A001607 -id:A001607 - cp's OEIS frontend

A101088 Prime subsequence of A077020 (also A001607).

3, 5, 7, 3, 17, 11, 23, 89, 271, 457, 967, 4049, 181, 8641, 7193, 2603047, 68476319, 335257649, 5554901257, 31797598073, 23369856751, 77031318395801969, 3293187233103900007, 637758902554659731714245315207

Offset: 1

Gerard P. Michon, Dec 01 2004

nonn

			a(13)=181 because A077020(26)=181 is the 13th prime value found in that sequence; A001607(26)=-181.

G. P. Michon, Never Back to -1.

A177693 Triangle, read by rows, T(n, k) = p(n)/(p(k)*p(n-k)), where p(n) = Product_{j=1..n} A001607(j).

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 3, 3, 3, 1, 1, -1, 3, 3, -1, 1, 1, -5, -5, 15, -5, -5, 1, 1, 7, 35, 35, 35, 35, 7, 1, 1, 3, -21, -105, 35, -105, -21, 3, 1, 1, -17, 51, -357, 595, 595, -357, 51, -17, 1, 1, 11, 187, -561, -1309, -6545, -1309, -561, 187, 11, 1

Offset: 0

Roger L. Bagula, May 11 2010

			Triangle begins as:
  1;
  1,   1;
  1,  -1,   1;
  1,  -1,  -1,    1;
  1,   3,   3,    3,     1;
  1,  -1,   3,    3,    -1,     1;
  1,  -5,  -5,   15,    -5,    -5,     1;
  1,   7,  35,   35,    35,    35,     7,    1;
  1,   3, -21, -105,    35,  -105,   -21,    3,   1;
  1, -17,  51, -357,   595,   595,  -357,   51, -17,  1;
  1,  11, 187, -561, -1309, -6545, -1309, -561, 187, 11,  1;

Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Page 47ff.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001607.

A001607:=[n le 2 select n-1 else -Self(n-1)-2*Self(n-2): n in [1..100]];
p:= func< n | n eq 0 select 1 else (&*[A001607[j+1]: j in [1..n]]) >;
A177693:= func< n,k | p(n)/(p(k)*p(n-k)) >;
[A177693(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 08 2024

A001607:= LinearRecurrence[{-1,-2}, {0,1}, 100];
p[n_]:= Product[A001607[[i+1]], {i,n}];
T[n_,k_]:= p[n]/(p[k]*p[n-k]);
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

A001607=BinaryRecurrenceSequence(-1,-2,0,1)
def p(n): return product(A001607(j) for j in range(1,n+1))
def A177693(n,k): return p(n)/(p(k)*p(n-k))
flatten([[A177693(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 08 2024

T(n, k) = p(n)/(p(k)*p(n-k)), where p(n) = Product_{j=1..n} A001607(j) and p(0) = 1.

T(n, n-k) = T(n, k).

Edited by G. C. Greubel, Apr 08 2024

A101087 Indices of prime values of A077020 (also A001607).

Original entry on oeis.org

4, 6, 7, 8, 9, 10, 11, 15, 17, 19, 23, 25, 26, 29, 31, 47, 53, 65, 67, 71, 73, 113, 127, 199, 257, 349, 421, 433, 449, 691, 761, 823, 991, 1237, 1277, 1399, 1531, 1571, 3461, 3697, 4933, 6199, 7351

Offset: 1

Gerard P. Michon, Dec 01 2004

nonn

There are only 9 composite numbers in the entire sequence, namely: 4, 6, 8, 9, 10, 15, 25, 26 and 65.

If we are prepared to accept probable primes, then the sequence continues as follows: 9551, 9719, 11681, 12037, 14629, 14951, 19079, 20327, 22549, 30517, 51511, 52813, 60923, 73943, 79687, 91249, 115321, 117017, 169493, 172411, 174413, 237053, 285631, 318751, 327433. - David Broadhurst, May 23 2007

G. P. Michon, Never Back to -1.

Cf. A001607, A077020 (all values) and A101088 (prime values only). Similar to A001605 (Fibonacci primes), A000043 (Mersenne primes), A096650 (Pell primes), etc.

A107920 Lucas and Lehmer numbers with parameters (1 +- sqrt(-7))/2.

Original entry on oeis.org

0, 1, 1, -1, -3, -1, 5, 7, -3, -17, -11, 23, 45, -1, -91, -89, 93, 271, 85, -457, -627, 287, 1541, 967, -2115, -4049, 181, 8279, 7917, -8641, -24475, -7193, 41757, 56143, -27371, -139657, -84915, 194399, 364229, -24569, -753027, -703889, 802165, 2209943, 605613, -3814273

Offset: 0

Michael Somos, May 28 2005

The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.
This is an example of a sequence of Lehmer numbers. In this case, the two parameters, alpha and beta, are (1 +- i*sqrt(7))/2. Bilu, Hanrot, Voutier and Mignotte show that all terms of a Lehmer sequence a(n) have a primitive factor for n > 30. Note that for this sequence, a(30) = 24475 = 5*5*11*89 has no primitive factors. - T. D. Noe, Oct 29 2003
Row sums of Riordan array (1/(1+2*x^2), x/(1+2*x^2)). - Paul Barry, Sep 10 2005
Pisano period lengths: 1, 1, 8, 2, 24, 8, 21, 2, 24, 24, 10, 8, 168, 21, 24, 4, 144, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012
This is the Lucas Sequence U_n(P, Q) = U_n(1, 2). V_n(1, 2) = A002249(n). - Raphie Frank, Dec 25 2013
Note that (A002249(n)/2)^2 + 7*(a(n)/2)^2 = 2^n for all n in N. This is a specific case of the Lucas sequence identity (V_n/2)^2 - D*(U_n/2)^2 = Q^n where V_n = (a^n + b^n), U_n = (a^n - b^n)/(a - b), Q = (a*b) = 2 and D = (a - b)^2 = -7; a = (1 + sqrt(-7))/2 and b = (1 - sqrt(-7))/2. - Raphie Frank, Nov 26 2015
For the special case where |a(n)| = 1, true for n if and only if n is in {1, 2, 3, 5, 13} = {A215795(n) + 1} = {A060728(n) - 2}, then, additionally, by the Lucas sequence identity (U_2n = U_n*V_n), we have (a(2n)/2)^2 + 7*(a(n)/2)^2 = 2^n. - Raphie Frank, Nov 26 2015

			G.f. = x + x^2 - x^3 - 3*x^4 - x^5 + 5*x^6 + 7*x^7 - 3*x^8 - 17*x^9 - 11*x^10 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Christian Ballot, Lucasnomial Fuss-Catalan Numbers and Related Divisibility Questions, J. Int. Seq., Vol. 21 (2018), Article 18.6.5.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
F. Beukers, The multiplicity of binary recurrences, Compositio Mathematica, Tome 40 (1980) no. 2 , p. 251-267. See Theorem 2 p. 259.
Y. Bilu, G. Hanrot, P. M. Voutier and M. Mignotte, Existence of primitive divisors of Lucas and Lehmer numbers, [Research Report] RR-3792, INRIA. 1999, pp.41, HAL Id : inria-00072867.
M. Mignotte, Propriétés arithmétiques des suites récurrentes, Besançon, 1988-1989, see p. 14. In French.
Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Lehmer Number
Wikipedia, Lucas Sequence
Index entries for linear recurrences with constant coefficients, signature (1,-2).
Index entries for sequences related to Chebyshev polynomials.

Similar sequences: A001607, A077020, A107920, A167433, A169998.

Cf. A002249, A049310, A060728, A109466, A215795, A227078.

[0] cat [n le 2 select 1 else Self(n-1)-2*Self(n-2): n in [1..45]]; // Vincenzo Librandi, Nov 27 2015

a:= n-> (Matrix([[1,1],[ -2,0]])^n)[1,2]: seq(a(n), n=0..45); # Alois P. Heinz, Sep 03 2008

LinearRecurrence[{1, -2}, {0, 1}, 50] (* Vincenzo Librandi, Nov 27 2015 *)
a[ n_] := Im[ ((1 + Sqrt[-7]) / 2)^n // FullSimplify] 2 / Sqrt[7]; (* Michael Somos, Jan 19 2017 *)
a[n_] := If[n < 2, n, Hypergeometric2F1[1 - n/2, (1 - n)/2, 1 - n, 8]];
Table[a[n], {n, 0, 45}] (* Peter Luschny, Feb 23 2018 *)

PARI
```
{a(n) = imag(quadgen(-7)^n)};
```

my(x='x+O('x^100)); concat(0, Vec(x/(1-x+2*x^2))) \\ Altug Alkan, Dec 04 2015

[lucas_number1(n,1,2) for n in range(0, 46)] # Zerinvary Lajos, Apr 22 2009

G.f.: x / (1 - x + 2*x^2).
a(n) = a(n-1) - 2*a(n-2).
a(n) = -(-1)^n*A001607(n).
From Paul Barry, Sep 10 2005: (Start)
a(n+1) = Sum_{k=0..n} C((n+k)/2, k)*(-2)^((n-k)/2)*(1+(-1)^(n-k))/2.
a(n+1) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-2)^k. (End)
a(n+1) = Sum_{k=0..n} A109466(n,k)*2^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = ((1 - i*sqrt(7))^n - (1 + i*sqrt(7))^n)*i/(2^n*sqrt(7)), where i=sqrt(-1). - Bruno Berselli, Jul 01 2011
(a(2*(A060728(n)) - 4))^2 = (A002249(A060728(n) - 2))^2 = 2^(A060728(n)) - 7 = A227078(n), the Ramanujan-Nagell squares. - Raphie Frank, Dec 25 2013
a(n) = -a(-n) * 2^n for all n in Z. - Michael Somos, Jan 19 2017
G.f.: x / (1 - x / (1 + 2*x / (1 - 2*x))). - Michael Somos, Jan 19 2017
a(n) = S(n-1, 1/sqrt(2))*(sqrt(2))^(n-1), n >= 0, with the Chebyshev S polynomials (coefficients in A049310), and S(-1, x) = 0. - Wolfdieter Lang, Feb 22 2018
a(n) = hypergeom([1-n/2, (1-n)/2], [1-n], 8) for n >= 2. - Peter Luschny, Feb 23 2018

A002249 a(n) = a(n-1) - 2*a(n-2) with a(0) = 2, a(1) = 1.

Original entry on oeis.org

2, 1, -3, -5, 1, 11, 9, -13, -31, -5, 57, 67, -47, -181, -87, 275, 449, -101, -999, -797, 1201, 2795, 393, -5197, -5983, 4411, 16377, 7555, -25199, -40309, 10089, 90707, 70529, -110885, -251943, -30173, 473713, 534059, -413367, -1481485

Offset: 0

N. J. A. Sloane, Iwan Duursma

4*2^n = A002249(n)^2 + 7*A001607(n)^2. See A077020, A077021.
Among presented initial elements of the sequence a(n), the maximal increasing or decreasing subsequences have length either 3 or 4. - Roman Witula, Aug 21 2012
This is the Lucas Sequence V_n(P, Q) = V_n(1, 2). U_n(1, 2) = A107920(n). - Raphie Frank, Dec 25 2013
The only numbers that occur more than once are 1=a(1)=a(4) and -5=a(3)=a(9).  See Noam D. Elkies's posting in the Mathematics Stack Exchange link. - Robert Israel, Dec 21 2016

			We have a(2)-a(7) = a(5)-a(4) = a(6)+a(4) = a(11)-a(10) = a(12)+a(10)=10. Further the following relations: ((1+i*sqrt(7))/2)^4 + ((1-i*sqrt(7))/2)^4 = 1 and ((1+i*sqrt(7))/2)^8 + ((1-i*sqrt(7))/2)^8 = -31. - _Roman Witula_, Aug 21 2012
G.f. = 2 + x - 3*x^2 - 5*x^3 + x^4 + 11*x^5 + 9*x^6 - 13*x^7 - 31*x^8 + ...
From _Raphie Frank_, Dec 05 2015: (Start)
V_n(1, 2) = a(1*n) = ((a(1) + sqrt(-7))/2)^n + ((a(1) - sqrt(-7))/2)^n; a(1) = 1.
V_n(-3, 4) = a(2*n) = ((a(2) + sqrt(-7))/2)^n + ((a(2) - sqrt(-7))/2)^n; a(2) = -3.
V_n(-5, 8) = a(3*n) = ((a(3) + sqrt(-7))/2)^n + ((a(3) - sqrt(-7))/2)^n; a(3) = -5.
V_n(11, 32) = a(5*n) = ((a(5) + sqrt(-7))/2)^n + ((a(5) - sqrt(-7))/2)^n; a(5) = 11.
V_n(-181, 8192) = a(13*n) = ((a(13) + sqrt(-7))/2)^n + ((a(13) - sqrt(-7))/2)^n; a(13) = -181.
(End)

T. D. Noe, Table of n, a(n) for n = 0..500
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Mathematics Stack Exchange, An exotic sequence, March 2014.
Mihai Prunescu, On other two representations of the C-recursive integer sequences by terms in modular arithmetic, arXiv:2406.06436 [math.NT], 2024. See p. 18.
Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 17.
Mihai Prunescu and Joseph M. Shunia, On modular representations of C-recursive integer sequences, arXiv:2502.16928 [math.NT], 2025. See p. 6.
Wikipedia, Lucas Sequence.
Index entries for linear recurrences with constant coefficients, signature (1,-2).

Cf. A014551, A107920, A060728.

I:=[2,1]; [n le 2 select I[n] else Self(n-1)-2*Self(n-2): n in [1..50]]; // Vincenzo Librandi, Nov 29 2015

A002249 := proc(n) option remember; >if n = 1 then 1 elif n = 2 then -3 else A002249(n-1>)-2*A002249(n-2); fi; end;

LinearRecurrence[{1,-2}, {2,1}, 50] (* Roman Witula, Aug 21 2012 *)
a[ n_] := 2^(n/2) ChebyshevT[ n, 8^(-1/2)] 2; (* Michael Somos, Jun 02 2014 *)
a[ n_] := 2^Min[0, n] SeriesCoefficient[ (2 - x) / (1 - x + 2 x^2), {x, 0, Abs @ n}]; (* Michael Somos, Jun 02 2014 *)
Table[2 Re[((1 + I Sqrt[7])/2)^n], {n, 0, 40}] (* Jean-François Alcover, Jun 02 2017 *)

{a(n) = if( n<0, 2^n * a(-n), polsym(2 - x + x^2, n)[n+1])}; /* Michael Somos, Jun 02 2014 */

{a(n) = 2 * real( ((1 + quadgen(-28)) / 2)^n )}; /* Michael Somos, Jun 02 2014 */

x='x+O('x^100); Vec((2-x)/(1-x+2*x^2)) \\ Altug Alkan, Dec 04 2015

from sympy import sqrt, re, I
def a(n): return 2*re(((1 + I*sqrt(7))/2)**n)
print([a(n) for n in range(40)]) # Indranil Ghosh, Jun 02 2017

[lucas_number2(n,1,2) for n in range(0, 40)] # Zerinvary Lajos, Apr 30 2009

G.f.: (2-x)/(1-x+2x^2). - Michael Somos, Oct 18 2002
a(n) = trace(A^n) for the square matrix A=[1, -2; 1, 0]. - Paul Barry, Sep 05 2003
a(n) = 2^((n+2)/2)*cos(-n*acot(sqrt(7)/7)). - Paul Barry, Sep 06 2003
a(n) = (-1)^n*(2*A110512(n) - A001607(n)) = ((1 + i*sqrt(7))/2)^n + ((1 - i*sqrt(7))/2)^n. - Roman Witula, Aug 21 2012
G.f.: G(0), where G(k) = 1 + 1/(1 - x*(7*k+1)/(x*(7*k+8) + 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013
(a(A060728(n) - 2))^2 = (A107920(2*(A060728(n)) - 4))^2 = 2^(A060728(n)) - 7 = A227078(n), the Ramanujan-Nagell squares. - Raphie Frank, Dec 25 2013
a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x - 7*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n) = (A107920(n+1) + 2*A107920(n+2) - A107920(n+3))/2. - Raphie Frank, Nov 28 2015
V_n(P,Q) = a(k*n) = ((a(k) + sqrt(-7))/2)^n + ((a(k) - sqrt(-7))/2)^n for k is in {1, 2, 3, 5, 13} = (A060728(n) - 2), P is in {1, -3, -5, 11, -181} = a(k), and Q is in {2, 4, 8, 32, 8192} = 2^k = (2*A076046(n) + 2) = (A227078(n) - 7)/4. P^2 - 4*Q = -7. - Raphie Frank, Dec 05 2015
From Peter Bala, Nov 16 2022: (Start)
The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k) for all positive integers n and k and all primes p.
A268924(n) == a(3^n) (mod 3^n). (End)

A077020 a(n) is the unique odd positive solution x of 2^n = 7x^2+y^2.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 7, 3, 17, 11, 23, 45, 1, 91, 89, 93, 271, 85, 457, 627, 287, 1541, 967, 2115, 4049, 181, 8279, 7917, 8641, 24475, 7193, 41757, 56143, 27371, 139657, 84915, 194399, 364229, 24569, 753027, 703889, 802165, 2209943, 605613

Offset: 3

Ed Pegg Jr, Oct 17 2002

nonn

The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.

			G.f. = x^3 + x^4 + x^5 + 3*x^6 + x^7 + 5*x^8 + 7*x^9 + 3*x^10 + 17*x^11 + ...
a(3)=1 since 2^3=8=7*1^2+1^2, a(6)=3 since 2^6=64=7*3^2+1^2.

A. Engel, Problem-Solving Strategies. p. 126.

T. D. Noe, Table of n, a(n) for n=3..500
Eric Weisstein's World of Mathematics, Diophantine Equations 2nd Powers

a(n)=abs(A001607(n-2)).

Cf. A077021.

a(n) = 2^(n-2) * a(4-n) for all n in Z. - Michael Somos, Jan 05 2017
0 = 8*a(n)^2 + 2*a(n+1)^2 - a(n+2)^2 - a(n+3)^2 for all n in Z. - Michael Somos, Jan 05 2017
2*a(n) + a(n+1) = a(n+2) or a(n+3). - Michael Somos, Jan 05 2017

A078050 Expansion of (1-x)/(1+x+2*x^2).

Original entry on oeis.org

1, -2, 0, 4, -4, -4, 12, -4, -20, 28, 12, -68, 44, 92, -180, -4, 364, -356, -372, 1084, -340, -1828, 2508, 1148, -6164, 3868, 8460, -16196, -724, 33116, -31668, -34564, 97900, -28772, -167028, 224572, 109484, -558628, 339660, 777596, -1456916, -98276, 3012108, -2815556, -3208660, 8839772

Offset: 0

N. J. A. Sloane, Nov 17 2002

INVERTi transform of A005408, (2n + 1) = A078050 signed: (1, 2, 0, -4, -4, 4, 12, 4, -20, -28, ...) = left border of triangle A144106. - Gary W. Adamson, Sep 11 2008

Index entries for linear recurrences with constant coefficients, signature (-1,-2).

Cf. A005408, A144106. - Gary W. Adamson, Sep 11 2008

Cf. A208904.

Vec((1-x)/(1+x+2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

a(n-1) = Sum_{k=1..n} (-1)^(n-k) * Sum_{i=0..n} binomial(k,n-i) * binomial(k+i-1, 2*k-1). - Vladimir Kruchinin, Mar 11 2013

a(n) = A001607(n+1)-A001607(n). - R. J. Mathar, Mar 19 2025

A078020 Expansion of (1-x)/(1-x+2*x^2).

Original entry on oeis.org

1, 0, -2, -2, 2, 6, 2, -10, -14, 6, 34, 22, -46, -90, 2, 182, 178, -186, -542, -170, 914, 1254, -574, -3082, -1934, 4230, 8098, -362, -16558, -15834, 17282, 48950, 14386, -83514, -112286, 54742, 279314, 169830, -388798, -728458, 49138, 1506054, 1407778, -1604330, -4419886, -1211226, 7628546

Offset: 0

N. J. A. Sloane, Nov 17 2002

Equals the INVERT transform of [1, -1, -1, 1, 1, -1, -1, 1, 1, ...], i.e., 1 followed by repeats of (-1, -1, 1, 1, ...). - Gary W. Adamson, Sep 16 2008

Pisano period lengths: 1, 1, 8, 1, 24, 8, 21, 2, 24, 24, 10, 8, 168, 21, 24, 2, 144, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-2).

Cf. A001607, A107920.

a:=[1,0];; for n in [2..50] do a[n]:=a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Jun 29 2019

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x+2*x^2) )); // G. C. Greubel, Jun 29 2019

LinearRecurrence[{1,-2}, {1,0}, 50] (* or *) CoefficientList[Series[(1 - x)/(1-x+2*x^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 29 2019 *)

Vec((1-x)/(1-x+2*x^2)+O(x^50)) \\ Charles R Greathouse IV, Sep 25 2012

((1-x)/(1-x+2*x^2)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 29 2019

a(n) = A107920(n+1) - A107920(n). - R. J. Mathar, Mar 14 2011

a(n) = (-1)^n*(A001607(n) + A001607(n-1)). - G. C. Greubel, Jun 29 2019

A087168 Expansion of (1 + 2x)/(1 + 3x + 4*x^2).

Original entry on oeis.org

1, -1, -1, 7, -17, 23, -1, -89, 271, -457, 287, 967, -4049, 8279, -8641, -7193, 56143, -139657, 194399, -24569, -703889, 2209943, -3814273, 2603047, 7447951, -32756041, 68476319, -74404793, -50690897, 449691863, -1146312001, 1640168551, -335257649, -5554901257

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 22 2003

For positive n, a(n) equals 2^n times the permanent of the (2n) X (2n) tridiagonal matrix with 1/sqrt(2)'s along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011

For n > 3, equals -1 times the determinant of the (n-2) X (n-2) matrix with 2^2's along the superdiagonal, 3^2's along the main diagonal, 4^2's along the subdiagonal, etc., and 0's everywhere else. - John M. Campbell, Dec 01 2011

			G.f. = 1 - x - x^2 + 7*x^3 - 17*x^4 + 23*x^5 - x^6 - 89*x^7 + 271*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 14.
Index entries for linear recurrences with constant coefficients, signature (-3,-4).

Cf. A001607, A049072.

A087168:= func< n | &+[ Binomial(n+k, 2*k)*(-2)^(n-k): k in [0..n] ] >;
[A087168(n): n in [0..35]];

CoefficientList[Series[(1+2x)/(1+3x+4x^2), {x, 0, 30}], x]
Table[-Det[Array[Sum[KroneckerDelta[#1, #2+q]*(q+3)^2, {q, -1, n-2}] &, {n-2, n-2}]], {n, 4, 50}] (* John M. Campbell, Dec 01 2011 *)
LinearRecurrence[{-3,-4},{1,-1},40] (* Harvey P. Dale, Apr 23 2014 *)

{a(n) = real( (-1 - quadgen(-7))^n )}; /* Michael Somos, Sep 19 2014 */

def A087168(n): return (-2)^(n-1)*(2*chebyshev_U(n-2, 3/4) -chebyshev_U(n-1, 3/4))
[A087168(n) for n in (0..50)] # G. C. Greubel, Jun 09 2022

G.f.: (1+2*x)/(1+3*x+4*x^2).
a(n) = -3*a(n-1) - 4*a(n-2); a(0)=1, a(1)=-1.
a(n) = Sum_{k=0..n} C(n+k,2*k)*(-2)^(n-k).
a(n) = -a(-1-n) * 2^(2*n+1) = A001607(2*n + 1) for all n in Z. - Michael Somos, Sep 19 2014
a(n) = (-2)^(n-1)*(2*ChebyshevU(n-2, 3/4) - ChebyshevU(n-1, 3/4)). - G. C. Greubel, Jun 09 2022

A110512 Expansion of (1 + x)/(1 + x + 2x^2).

Original entry on oeis.org

1, 0, -2, 2, 2, -6, 2, 10, -14, -6, 34, -22, -46, 90, 2, -182, 178, 186, -542, 170, 914, -1254, -574, 3082, -1934, -4230, 8098, 362, -16558, 15834, 17282, -48950, 14386, 83514, -112286, -54742, 279314, -169830, -388798, 728458

Offset: 0

Paul Barry, Jul 24 2005

Row sums of number triangle A110511.
The sequences A110512 and A001607 are conjugated by one of the relations ((-1 + i*sqrt(7))/2)^n = a(n) + A001607(n)*(-1 + i*sqrt(7))/2 or ((-1 - i*sqrt(7))/2)^n = a(n) + A001607(n)*(-1 - i*sqrt(7))/2. These relations are connected with the Gauss sums; for example, if e := exp(i*2Pi/7) then e + e^2 + e^4 = (-1 + i*sqrt(7))/2 and e^3 + e^5 + e^6 = (-1 - i*sqrt(7))/2 -- for details see Witula's book. We also have a(n+1) = -2*A001607(n), which implies the Binet formula for a(n) (from the respective Binet formula for A001607(n) given in A001607), and A001607(n+1) = a(n) - A001607(n). - Roman Witula, Jul 27 2012
Pisano period lengths: 1, 1, 8, 1, 24, 8, 42, 1, 24, 24, 10, 8, 168, 42, 24, 2, 144, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012

R. Witula, On some applications of formulas for unimodular complex numbers, Jacek Skalmierski's Press, Gliwice 2011 (in Polish).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tejo V. Madhavarapu, The Most Malicious Maître D', arXiv:2407.09000 [math.CO], 2024. See p. 3.
Index entries for linear recurrences with constant coefficients, signature (-1,-2).

CoefficientList[Series[(1 + x)/(1 + x + 2*x^2), {x,0,50}], x] (* G. C. Greubel, Aug 29 2017 *)
LinearRecurrence[{-1,-2},{1,0},40] (* Harvey P. Dale, Dec 30 2024 *)

my(x='x+O('x^50)); Vec((1 + x)/(1 + x + 2*x^2)) \\ G. C. Greubel, Aug 29 2017

a(n) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-2)^(j-k)*C(k, j-k).
a(n) = (-1)^n*A078020(n). - R. J. Mathar, Feb 04 2009
a(n+2) + a(n+1) + 2*a(n) = 0. - Roman Witula, Jul 27 2012
G.f.: 2 - x + 2*x^2 + 3*x/Q(0), where Q(k)= 1 - 1/(4^k - 2*x*16^k/(2*x*4^k + 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013
From Ammar Khatab, Jul 11 2025: (Start)
a(n) = ((-sqrt(2))^(n+3)/sqrt(7)) * sin((n-1) * arctan(sqrt(7))).
x^n = A001607(n) * x + a(n) in Z[x]/(x^2 + x + 2).
a(n) = -2 * A001607(n-1), for n > 0.  (End)

cp's OEIS Frontend

A101088 Prime subsequence of A077020 (also A001607).

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A177693 Triangle, read by rows, T(n, k) = p(n)/(p(k)*p(n-k)), where p(n) = Product_{j=1..n} A001607(j).

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A101087 Indices of prime values of A077020 (also A001607).

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A107920 Lucas and Lehmer numbers with parameters (1 +- sqrt(-7))/2.

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A002249 a(n) = a(n-1) - 2*a(n-2) with a(0) = 2, a(1) = 1.

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A077020 a(n) is the unique odd positive solution x of 2^n = 7x^2+y^2.

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A078050 Expansion of (1-x)/(1+x+2*x^2).

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A087168 Expansion of (1 + 2x)/(1 + 3x + 4*x^2).