A092886 -id:A092886 - cp's OEIS frontend

A105309 a(n) = |b(n)|^2 = x^2 + 3y^2 where (x,y,y,y) is the quaternion b(n) of the sequence b of quaternions defined by b(0)=1,b(1)=1, b(n) = b(n-1) + b(n-2)(0,c,c,c) where c = 1/sqrt(3).

1, 1, 2, 5, 9, 20, 41, 85, 178, 369, 769, 1600, 3329, 6929, 14418, 30005, 62441, 129940, 270409, 562725, 1171042, 2436961, 5071361, 10553600, 21962241, 45703841, 95110562, 197926885, 411889609, 857150100, 1783745641, 3712008565

Offset: 0

Gerald McGarvey, Apr 25 2005

Prepending 0 and keeping the offset at 0, turns this into a divisibility sequence with g.f. x(1-x^2)/(1-x-2x^2-x^3+x^4). - T. D. Noe, Dec 22 2008
Equals INVERT transform of (1, 1, 2, 0, 2, 0, 2, ...). - Gary W. Adamson, Apr 28 2009
Sequence gives the norm of the coefficients in 1/(1 - I*x - I*x^2), where I^2=-1. - Paul D. Hanna, Dec 06 2011
This is the case P1 = 1, P2 = -4, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 27 2014

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 9*x^4 + 20*x^5 + 41*x^6 + 85*x^7 + 178*x^8 + ...

Peter Bala, Linear divisibility sequences and Chebyshev polynomials
Ricky X. F. Chen and Louis W. Shapiro, On Sequences G(n) satisfying G(n) = (d+2)G(n-1)-G(n-2), J. Int. Seq. 10 (2007) 07.8.1, Theorem 16.
Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 8.
Eric Weisstein's World of Mathematics, Quaternion
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (1,2,1,-1).

Cf. A092886, A201837, A201838.

Cf. A006131, A100047, A240513

a[ n_] := (ChebyshevT[n + 1, (1 + Sqrt[17])/4] - ChebyshevT[n + 1, (1 - Sqrt[17])/4]) 2 / Sqrt[17] // Simplify; (* Michael Somos, Dec 20 2016 *)

{a(n) = my(A); n = abs(n+1)-1; if( n<2, n>=0, n++; A = vector(n, i, 1); for(i=3, n, A[i] = A[i-1] + A[i-2]*I); norm(A[n]))}; /* Michael Somos, Apr 28 2005 */

{a(n)=norm(polcoeff(1/(1-I*x-I*x^2+x*O(x^n)), n))} /* Paul D. Hanna */

{a(n)=polcoeff((1-x^2)/(1-x-2*x^2-x^3+x^4)+x*O(x^n),n)}

a(n) = A092886(n+1) - A092886(n-1), n > 0.
a(n) = A201837(n)^2 + A201838(n)^2. - Paul D. Hanna, Dec 06 2011
From Peter Bala, Mar 27 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(17))/4 and beta = (1 - sqrt(17))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 1/2].
a(n) = U(n-1,(1 + i)/sqrt(8))*U(n-1,(1 - i)/sqrt(8)), where U(n,x) denotes the Chebyshev polynomial of the second kind.
The o.g.f. is the Chebyshev transform of the rational function x/(1 - x + 4*x^2) = x + x^2 + 5*x^2 + 9*x^4 + 29*x^5 + ... (see A006131), where the Chebyshev transform takes the function A(x) to the function (1 - x^2)/(1 + x^2)*A(x/(1 + x^2)).
See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
a(n) = abs(((sqrt(4*i - 1) + i)^(n+1) - (i - sqrt(4*i - 1))^(n+1)) / 2^(n+1) / sqrt(4*i - 1))^2. - Daniel Suteu, Dec 20 2016
a(n) = a(-2-n) for all n in Z. - Michael Somos, Dec 20 2016
G.f.: (1+x)*(1-x)/(1-x-2*x^2-x^3+x^4). - R. J. Mathar, Apr 26 2024

A101400 a(n) = a(n-1) + 2*a(n-2) + a(n-3) - a(n-4).

Original entry on oeis.org

1, 2, 5, 10, 21, 44, 91, 190, 395, 822, 1711, 3560, 7409, 15418, 32085, 66770, 138949, 289156, 601739, 1252230, 2605915, 5422958, 11285279, 23484880, 48872481, 101704562, 211649125, 440445850, 916576181, 1907412444, 3969361531

Offset: 0

Jeroen F.J. Laros, Jan 15 2005

Lengths of successive words (starting with a) under the substitution: {a -> ab, b -> aac, c -> d, d -> b}.

			a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 10, a(4) = 21, a(5) = 44

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

Cf. A092886, A105309.

a:=[1,2,5,10];; for n in [5..35] do a[n]:=a[n-1]+2*a[n-2]+a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Apr 03 2018

I:=[1,2,5,10]; [n le 4 select I[n] else Self(n-1) + 2*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // G. C. Greubel, Apr 03 2018

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x+x^2)/(1-x-2*x^2-x^3+x^4))); // G. C. Greubel, Apr 03 2018

a[0] = 1; a[1] = 2; a[2] = 5; a[3] = 10; a[n_] := a[n] = a[n - 1] + 2a[n - 2] + a[n - 3] - a[n - 4]; Table[ a[n], {n, 0, 30}] (* Robert G. Wilson v, Jan 15 2005 *)
LinearRecurrence[{1,2,1,-1},{1,2,5,10},40] (* Harvey P. Dale, Oct 24 2017 *)

x='x+O('x^30); Vec((1+x+x^2)/(1-x-2*x^2-x^3+x^4)) \\ G. C. Greubel, Apr 03 2018

G.f.: (1+x+x^2)/(1-x-2*x^2-x^3+x^4). - G. C. Greubel, Apr 03 2018

More terms from Robert G. Wilson v, Jan 15 2005

A119749 Number of compositions of n into odd blocks with one element in each block distinguished.

Original entry on oeis.org

1, 1, 4, 7, 15, 32, 65, 137, 284, 591, 1231, 2560, 5329, 11089, 23076, 48023, 99935, 207968, 432785, 900633, 1874236, 3900319, 8116639, 16890880, 35150241, 73148321, 152223044, 316779047, 659223215, 1371856032, 2854858465

Offset: 1

Louis Shapiro, Jul 30 2006

The sequence is the INVERT transform of the aerated odd integers. - Gary W. Adamson, Feb 02 2014

Number of compositions of n into odd parts where there is 1 sort of part 1, 3 sorts of part 3, 5 sorts of part 5, ... , 2*k-1 sorts of part 2*k-1. - Joerg Arndt, Aug 04 2014

			a(3) = 4 since Abc, aBc, abC come from one block of size 3 and A/B/C comes from having three blocks. The capital letters are the distinguished elements.

R. X. F. Chen and L. W. Shapiro, On Sequences G(n) satisfying G(n)=(d+2)*G(n-1)-G(n-2), J. Int. Seq. 10 (2007) #07.8.1, Theorem 16.
Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (6).
Y.-h. Guo, n-Color Odd Self-Inverse Compositions, J. Int. Seq. 17 (2014) # 14.10.5, eq. (2).
B. Hopkins, Spotted tilings and n-color compositions, INTEGERS 12B (2012/2013), #A6.
Index entries for linear recurrences with constant coefficients, signature (1,2,1,-1).

Cf. A105309, A052530, A000045, A030267. Row sums of A292835.

Rest@ CoefficientList[ Series[x(1 + x^2)/(x^4 - x^3 - 2x^2 - x + 1), {x, 0, 50}], x] (* Robert G. Wilson v *)

G.f.: (x+x^3)/(x^4 - x^3 -2x^2 -x +1).
a(n) = A092886(n)+A092886(n-2). - R. J. Mathar, Mar 08 2018
Sum_{k=0..n} a(k) = (3*a(n) + 2*a(n-1) - a(n-3))/2 - 1. - Xilin Wang and Greg Dresden, Aug 27 2020

A138269 a(n+1) is the Hankel transform of C(n)+C(n+2), where C(n) = A000108(n).

Original entry on oeis.org

1, 3, 12, 53, 231, 1000, 4329, 18747, 81188, 351597, 1522639, 6594000, 28556241, 123666803, 535556412, 2319302053, 10044062391, 43497219000, 188370799289, 815766130347, 3532789487188, 15299239691997

Offset: 0

Paul Barry, Mar 10 2008

Index entries for linear recurrences with constant coefficients, signature (5,-4,5,-1).

Cf. A000108, A092886, A138268.

LinearRecurrence[{5,-4,5,-1},{1,3,12,53},30] (* Harvey P. Dale, Dec 04 2022 *)

G.f.: (1-x)^2/(1-5*x+4*x^2-5*x^3+x^4).
a(n) = Re(P(2n,i)*CONJ(P(2n+1,i))) where i=sqrt(-1) and P(n,x) = Sum_{k=0..floor(n/2)} binomial(n-k,k) x^k.
a(n) = 5*a(n-1)-4*a(n-2)+5*a(n-3)-a(n-4). - Wesley Ivan Hurt, Mar 07 2022

cp's OEIS Frontend

A105309 a(n) = |b(n)|^2 = x^2 + 3*y^2 where (x,y,y,y) is the quaternion b(n) of the sequence b of quaternions defined by b(0)=1,b(1)=1, b(n) = b(n-1) + b(n-2)*(0,c,c,c) where c = 1/sqrt(3).

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A101400 a(n) = a(n-1) + 2*a(n-2) + a(n-3) - a(n-4).

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A119749 Number of compositions of n into odd blocks with one element in each block distinguished.

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Author

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Examples

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Crossrefs

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Mathematica

Formula

A138269 a(n+1) is the Hankel transform of C(n)+C(n+2), where C(n) = A000108(n).

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Programs

Mathematica

Formula

A105309 a(n) = |b(n)|^2 = x^2 + 3y^2 where (x,y,y,y) is the quaternion b(n) of the sequence b of quaternions defined by b(0)=1,b(1)=1, b(n) = b(n-1) + b(n-2)(0,c,c,c) where c = 1/sqrt(3).