k = 0 column we get A210039.
Row maxima appear to be A214282.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.
Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.

Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jul 08 2025 *)

T(n,k) = binomial(n,2k).
G.f.: (1-x)/((1-x)^2-y*x^2).
T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if nT(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Nov 10 2013

A208343 Triangle of coefficients of polynomials v(n,x) jointly generated with A208342; see the Formula section.

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 1, 2, 5, 0, 1, 2, 5, 8, 0, 1, 2, 6, 10, 13, 0, 1, 2, 7, 13, 20, 21, 0, 1, 2, 8, 16, 29, 38, 34, 0, 1, 2, 9, 19, 39, 60, 71, 55, 0, 1, 2, 10, 22, 50, 86, 122, 130, 89, 0, 1, 2, 11, 25, 62, 116, 187, 241, 235, 144, 0, 1, 2, 12, 28, 75, 150, 267, 392, 468

Offset: 1

Clark Kimberling, Feb 25 2012

u(n,n) = A000045(n+1) (Fibonacci numbers).
n-th row sum:  2^(n-1)
As triangle T(n,k) with 0 <= k <= n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

			First five rows:
  1;
  0, 2;
  0, 1, 3;
  0, 1, 2, 5;
  0, 1, 2, 5, 8;
First five polynomials v(n,x):
  1
     2x
      x + 3x^2
      x + 2x^2 + 5x^3
      x + 2x^2 + 5x^3 + 8x^4.

Cf. A208343.

Cf. A084938, A000045, A000079.

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
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[%]  (* A208342 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208343 *)

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = x*u(n-1,x) + x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 26 2012: (Start)
As triangle T(n,k) with 0 <= k <= n:
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k > n or if k < 0.
G.f.: (1-(1-y)*x)/(1-(1+y)*x+y*(1-y)*x^2).
Sum_{k=0..n} T(n,k)*x^k = (-1)*A091003(n+1), A152166(n), A000007(n), A000079(n), A055099(n), A152224(n) for x = -2, -1, 0, 1, 2, 3 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A087205(n), A140165(n+1), A016116(n+1), A000045(n+2), A000079(n), A122367(n), A006012(n), A052961(n), A154626(n) for x = -3, -2, -1, 0, 1, 2, 3, 4 respectively. (End)
T(n,k) = A208748(n,k)/2^k. - Philippe Deléham, Mar 05 2012

A265185 Non-vanishing traces of the powers of the adjacency matrix for the simple Lie algebra B_4: 2 * ((2 + sqrt(2))^n + (2 - sqrt(2))^n).

Original entry on oeis.org

4, 8, 24, 80, 272, 928, 3168, 10816, 36928, 126080, 430464, 1469696, 5017856, 17132032, 58492416, 199705600, 681837568, 2327939072, 7948081152, 27136446464, 92649623552, 316325601280, 1080003158016, 3687361429504, 12589439401984, 42983034748928

Offset: 0

Tom Copeland, Dec 04 2015

a(n) is the trace of the 2*n-th power of the adjacency matrix M for the simple Lie algebra B_4, given in the Damianou link. M = Matrix[row 1; row 2; row 3; row 4] = Matrix[0,1,0,0; 1,0,1,0; 0,1,0,2; 0,0,1,0]. Equivalently, the trace tr(M^(2*k)) is the sum of the 2*n-th powers of the eigenvalues of M. The eigenvalues are the zeros of the characteristic polynomial of M, which is det(x*I - M) = x^4 - 4*x^2 + 2 = A127672(4,x), and are (+-) sqrt(2 + sqrt(2)) and (+-) sqrt(2 - sqrt(2)), or the four unique values generated by 2*cos((2*n+1)*Pi/8). Compare with A025192 for B_3. The odd power traces vanish.
-log(1 - 4*x^2 + 2*x^4) = 8*x^2/2 + 24*x^4/4 + 80*x^6/6 + ... = Sum_{n>0} tr(M^k) x^k / k = Sum_{n>0} a(n) x^(2k) / 2k gives an aerated version of the sequence a(n), excluding a(0), and exp(-log(1 - 4*x + 2*x^2)) = 1 / (1 - 4*x + 2*x^2) is the e.g.f. for A007070.
As in A025192, the cycle index partition polynomials P_k(x[1],...,x[k]) of A036039 evaluated with the negated power sums, the aerated a(n), are P_2(0,-a(1)) = P_2(0,-8) = -8, P_4(0,-a(1),0,-a(2)) = P_4(0,-8,0,-24) = 48, and all other P_k(0,-a(1),0,-a(2),0,...) = 0 since 1 - 4*x^2 + 2*x^4 = 1 - 8*x^2/2! + 48*x^4/4! = det(I - x M) = exp(-Sum_{k>0} tr(M^k) x^k / k) = exp[P.(-tr(M),-tr(M^2),...)x] = exp[P.(0,-a(1),0,-a(2),...)x].
Because of the inverse relation between the Faber polynomials F_n(b1,b2,...,bn) of A263916 and the cycle index polynomials, F_n(0,-4,0,2,0,0,0,...) = tr(M^n) gives aerated a(n), excluding a(0). E.g., F_2(0,-4) = -2 * -4 = 8, F_4(0,-4,0,2) = -4 * 2 + 2 * (-4)^2 = 24, and F_6(0,-4,0,2,0,0) = -2*(-4)^3 + 6*(-4)*2 = 80.

G. C. Greubel, Table of n, a(n) for n = 0..1000
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2011-2014.
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A006012, A007052, A007070, A025192, A036039, A056236, A127672, A189315, A263916.

[Floor(2 * ((2 + Sqrt(2))^n + (2 - Sqrt(2))^n)): n in [0..30]]; // Vincenzo Librandi, Dec 06 2015

4 LinearRecurrence[{4, -2}, {1, 2}, 30] (* Vincenzo Librandi, Dec 06 2015 and slightly modified by Robert G. Wilson v, Feb 13 2018 *)

my(x='x+O('x^30)); Vec((4-8*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Feb 12 2018

a(n) = 2 * ((2 + sqrt(2))^n + (2 - sqrt(2))^n) = Sum_{k=0..3} 2^(2n) (cos((2k+1)*Pi/8))^(2n) = 2*2^(2n) (cos(Pi/8)^(2n) + cos(3*Pi/8)^(2n)) = 2 Sum_{k=0..1} (exp(i(2k+1)*Pi/8) + exp(-i*(2k+1)*Pi/8))^(2n).
E.g.f.: 2 * e^(2*x) * (e^(sqrt(2)*x) + e^(-sqrt(2)*x)) = 4*e^(2*x)*cosh(sqrt(2)*x) = 2*(exp(4*x*cos(Pi/8)^2) + exp(4*x cos(3*Pi/8)^2) ).
a(n) = 4*A006012(n) = 8*A007052(n-1) = 2*A056236(n).
G.f.: (4-8*x)/(1-4*x+2*x^2). - Robert Israel, Dec 07 2015
Note the preceding o.g.f. is four times that of A006012 and the denominator is y^4 * A127672(4,1/y) with y = sqrt(x). Compare this with those of A025192 and A189315. - Tom Copeland, Dec 08 2015

More terms from Vincenzo Librandi, Dec 06 2015

A083880 a(0)=1, a(1)=5, a(n) = 10a(n-1) - 23a(n-2), n >= 2.

Original entry on oeis.org

1, 5, 27, 155, 929, 5725, 35883, 227155, 1446241, 9237845, 59114907, 378678635, 2427143489, 15561826285, 99793962603, 640017621475, 4104915074881, 26328745454885, 168874407826587, 1083182932803515, 6947717948023649

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A083879.

Inverse binomial transform of A147957. 5th binomial transform of A077957. - Philippe Deléham, Nov 30 2008

Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A083878, A006012.

[ n eq 1 select 1 else n eq 2 select 5 else 10*Self(n-1)-23*Self(n-2): n in [1..21] ]; // Klaus Brockhaus, Dec 16 2008

LinearRecurrence[{10,-23},{1,5},30] (* Harvey P. Dale, May 14 2018 *)

a(n)=if(n<0,0,polsym(23-10*x+x^2,n)[n+1]/2)

G.f.: (1-5x)/(1-10x+23x^2).
E.g.f.: exp(5x)cosh(x*sqrt(2)).
a(n) = ((5-sqrt(2))^n + (5+sqrt(2))^n)/2;
a(n) = Sum_{k=0..n} C(n, 2k)*5^(n-2k)*2^k.
a(n) = (Sum_{k=0..n} A098158(n,k)*5^(2k)*2^(n-k))/5^n. - Philippe Deléham, Nov 30 2008

Typo in definition corrected by Klaus Brockhaus, Dec 16 2008

A094811 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 1, s(2n+1) = 6.

Original entry on oeis.org

1, 6, 26, 100, 364, 1288, 4488, 15504, 53296, 182688, 625184, 2137408, 7303360, 24946816, 85196928, 290926848, 993379072, 3391793664, 11580678656, 39539651584, 134998297600, 460915984384, 1573671536640, 5372862566400, 18344123969536

Offset: 2

Herbert Kociemba, Jun 11 2004

In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n+1) counts (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = j, s(2n+1) = k.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Andrei Asinowski and Michaela A. Polley, Patterns in rectangulations. Part I: T-like patterns, inversion sequence classes I(010, 101, 120, 201) and I(011, 201), and rushed Dyck paths, arXiv:2501.11781 [math.CO], 2025. See p. 26.
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Xavier Gérard Viennot, A Strahler bijection between Dyck paths and planar trees. Formal power series and algebraic combinatorics (Barcelona, 1999). Discrete Math. 246 (2002), no. 1-3, 317--329. MR1887493 (2003b:05013)
Index entries for linear recurrences with constant coefficients, signature (6,-10,4).

See A005022 for another version.

I:=[1, 6, 26]; [n le 3 select I[n] else 6*Self(n-1) - 10*Self(n-2) + 4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 21 2012

CoefficientList[Series[1/((1 - 2x)(1 - 4x + 2x^2)), {x, 0, 200}], x] (* Vincenzo Librandi, Oct 21 2012 *)
Table[FullSimplify[TrigToExp[(1/4) Sum[Sin[r*Pi/8] Sin[3 r Pi/4] (2 Cos[r Pi/8])^(2 n + 1), {r, 7}]]], {n, 2, 26}] (* Michael De Vlieger, Apr 27 2016 *)

a(n) = (1/4)*Sum_{r=1..7} sin(r*Pi/8)*sin(r*3*Pi/4)*(2*cos(r*Pi/8))^(2n+1).
G.f.: x^2/((1-2*x)*(1-4*x+2*x^2)).
a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3).
a(n) = A005022(n-2), n>2. - R. J. Mathar, Sep 05 2008
The g.f. x^3/(1 - 6x + 10x^2 - 4x^3) occurs on page 320 of Viennot, 2002.
a(n) = (A006012(n) - 2^n)/2. - R. J. Mathar, Jun 29 2012
a(n) = (-2^(1+n) + (2-sqrt(2))^n + (2+sqrt(2))^n)/4. - Colin Barker, Apr 27 2016
E.g.f.: exp(2*x)*sinh(x/sqrt(2))^2. - Ilya Gutkovskiy, Apr 27 2016

Additional comments from N. J. A. Sloane, May 01 2012

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cp's OEIS Frontend

A084130 a(n) = 8*a(n-1) - 8*a(n-2), a(0)=1, a(1)=4.

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A094803 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 1, s(2n) = 3.

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A111567 Binomial transform of A048654: generalized Pellian with second term equal to 4.

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A081704 Let f(0)=1, f(1)=t, f(n+1) = (f(n)^2+t^n)/f(n-1). f(t) is a polynomial with integer coefficients. Then a(n) = f(n) when t=3.

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A111566 a(n) = ((1+sqrt(8))*(2+sqrt(2))^n + (1-sqrt(8))*(2-sqrt(2))^n)/2.

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A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A208343 Triangle of coefficients of polynomials v(n,x) jointly generated with A208342; see the Formula section.

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A084130 a(n) = 8a(n-1) - 8a(n-2), a(0)=1, a(1)=4.

A111566 a(n) = ((1+sqrt(8))(2+sqrt(2))^n + (1-sqrt(8))(2-sqrt(2))^n)/2.

A083880 a(0)=1, a(1)=5, a(n) = 10a(n-1) - 23a(n-2), n >= 2.