A002203 -id:A002203 - cp's OEIS frontend

A164300 a(n) = ((1+4sqrt(2))(4+sqrt(2))^n + (1-4sqrt(2))(4-sqrt(2))^n)/2.

1, 12, 82, 488, 2756, 15216, 83144, 452128, 2453008, 13294272, 72012064, 389976704, 2111644736, 11433484032, 61904845952, 335169991168, 1814692086016, 9825156811776, 53195565289984, 288012326955008

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

Binomial transform of A164299. Fourth binomial transform of A164587. Inverse binomial transform of A164301.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 -2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-14).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), this sequence (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

Cf. A081180, A083879, A164587.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(4+r)^n+(1-4*r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

LinearRecurrence[{8,-14},{1,12},30] (* Harvey P. Dale, Apr 13 2012 *)

my(x='x+O('x^50)); Vec((1+4*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Sep 13 2017

[( (1+4*x)/(1-8*x+14*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 12.
G.f.: (1+4*x)/(1-8*x+14*x^2).
E.g.f.: (4*sqrt(2)*sinh(sqrt(2)*x) + cosh(sqrt(2)*x))*exp(4*x). - Ilya Gutkovskiy, Jun 24 2016
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = 2*A083879(n) + 8*A081180(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*3^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 17 2009

A164301 a(n) = ((1+4sqrt(2))(5+sqrt(2))^n + (1-4sqrt(2))(5-sqrt(2))^n)/2.

Original entry on oeis.org

1, 13, 107, 771, 5249, 34757, 226843, 1469019, 9472801, 60940573, 391531307, 2513679891, 16131578849, 103501150997, 663985196443, 4259325491499, 27321595396801, 175251467663533, 1124117982508907, 7210396068827811, 46249247090573249, 296653361322692837

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

nonn

Binomial transform of A164300. Fifth binomial transform of A164587. Inverse binomial transform of A164598.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (10,-23).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), this sequence (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

Cf. A081182, A083880, A164587.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(5+r)^n+(1-4*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

LinearRecurrence[{10,-23},{1,13},20] (* Harvey P. Dale, Oct 15 2015 *)

my(x='x+O('x^50)); Vec((1+3*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Sep 13 2017

[( (1+3*x)/(1-10*x+23*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 13.
G.f.: (1+3*x)/(1-10*x+23*x^2).
E.g.f.: ( cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x) )*exp(5*x). - G. C. Greubel, Sep 13 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = 2*A083880(n) + 8*A081182(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*4^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 17 2009

A164598 a(n) = 12a(n-1) - 34a(n-2), for n > 1, with a(0) = 1, a(1) = 14.

Original entry on oeis.org

1, 14, 134, 1132, 9028, 69848, 531224, 3999856, 29936656, 223244768, 1661090912, 12342768832, 91636134976, 679979479424, 5044125163904, 37410199666432, 277422140424448, 2057118896434688, 15253073982785024, 113094845314640896

Offset: 0

Klaus Brockhaus, Aug 17 2009

Binomial transform of A164301. Sixth binomial transform of A164587. Inverse binomial transform of A164599.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 -2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 -2)*x^2) and have a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 11 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (12,-34).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), this sequence (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

Cf. A081183, A147957, A164587.

[ n le 2 select 13*n-12 else 12*Self(n-1)-34*Self(n-2): n in [1..30] ];

m:=30; S:=series( (1+2*x)/(1-12*x+34*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 11 2021

LinearRecurrence[{12,-34}, {1,14}, 30] (* G. C. Greubel, Aug 11 2017 *)

my(x='x+O('x^30)); Vec((1+2*x)/(1-12*x+34*x^2)) \\ G. C. Greubel, Aug 11 2017

[( (1+2*x)/(1-12*x+34*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 11 2021

a(n) = ((1+4*sqrt(2))*(6+sqrt(2))^n + (1-4*sqrt(2))*(6-sqrt(2))^n)/2.
G.f.: (1+2*x)/(1-12*x+34*x^2).
E.g.f.: exp(6*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - G. C. Greubel, Aug 11 2017
From G. C. Greubel, Mar 11 2021: (Start)
a(n) = A147957(n) + 8*A081183(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*5^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

A164599 a(n) = 14a(n-1) - 47a(n-2), for n > 1, with a(0) = 1, a(1) = 15.

Original entry on oeis.org

1, 15, 163, 1577, 14417, 127719, 1110467, 9543745, 81420481, 691330719, 5851867459, 49433600633, 417032638289, 3515077706295, 29610553888547, 249339102243793, 2099051398651393, 17667781775661231, 148693529122641763

Offset: 0

Klaus Brockhaus, Aug 17 2009

nonn

Binomial transform of A164598. Seventh binomial transform of A164587. Inverse binomial transform of A081185 without initial term 0.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 -2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 -2)*x^2) and have a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 11 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (14,-47).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), this sequence (m=6), A081185 (m=7), A164600 (m=8).

Cf. A002203, A081184, A081185, A147958, A164587.

[ n le 2 select 14*n-13 else 14*Self(n-1)-47*Self(n-2): n in [1..30] ];

m:=30; S:=series( (1+x)/(1-14*x+47*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 11 2021

LinearRecurrence[{14,-47}, {1,15}, 30] (* G. C. Greubel, Aug 11 2017 *)

my(x='x+O('x^30)); Vec((1+x)/(1-14*x+47*x^2)) \\ G. C. Greubel, Aug 11 2017

def A164599_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)/(1-14*x+47*x^2) ).list()
A164599_list(30) # G. C. Greubel, Mar 11 2021

a(n) = ((1+4*sqrt(2))*(7+sqrt(2))^n + (1-4*sqrt(2))*(7-sqrt(2))^n)/2.
G.f.: (1+x)/(1-14*x+47*x^2).
E.g.f.: exp(7*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - G. C. Greubel, Aug 11 2017
From G. C. Greubel, Mar 11 2021: (Start)
a(n) = A147958(n) + 8*A081184(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*6^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

A054489 Expansion of (1+4x)/(1-6x+x^2).

Original entry on oeis.org

1, 10, 59, 344, 2005, 11686, 68111, 396980, 2313769, 13485634, 78600035, 458114576, 2670087421, 15562409950, 90704372279, 528663823724, 3081278570065, 17959007596666, 104672767009931, 610077594462920

Offset: 0

Barry E. Williams, May 04 2000

Numbers n such that 8*n^2 + 41 is a square.

(x, y) = (a(n), a(n+1)) are solutions to x^2 + y^2 - 6*x*y = 41. - John O. Oladokun, Mar 17 2021

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pp. 122-125, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A000129, A002203, A038761, A054488.

a:=[1,10];; for n in [3..30] do a[n]:=6*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 19 2020

I:=[1,10]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2020

a[0]:=1: a[1]:=10: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..19); # Zerinvary Lajos, Jul 26 2006

Table[(LucasL[2*n+1, 2] + 3*Fibonacci[2*n, 2])/2, {n,0,30}] (* G. C. Greubel, Jan 19 2020 *)
LinearRecurrence[{6,-1},{1,10},20] (* Harvey P. Dale, Jun 11 2024 *)

vector(31, n, polchebyshev(n-1,2,3) +4*polchebyshev(n-2,2,3) ) \\ G. C. Greubel, Jan 19 2020

[chebyshev_U(n,3) +4*chebyshev_U(n-1,3) for n in (0..30)] # G. C. Greubel, Jan 19 2020

a(n) = 6*a(n-1) - a(n-2), a(0)=1, a(1)=10.
a(n) = (10*((3+2*sqrt(2))^n - (3-2*sqrt(2))^n) - ((3+2*sqrt(2))^(n-1) - (3-2*sqrt(2))^(n-1)))/(4*sqrt(2)).
From G. C. Greubel, Jan 19 2020: (Start)
a(n) = ChebyshevU(n,3) + 4*ChebyshevU(n-1,3).
a(n) = (Pell(2*n+2) + 4*Pell(2*n))/2 = (Pell-Lucas(2*n+1) + 3*Pell(2*n))/2.
E.g.f.: exp(3*x)*( cosh(2*sqrt(2)*x) + 7*sinh(2*sqrt(2)*x)/(2*sqrt(2)) ). (End)

More terms from James Sellers, May 05 2000

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

Original entry on oeis.org

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)

A102413 Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 8, 16, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 48, 76, 48, 12, 1, 1, 14, 70, 154, 154, 70, 14, 1, 1, 16, 96, 272, 384, 272, 96, 16, 1, 1, 18, 126, 438, 810, 810, 438, 126, 18, 1, 1, 20, 160, 660, 1520, 2004, 1520, 660, 160, 20, 1, 1, 22, 198, 946, 2618, 4334, 4334, 2618, 946, 198, 22, 1

Offset: 0

Emeric Deutsch, Jan 07 2005

The n-sunlet graph is the corona C'(n) of the cycle graph C(n) and the complete graph K(1); in other words, C'(n) is the graph constructed from C(n) to which for each vertex v a new vertex v' and the edge vv' is added.
Row n contains n+1 terms. Row sums yield A099425. T(n,k) = T(n,n-k).
For n > 2: same recurrence as A008288 and A128966. - Reinhard Zumkeller, Apr 15 2014

			T(3,2) = 6 because in the graph with vertex set {A,B,C,a,b,c} and edge set {AB,AC,BC,Aa,Bb,Cc} we have the following six 2-matchings: {Aa,BC}, {Bb,AC}, {Cc,AB}, {Aa,Bb}, {Aa,Cc} and {Bb,Cc}.
The triangle starts:
  1;
  1, 1;
  1, 4,  1;
  1, 6,  6, 1;
  1, 8, 16, 8, 1;
From _Eric W. Weisstein_, Apr 03 2018: (Start)
Rows as polynomials:
  1
  1 +    x,
  1 +  4*x +    x^2,
  1 +  6*x +  6*x^2 +    x^3,
  1 +  8*x + 16*x^2 +  8*x^3 +    x^4,
  1 + 10*x + 30*x^2 + 30*x^3 + 10*x^4 + x^5,
  ... (End)

J. L. Gross and J. Yellen, Handbook of Graph Theory, CRC Press, Boca Raton, 2004, p. 894.
F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969, p. 167.

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Frédéric Bihan, Francisco Santos, and Pierre-Jean Spaenlehauer, A Polyhedral Method for Sparse Systems with many Positive Solutions, arXiv:1804.05683 [math.CO], 2018.
A. F. Horadam, Chebyshev and Pell connections, Fib. Quart. 43 (2) (2005) 108-121, table (6.11)
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Eric Weisstein's World of Mathematics, Sunlet Graph

Cf. A002203 or A099425 (row sums), A006318, A008288.

Cf. A241023 (central terms).

a102413 n k = a102413_tabl !! n !! k
a102413_row n = a102413_tabl !! n
a102413_tabl = [1] : [1,1] : f [2] [1,1] where
   f us vs = ws : f vs ws where
             ws = zipWith3 (((+) .) . (+))
                  ([0] ++ us ++ [0]) ([0] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Apr 15 2014

G:=(1+t*z^2)/(1-(1+t)*z-t*z^2): Gser:=simplify(series(G,z=0,38)): P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od:for n from 0 to 11 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form

CoefficientList[Table[2^-n ((1 + x - Sqrt[1 + x (6 + x)])^n + (1 + x + Sqrt[1 + x (6 + x)])^n), {n, 10}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
LinearRecurrence[{1 + x, x}, {1, 1 + x, 1 + 4 x + x^2}, 10] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
Join[{1}, CoefficientList[CoefficientList[Series[(-1 - x - 2 x z)/(-1 + z + x z + x z^2), {z, 0, 10}], z], x]] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)

G.f.: G(t,z) = (1 + t*z^2) / (1 - (1+t)*z - t*z^2).
For n > 2: T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1), 0 < k < n. - Reinhard Zumkeller, Apr 15 2014 (corrected by Andrew Woods, Dec 08 2014)
From Peter Bala, Jun 25 2015: (Start)
The n-th row polynomial R(n, t) = [z^n] F(z, t)^n, where F(z, t) = 1/2*( 1 + (1 + t)*z + sqrt(1 + 2*(1 + t)*z + (1 + 6*t + t^2)*z^2) ).
exp( Sum_{n >= 1} R(n, t)*z^n/n ) = 1 + (1 + t)*z + (1 + 3*t + t^2)*z^2 + (1 + 5*t + 5*t^2 + t^3)*z^3 + ... is the o.g.f for A008288 read as a triangular array. (End)
From Peter Bala, Aug 01 2024: (Start)
T(n, k) = A008288(n-k, k) + A008288(n-k-1, k-1) (Bihan et al., Proposition 6.6).
T(n, k) = 1 if n = 0 or k = n, else for 1 <= k <= n-1, T(n, k) = Sum_{j = 0..min(n-k, k)} (2^j)*(binomial(n-k, j)*binomial(k, j) + binomial(n-k-1, j)*binomial(k-1, j)).
Let S(x) = (1 - x - (1 - 6*x + x^2)^(1/2))/(2*x) denote the g.f. of the sequence of large Schröder numbers A006318. The signed n-th row polynomial R(n, -x) = 1/S(x)^n + (x*S(x))^n. (End)

Row 0 in polynomials and Mathematica programs added by Georg Fischer, Apr 01 2019

A026933 Self-convolution of array T given by A008288.

Original entry on oeis.org

1, 2, 11, 52, 269, 1414, 7575, 41064, 224665, 1237898, 6859555, 38187164, 213408805, 1196524814, 6727323439, 37915058384, 214140178225, 1211694546194, 6867622511675, 38981807403268, 221562006394173, 1260814207833750, 7182599953332423, 40958645048598840, 233779564099963081

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 15.

Cf. A002203, A008288, A204061, A204062.

Table[SeriesCoefficient[1/(1+x)/Sqrt[1-6*x+x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[ n_]:= Sum[ SeriesCoefficient[ SeriesCoefficient[1/(1-x-y-x*y) , {x,0,n-k}] , {y, 0, k}]^2, {k, 0, n}]; (* Michael Somos, Jun 27 2017 *)
A026933[n_]:= Sum[(Binomial[n, k]*Hypergeometric2F1[-k,k-n,-n,-1])^2, {k,0,n}];
Table[A026933[n], {n, 0, 40}] (* G. C. Greubel, May 25 2021 *)

/* Sum of squares of Delannoy numbers: */
{a(n)=sum(k=0,n,polcoeff(polcoeff(1/(1-x-y-x*y +x*O(x^n)+y*O(y^k)),n-k,x),k,y)^2)} \\ Paul D. Hanna, Jan 10 2012

/* Involving squares of companion Pell numbers: */
{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^2/2*x^k/k)+x*O(x^n)), n)}
\\ Paul D. Hanna, Jan 10 2012

my(x='x+O('x^66)); Vec( 1/(1+x)/sqrt(1-6*x+x^2) ) \\ Joerg Arndt, May 04 2013

def A026933_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*sqrt(1-6*x+x^2)) ).list()
A026933_list(40) # G. C. Greubel, May 25 2021

a(n) = Sum_{k=0..n} D(n-k,k)^2 where D(n,k) = A008288(n,k) are the Delannoy numbers. - Paul D. Hanna, Jan 10 2012
G.f.: 1/((1+x)*sqrt(1-6*x+x^2)). - Vladeta Jovovic, May 13 2003
a(n) = (-1)^n*Sum_{k=0...n} (-1)^k*A001850(k). - Benoit Cloitre, Sep 28 2005
G.f.: exp( Sum_{n>=1} A002203(n)^2/2 * x^n/n ), where A002203 are the companion Pell numbers. - Paul D. Hanna, Jan 10 2012
Self-convolution yields A204062; self-convolution of A204061. - Paul D. Hanna, Jan 10 2012
From Vaclav Kotesovec, Oct 08 2012: (Start)
Recurrence: n*a(n) = (5*n-3)*a(n-1) + (5*n-2)*a(n-2) - (n-1)*a(n-3).
a(n) ~ sqrt(24+17*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). (End)
0 = +a(n)*(+a(n+1) -8*a(n+2) -7*a(n+3) +2*a(n+4)) +a(n+1)*(-2*a(n+1) +22*a(n+2) +20*a(n+3) -7*a(n+4)) +a(n+2)*(+30*a(n+2) +22*a(n+3) -8*a(n+4)) +a(n+3)*(-2*a(n+3) +a(n+4)) for all n in Z. - Michael Somos, Jun 27 2017

More terms from Vladeta Jovovic, May 13 2003

A099425 Expansion of (1+x^2)/(1-2*x-x^2).

Original entry on oeis.org

1, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, 39202, 94642, 228486, 551614, 1331714, 3215042, 7761798, 18738638, 45239074, 109216786, 263672646, 636562078, 1536796802, 3710155682, 8957108166, 21624372014, 52205852194, 126036076402, 304278004998

Offset: 0

Paul Barry, Oct 15 2004

Binomial transform of A094024(n+1).
a(n) is the number of matchings of the corona C'(n) of the cycle graph C(n) and the complete graph K(1); in other words, C'(n) is the graph constructed from C(n) to which for each vertex v a new vertex v' and the edge vv' is added. Example: a(3)=14 because in the graph with vertex set {A,B,C,a,b,c} and edge set {AB,AC,BC,Aa,Bb,Cc} we have the following 14 matchings: the empty set, the six singletons containing one of the edges, {Aa,BC}, {Bb,AC}, {Cc,AB}, {Aa,Bb}, {Aa,Cc}, {Bb,Cc} and {Aa,Bb,Cc}. Row sums of A102413. - Emeric Deutsch, Jan 07 2005
Apart from first term, same as A002203. - Peter Shor, May 12 2005
Equals the INVERT transform of integers with repeats. Example: a(4) = 34 = (1, 1, 2, 6, 14) dot (5, 3, 3, 1, 1) = (5 + 3 + 6 + 6 + 14) = 34.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A000129, A001333, A002203, A094024, A102413.

Cf. A014176 (silver mean).

a099425 = sum . a102413_row  -- Reinhard Zumkeller, Apr 15 2014

a:= n-> (<<0|1>, <1|2>>^n. <<2, 2>>)[1, 1]-0^n:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 26 2018

CoefficientList[Series[(1+x^2)/(1-2x-x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,1},{1,2,6},40] (* Harvey P. Dale, Mar 23 2020 *)

a(n) = (1+sqrt(2))^n + (1-sqrt(2))^n - 0^n see silver mean (A014176).
a(n) = Sum_{k=0..n} A000129(n+1-k)*C(1, k/2)*(1+(-1)^k)/2.
a(n) = 2*A001333(n) - 0^n.
a(n) = round((1+sqrt(2))^n). - Bruno Berselli, Feb 04 2013
G.f.: G(0) - 1, where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 30 2013
a(n) = A000129(n+1) + A000129(n-1). - Vladimir Kruchinin, Apr 19 2024

A204274 G.f.: Sum_{n>=1} Pell(n^2)*x^(n^2).

Original entry on oeis.org

1, 0, 0, 12, 0, 0, 0, 0, 985, 0, 0, 0, 0, 0, 0, 470832, 0, 0, 0, 0, 0, 0, 0, 0, 1311738121, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21300003689580, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2015874949414289041, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Paul D. Hanna, Jan 14 2012

nonn

Compare g.f. to the Lambert series identity: Sum_{n>=1} lambda(n)*x^n/(1-x^n) = Sum_{n>=1} x^(n^2); Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).

			G.f.: A(x) = x + 12*x^4 + 985*x^9 + 470832*x^16 + 1311738121*x^25 +...
where A(x) = x/(1-2*x-x^2) + (-1)*2*x^2/(1-6*x^2+x^4) + (-1)*5*x^3/(1-14*x^3-x^6) + (+1)*12*x^4/(1-34*x^4+x^8) + (-1)*29*x^5/(1-82*x^5-x^10) + (+1)*70*x^6/(1-198*x^6+x^12) +...+ lambda(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A204327, A204060, A204270, A204271, A204272, A204273, A204275, A008836 (lambda), A002203, A000045.

pell:= gfun:-rectoproc({a(0)=0,a(1)=1,a(n)=2*a(n-1)+a(n-2)},a(n),remember):
seq(`if`(issqr(n),pell(n),0), n=1..100); # Robert Israel, Nov 24 2015

CoefficientList[Sum[Fibonacci[n^2, 2] x^n^2/x, {n, 1, 8}], x] (* Jean-François Alcover, Mar 25 2019 *)

/* Subroutines used in PARI programs below: */
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

PARI
```
{a(n)=issquare(n)*Pell(n)}
```

{lambda(n)=local(F=factor(n));(-1)^sum(i=1,matsize(F)[1],F[i,2])}
{a(n)=polcoeff(sum(m=1,n,lambda(m)*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

G.f.: Sum_{n>=1} lambda(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where lambda(n) = A008836(n), Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

cp's OEIS Frontend

A164300 a(n) = ((1+4*sqrt(2))*(4+sqrt(2))^n + (1-4*sqrt(2))*(4-sqrt(2))^n)/2.

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A164301 a(n) = ((1+4*sqrt(2))*(5+sqrt(2))^n + (1-4*sqrt(2))*(5-sqrt(2))^n)/2.

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A164598 a(n) = 12*a(n-1) - 34*a(n-2), for n > 1, with a(0) = 1, a(1) = 14.

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A164599 a(n) = 14*a(n-1) - 47*a(n-2), for n > 1, with a(0) = 1, a(1) = 15.

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A054489 Expansion of (1+4*x)/(1-6*x+x^2).

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

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A164300 a(n) = ((1+4sqrt(2))(4+sqrt(2))^n + (1-4sqrt(2))(4-sqrt(2))^n)/2.

A164301 a(n) = ((1+4sqrt(2))(5+sqrt(2))^n + (1-4sqrt(2))(5-sqrt(2))^n)/2.

A164598 a(n) = 12a(n-1) - 34a(n-2), for n > 1, with a(0) = 1, a(1) = 14.

A164599 a(n) = 14a(n-1) - 47a(n-2), for n > 1, with a(0) = 1, a(1) = 15.

A054489 Expansion of (1+4x)/(1-6x+x^2).

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