A111566 -id:A111566 - cp's OEIS frontend

A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).

-1, 2, 3, 8, 19, 46, 111, 268, 647, 1562, 3771, 9104, 21979, 53062, 128103, 309268, 746639, 1802546, 4351731, 10506008, 25363747, 61233502, 147830751, 356895004, 861620759, 2080136522, 5021893803, 12123924128, 29269742059, 70663408246, 170596558551, 411856525348

Offset: 0

Benoit Cloitre, Nov 22 2002

Inverse binomial transform of -1, 1, 6, 22, 76, 260, ... (see A111566). Binomial transform of -1, 3, -2, 6, -4, 12, -8, 24, -16, ... (see A162255). - R. J. Mathar, Oct 02 2012

			G.f. = -1 + 2*x + 3*x^2 + 8*x^3 + 19*x^4 + 46*x^5 + 111*x^6 + ... - _Michael Somos_, Jun 30 2022

H. S. M. Coxeter, 1998, Numerical distances among the circles in a loxodromic sequence, Nieuw Arch. Wisk, 16, pp. 1-9.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
H. S. M. Coxeter, Numerical distances among the spheres in a loxodromic sequence, Math. Intell. 19(4) 1997 pp. 41-47. See page 41. See pp. 46-47.
Tanya Khovanova, Recursive Sequences.
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv:1212.1368 [cs.DM], 2012.
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A000129, A048654, A048655, A048746, A111566, A162255, A221172, A221173, A221174, A221175.

a078343 n = a078343_list !! n
a078343_list = -1 : 2 : zipWith (+)
                        (map (* 2) $ tail a078343_list) a078343_list
-- Reinhard Zumkeller, Jan 04 2013

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)/(-1+2*x+x^2))); // G. C. Greubel, Jul 26 2018

f:=proc(n) option remember; if n=0 then RETURN(-1); fi; if n=1 then RETURN(2); fi; 2*f(n-1)+f(n-2); end;

Table[4 Fibonacci[n, 2] - Fibonacci[n + 1, 2], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 27 2016 *)
LinearRecurrence[{2,1},{-1,2},40] (* Harvey P. Dale, Apr 15 2019 *)

a(n)=([0,1;1,2]^n*[-1;2])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

For the unsigned version: a(1)=1; a(2)=2; a(n) = Sum_{k=2..n-1} (a(k) + a(k-1)).
a(n) is asymptotic to (1/4)*(-2+3*sqrt(2))*(1+sqrt(2))^n.
a(n) = A048746(n-3) + 2, for n > 2. - Ralf Stephan, Oct 17 2003
a(n) = 2*A000129(n) - A000129(n-1) if n > 0; abs(a(n)) = Sum_{k=0..floor(n/2)} (C(n-k-1, k) - C(n-k-1, k-1))2^(n-2k). - Paul Barry, Dec 23 2004
O.g.f.: (1-4*x)/(-1 + 2*x + x^2). - R. J. Mathar, Feb 15 2008
a(n) = 4*Pell(n) - Pell(n+1), where Pell = A000129. - Vladimir Reshetnikov, Sep 27 2016
a(n) = -(-1)^n * A048654(-n) = ( (-2+3*sqrt(2))*(1+sqrt(2))^n + (-2-3*sqrt(2))*(1-sqrt(2))^n )/4 for all n in Z. - Michael Somos, Jun 30 2022
2*a(n+1)^2 = A048655(n)^2 + (-1)^n*7. - Philippe Deléham, Mar 07 2023
E.g.f.: 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2) - exp(x)*cosh(sqrt(2)*x). - Stefano Spezia, May 26 2024

Entry revised by N. J. A. Sloane, Apr 29 2004

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

A207543 Triangle read by rows, expansion of (1+yx)/(1-2y*x+y(y-1)x^2).

Original entry on oeis.org

1, 0, 3, 0, 1, 5, 0, 0, 5, 7, 0, 0, 1, 14, 9, 0, 0, 0, 7, 30, 11, 0, 0, 0, 1, 27, 55, 13, 0, 0, 0, 0, 9, 77, 91, 15, 0, 0, 0, 0, 1, 44, 182, 140, 17, 0, 0, 0, 0, 0, 11, 156, 378, 204, 19, 0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21, 0

Offset: 0

Philippe Deléham, Feb 24 2012

Previous name was: "A scaled version of triangle A082985."

Triangle, read by rows, given by (0, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

			Triangle begins :
1
0, 3
0, 1, 5
0, 0, 5, 7
0, 0, 1, 14, 9
0, 0, 0, 7, 30, 11
0, 0, 0, 1, 27, 55, 13
0, 0, 0, 0, 9, 77, 91, 15
0, 0, 0, 0, 1, 44, 182, 140, 17
0, 0, 0, 0, 0, 11, 156, 378, 204, 19
0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21
0, 0, 0, 0, 0, 0, 13, 275, 1122, 1254, 385, 23

R. Zielinski, Induction and Analogy in a Problem of Finite Sums arXiv:1608.04006 [math.GM], 2016.

Cf. A082985 which is another version of this triangle.

Cf. Diagonals : A005408, A000330, A005585, A050486, A054333, A057788. Cf. A119900.

s := (1+y*x)/(1-2*y*x+y*(y-1)*x^2): t := series(s,x,12):
seq(print(seq(coeff(coeff(t,x,n),y,m),m=0..n)),n=0..11); # Peter Luschny, Aug 17 2016

T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 3.
G.f.: (1+y*x)/(1-2*y*x+y*(y-1)*x^2).
Sum_{i, i>=0} T(n+i,n) = A000204(2*n+1).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A078069(n), A000007(n), A003945(n), A111566(n) for x = -1, 0, 1, 2 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A090131(n), A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively.
From Peter Bala, Aug 17 2016: (Start)
Let S(k,n) = Sum_{i = 1..n} i^k. Calculations in Zielinski 2016 suggest the following identity holds involving the p-th row elements of this triangle:
Sum_{k = 0..p} T(p,k)*S(2*k,n) = 1/2*(2*n + 1)*(n*(n + 1))^p.
For example, for row 6 we find S(6,n) + 27*S(8,n) + 55*S(10,n) + 13*S(12,n) = 1/2*(2*n + 1)*(n*(n + 1))^6.
There appears to be a similar result for the odd power sums S(2*k + 1,n) involving A119900. (End)

New name using a formula of the author from Peter Luschny, Aug 17 2016

A163348 a(n) = 6a(n-1) - 7a(n-2) for n > 1; a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 35, 161, 721, 3199, 14147, 62489, 275905, 1218007, 5376707, 23734193, 104768209, 462469903, 2041441955, 9011362409, 39778080769, 175588947751, 775087121123, 3421400092481, 15102790707025, 66666943594783

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009

Binomial transform of A111566. Third binomial transform of A143095. Inverse binomial transform of A081180 without initial 0.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A111566, A143095 (1,4,2,8,4,16,...), A081180.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+2*r)*(3+r)^n+(1-2*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 26 2009

LinearRecurrence[{6, -7}, {1, 7}, 50] (* G. C. Greubel, Dec 19 2016 *)

Vec((1+x)/(1-6*x+7*x^2) + O(x^50)) \\ G. C. Greubel, Dec 19 2016

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7.
a(n) = ((1+2*sqrt(2))*(3+sqrt(2))^n + (1-2*sqrt(2))*(3-sqrt(2))^n)/2.
G.f.: (1+x)/(1-6*x+7*x^2).
E.g.f.: exp(3*x)*( cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Dec 19 2016
a(n) = A081179(n)+A081179(n+1). - R. J. Mathar, Feb 04 2021

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 26 2009

cp's OEIS Frontend

A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).

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

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A207543 Triangle read by rows, expansion of (1+y*x)/(1-2*y*x+y*(y-1)*x^2).

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Programs

Maple

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A163348 a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A207543 Triangle read by rows, expansion of (1+yx)/(1-2y*x+y(y-1)x^2).

A163348 a(n) = 6a(n-1) - 7a(n-2) for n > 1; a(0) = 1, a(1) = 7.