A100047 -id:A100047 - cp's OEIS frontend

A049678 a(n) = F(8*n+4)/3, where F=A000045 (the Fibonacci sequence).

1, 48, 2255, 105937, 4976784, 233802911, 10983760033, 516002918640, 24241153416047, 1138818207635569, 53500214605455696, 2513371268248782143, 118074949393087305025, 5547009250206854554032, 260591359810329076734479, 12242246901835259751966481

Offset: 0

Clark Kimberling

			a(2) = F(8 * 2 + 4) / 3 = F(20) / 3 = 6765 / 3 = 2255. - _Indranil Ghosh_, Feb 04 2017

Indranil Ghosh, Table of n, a(n) for n = 0..596
Tanya Khovanova, Recursive Sequences
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 (47,-1).

Cf. A004187, A099483, A100047.

[Fibonacci(8*n+4)/3: n in [0..30]]; // G. C. Greubel, Dec 02 2017

CoefficientList[Series[(1+x)/(1-47x+x^2),{x,0,20}],x]  (* Harvey P. Dale, Feb 18 2011 *)
Table[Fibonacci[8*n+4]/3, {n,0,30}] (* G. C. Greubel, Dec 02 2017 *)

for(n=0,30, print1(fibonacci(8*n+4)/3, ", ")) \\ G. C. Greubel, Dec 02 2017

a(n) = 47*a(n-1) - a(n-2), n>1. a(0)=1, a(1)=48.
G.f.: (1+x)/(1-47*x+x^2).
From Peter Bala, Mar 23 2015: (Start)
a(n) = A004187(2*n + 1); a(n) = A099483(4*n + 1).
a(n) = ( Fibonacci(8*n + 8 - 2*k) + Fibonacci(8*n + 2*k) )/( Fibonacci(8 - 2*k) + Fibonacci(2*k) ), for k an arbitrary integer.
a(n) = ( Fibonacci(8*n + 8 - 2*k - 1) - Fibonacci(8*n + 2*k + 1) )/( Fibonacci(8 - 2*k - 1) - Fibonacci(2*k + 1) ), for k an arbitrary integer.
The aerated sequence (b(n))n>=1 = [1, 0, 48, 0, 2255, 0, 105937, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -45, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)

Better description and more terms from Michael Somos

2 more terms from Indranil Ghosh, Feb 04 2017

A161498 Expansion of x(1-x)(1+x)/(1-13x+36x^2-13*x^3+x^4).

Original entry on oeis.org

1, 13, 132, 1261, 11809, 109824, 1018849, 9443629, 87504516, 810723277, 7510988353, 69584925696, 644660351425, 5972359368781, 55329992188548, 512595960817837, 4748863783286881, 43995092132369664, 407585519020921249

Offset: 1

R. J. Mathar, Jun 11 2009

Proposed by R. Guy in the seqfan list, Mar 29 2009.

The sequence is the case P1 = 13, P2 = 34, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 03 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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 (13,-36,13,-1)

Cf. A006238, A100047.

I:=[1,13,132,1261]; [n le 4 select I[n] else 13*Self(n-1)-36*Self(n-2)+13*Self(n-3)-Self(n-4): n in [1..20]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[(1 - x)*(1 + x)/(1 - 13*x + 36*x^2 - 13*x^3 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 19 2012 *)

a(n) = A139400(n) / ( A001906(n)*A001353(n)*A004254(n) ).
a(n) = 13*a(n-1)-36*a(n-2)+13*a(n-3)-a(n-4).
a(n) = A187732(n)-A187732(n-2). - R. J. Mathar, Mar 18 2011
From Peter Bala, Apr 03 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = 1/4*(13 + sqrt(33)), beta = 1/4*(13 - sqrt(33)) and where T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = U(n-1,1/2*(4 + sqrt(3) ))*U(n-1,1/2*(4 - sqrt(3))) for n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -17/2; 1, 13/2].
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

A192422 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

0, 1, 1, 5, 7, 20, 35, 83, 161, 355, 720, 1541, 3185, 6733, 14027, 29500, 61663, 129403, 270865, 567911, 1189440, 2492905, 5222449, 10943813, 22928815, 48044900, 100665083, 210927155, 441948689, 926020171, 1940274000, 4065458669, 8518311809

Offset: 0

Clark Kimberling, Jun 30 2011

nonn

The polynomial p(n,x) is defined by ((x+d)/2)^n + ((x-d)/2)^n, where d=sqrt(x^2+4). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

Assuming the o.g.f. given below, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. It is the case P1 = 1, P2 = -1, Q = -1 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. Cf. A100047. - Peter Bala, Aug 28 2019

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x) = 2 -> 2
  p(1,x) = x -> x
  p(2,x) = 2 + x^2 -> 3 + x
  p(3,x) = 3*x + x^3 -> 1 + 5*x
  p(4,x) = 2 + 4*x^2 + x^4 -> 8 + 7*x.
From these, read A192421 = (2, 0, 3, 1, 8, ...) and a(n) = (0, 1, 1, 5, 7, ...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1).

Cf. A000045, A100047, A192232, A192421.

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x^2)/(1-x-3*x^2+x^3+x^4) )); // G. C. Greubel, Jul 11 2023

(See A192421.)
LinearRecurrence[{1,3,-1,-1}, {0,1,1,5}, 40] (* G. C. Greubel, Jul 11 2023 *)

a(n):=n*sum((binomial(n-i-1,i))/(n-2*i)*fib(n-2*i),i,0,(n-1)/2); /* Vladimir Kruchinin, Mar 20 2016 */

@CachedFunction
def a(n): # a = A192422
    if (n<4): return (0,1,1,5)[n]
    else: return a(n-1) +3*a(n-2) -a(n-3) -a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 11 2023

From Colin Barker, May 12 2014: (Start)
a(n) = a(n-1) + 3*a(n-2) - a(n-3) - a(n-4).
G.f.: x*(1 + x^2)/(1 - x - 3*x^2 + x^3 + x^4). (End)
From Vladimir Kruchinin, Mar 20 2016: (Start)
G.f.: ((1+x^2)/(1-x^2)) * F(x/(1-x^2)), where F(x) is g.f. of Fibonacci numbers (A000045).
a(n) = n*Sum_{i=0..floor((n-1)/2)} (binomial(n-i-1,i)/(n-2*i))*Fibonacci(n-2*i). (End)
a(n) = Sum_{j=0..n} T(n, j)*Fibonacci(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n. - G. C. Greubel, Jul 11 2023

A241606 A linear divisibility sequence of the fourth order related to A003779.

Original entry on oeis.org

1, 11, 95, 781, 6336, 51205, 413351, 3335651, 26915305, 217172736, 1752296281, 14138673395, 114079985111, 920471087701, 7426955448000, 59925473898301, 483517428660911, 3901330906652795, 31478457514091281, 253988526230055936

Offset: 1

Peter Bala, Apr 26 2014

A003779, which counts spanning trees in the graph P_5 x P_n, is a linear divisibility sequence of order 16. It factors into two fourth-order linear divisibility sequences; this sequence is one of the factors, the other is A143699.

The present sequence is the case P1 = 11, P2 = 23, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy.

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 (11,-25,11,-1).

Cf. A003779, A100047, A143699.

a[n_] := ChebyshevU[n-1, 1/4*(7-Sqrt[5])]*ChebyshevU[n-1, 1/4*(7+Sqrt[5])]; Table[a[n]//Round, {n, 1, 20}] (* Jean-François Alcover, Apr 28 2014, after Peter Bala *)

O.g.f. x*(1 - x^2)/(1 - 11*x + 25*x^2 - 11*x^3 + x^4).
a(n) = A003779(n)/A143699(n).
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), n >= 1, where alpha = 1/4*(11 + sqrt(29)), beta = 1/4*(11 - sqrt(29)) and where T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n)= U(n-1,1/4*(7 - sqrt(5)))*U(n-1,1/4*(7 + sqrt(5))), n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = the bottom left entry of the 2X2 matrix T(n,M), where M is the 2 X 2 matrix [0, -23/4; 1, 11/2].
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences.
a(n) = 11*a(n-1) - 25*a(n-2) + 11*a(n-3) - a(n-4). - Vaclav Kotesovec, Apr 28 2014

A244895 Period 5: repeat [0, 1, 1, -1, -1].

Original entry on oeis.org

0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1

Offset: 0

Michael Somos, Jul 07 2014

This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = z = -1.

			G.f. = x + x^2 - x^3 - x^4 + x^6 + x^7 - x^8 - x^9 + x^11 + x^12 + ...

C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
Index entries for linear recurrences with constant coefficients, signature (-1, -1, -1, -1).

Cf. A011558, A080891, A100047.

a[ n_] := {1, 1, -1, -1, 0}[[Mod[ n, 5, 1]]]; (* Michael Somos, Jan 08 2015 *)
a[ n_] := Sign[ Mod[ n, 5, -2]]; (* Michael Somos, Jan 08 2015 *)
PadRight[{},120,{0,1,1,-1,-1}] (* Harvey P. Dale, Nov 11 2020 *)

PARI
```
{a(n) = [0, 1, 1, -1, -1][n%5 + 1]};
```
PARI
```
{a(n) = sign( centerlift( Mod(n, 5)))};
```

G.f.: x * (1 + x) * (1 - x^2) / (1 - x^5).
Euler transform of length 5 sequence [ 1, -2, 0, 0, 1].
a(n) = -a(-n) = a(n + 5) for all n in Z.
0 = (a(n) + a(n+2)) * (a(n) - a(n+1) + a(n+2)) for all n in Z.
0 = a(n)*a(n+4) - a(n+1)*a(n+3) - a(n+2)*a(n+2) for all n in Z.
0 = a(n)*a(n+5) + a(n+1)*a(n+4) - a(n+2)*a(n+3) for all n in Z.
|A011558(n)| = |A080891(n)| = |A100047(n)| = |a(n)|. - Michael Somos, May 24 2015
a(5*n) = 0, a(5*n + 1) = a(5*n + 2) = 1, a(5*n + 3) = a(5*n + 4) = -1 for all n in Z. -Michael Somos, Nov 27 2019

A100049 A Chebyshev transform of the Padovan numbers.

Original entry on oeis.org

1, 0, -1, 1, -1, -3, 3, 3, -6, 2, 10, -13, -9, 29, -9, -43, 55, 32, -126, 48, 183, -243, -121, 541, -241, -765, 1082, 450, -2326, 1171, 3179, -4803, -1617, 9993, -5601, -13168, 21250, 5552, -42849, 26489, 54351, -93763, -17765, 183347, -124086, -223422, 412698, 49827, -782881, 576541, 914279

Offset: 0

Paul Barry, Oct 31 2004

A Chebyshev transform of the Padovan numbers A000931(n+3): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).

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

Cf. A099443, A011655, A100047, A100048.

LinearRecurrence[{0, -2, 1, -2, 0, -1}, {1, 0, -1, 1, -1, -3, 3}, 50] (* G. C. Greubel, Aug 08 2017 *)

x='x+O('x^50); Vec((1-x^2)*(1+x^2)^2/(1+2*x^2-x^3+2*x^4+x^6)) \\ G. C. Greubel, Aug 08 2017

G.f.: (1-x^2)*(1+x^2)^2/(1+2*x^2-x^3+2*x^4+x^6).
a(n) = -2*a(n-2) +a(n-3) -2*a(n-4) -a(n-6).
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k,k)*A000931(n-2*k+3)/(n-k).

A140824 Expansion of (x-x^3)/(1-3x+2x^2-3*x^3+x^4).

Original entry on oeis.org

0, 1, 3, 6, 15, 41, 108, 281, 735, 1926, 5043, 13201, 34560, 90481, 236883, 620166, 1623615, 4250681, 11128428, 29134601, 76275375, 199691526, 522799203, 1368706081, 3583319040, 9381251041, 24560434083, 64300051206, 168339719535, 440719107401, 1153817602668

Offset: 0

N. J. A. Sloane, Sep 07 2009, based on email from R. K. Guy, Mar 09 2009

Case P1 = 3, P2 = 0, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Bala, Linear divisibility sequences and Chebyshev polynomials
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 (3,-2,3,-1).

Cf. A006238, A005248, A054493, A078070, A092184, A098306, A100047, A100048, A108196, A138573, A152090, A218134.

LinearRecurrence[{3, -2, 3, -1}, {0, 1, 3, 6}, 50] (* G. C. Greubel, Aug 08 2017 *)

x='x+O('x^50); concat([0], Vec((x-x^3)/(1-3*x+2*x^2-3*x^3+x^4))) \\ G. C. Greubel, Aug 08 2017

a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 6, a(n) - 3 a(n + 1) + 2 a(n + 2) - 3 a(n + 3) + a(n + 4) = 0.
From Peter Bala, Mar 25 2014: (Start)
a(n) = 2/3*( T(n,3/2) - T(n,0) ), where T(n,x) is a Chebyshev polynomial of the first kind.
a(n) = 1/3 * (A005248(n) - (i^n + (-i)^n)) = 1/3 * (Fibonacci(2*n-1) + Fibonacci(2*n+1) - (i^n + (-i)^n)).
a(n) = bottom left entry of the 2 X 2 matrix 2*T(n, 1/2*M), where M is the 2 X 2 matrix [0, 0; 1, 3].
The o.g.f. is the Hadamard product of the rational functions x/(1 - 1/sqrt(2)*(sqrt(5) + i)*x + x^2) and x/(1 - 1/sqrt(2)*(sqrt(5) - i)*x + x^2). See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
a(n) = A099483(n) - A099483(n-2). - R. J. Mathar, Feb 10 2016

A171065 G.f. -x(x-1)(1+x)/(1-x-8*x^2-x^3+x^4).

Original entry on oeis.org

0, 1, 1, 8, 17, 81, 224, 881, 2737, 9928, 32481, 113761, 380800, 1313441, 4441121, 15215688, 51677297, 176530481, 600723424, 2049428881, 6980069457, 23799693448, 81088954561, 276417142721, 941948403200, 3210574806081

Offset: 0

R. J. Mathar, at the request of R. K. Guy, Sep 03 2010

The member k=8 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).

This is the case P1 = 1, P2 = -10, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 31 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 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 (1,8,1,-1).

Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6). A100047.

I:=[0, 1, 1, 8]; [n le 4 select I[n] else Self(n-1) + 8*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[-x*(x - 1)*(1 + x)/(1 - x - 8*x^2 - x^3 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{1,8,1,-1},{0,1,1,8},30] (* Harvey P. Dale, Dec 27 2017 *)

a(n)= +a(n-1) +8*a(n-2) +a(n-3) -a(n-4).
From Peter Bala, Mar 31 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(41))/4 and beta = (1 - sqrt(41))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 5/2; 1, 1/2].
a(n) = U(n-1,i*(1 + sqrt(2))/2)*U(n-1,i*(1 + sqrt(2))/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)

A248848 Norm of coefficients in the expansion of 1/(1 - 3x - Ix^2), where I^2=-1.

Original entry on oeis.org

1, 9, 82, 765, 7129, 66420, 618841, 5765805, 53720578, 500519961, 4663394209, 43449307200, 404821512289, 3771762252921, 35141883671458, 327420421852365, 3050608602778201, 28422823459498740, 264818270254044889, 2467338136208552925, 22988434568917776562, 214185529001504000169

Offset: 0

Paul D. Hanna, Nov 02 2014

Working with an offset of 1, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. It is the case P1 = 9, P2 = -4, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Dec 02 2014

			G.f.: A(x) = 1 + 9*x + 82*x^2 + 765*x^3 + 7129*x^4 + 66420*x^5 +...
If we expand the complex series:
1/(1 - 3*x + I*x^2) = 1 + 3*x + (9 - I)*x^2 + (27 - 6*I)*x^3 + (80 - 27*I)*x^4 + (234 - 108*I)*x^5 + (675 - 404*I)*x^6 + (1917 - 1446*I)*x^7 + (5347 - 5013*I)*x^8 + (14595 - 16956*I)*x^9 +...
then the terms of this sequence equals the norm of the above coefficients:
a(0) = 1^2 = 1;
a(1) = 3^2 = 9;
a(2) = 9^2 + (-1)^2 = 82;
a(3) = 27^2 + (-6)^2 = 765;
a(4) = 80^2 + (-27)^2 = 7129;
a(5) = 234^2 + (-108)^2 = 66420; ...

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 (9,2,9,-1).

Cf. A100047.

I:=[1,9,82,765]; [n le 4 select I[n] else 9*Self(n-1)+2*Self(n-2)+9*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Mar 22 2015

Abs[CoefficientList[Series[1/(1 - 3*x - I*x^2), {x, 0, 20}], x]]^2 (* Vaclav Kotesovec, Nov 09 2014 *)

{a(n)=(polcoeff(1/(1-3*x+I*x^2 +x*O(x^n)), n))}
for(n=0, 31, print1(norm(a(n)), ", "))

G.f.: (1-x^2)/(1 - 9*x - 2*x^2 - 9*x^3 + x^4).
a(n) = 9*a(n-1) + 2*a(n-2) + 9*a(n-3) - a(n-4). - Vaclav Kotesovec, Nov 09 2014
a(n) ~ (1 + 9/sqrt(97) + 3*sqrt((18+2*sqrt(97))/97)) * (9 + sqrt(97) + 3*sqrt(18+2*sqrt(97)))^n / 4^(n+1). - Vaclav Kotesovec, Nov 09 2014
From Peter Bala, Dec 02 2014: (Start)
The following remarks assume an offset of 1:
a(n) = ( T(n,a) - T(n,b) )/(a - b), where T(n,x) denotes the Chebyshev polynomial of the first kind and where a = ( 9 + sqrt(97) )/4 and b = ( 9 - sqrt(97) )/4 denote the roots of the quadratic equation x^2 - 9/2*x - 1 = 0.
a(n) = the bottom left entry of the 2 X 2 matrix 2*T(n,1/2*M), where M is the 2 X 2 matrix [0, 4; 1, 9]. See A100047. (End)

A351683 Squares that are also 4-dimensional pyramidal numbers.

Original entry on oeis.org

0, 1, 196, 38025, 7376656, 1431033241, 277613072100, 53855504954161, 10447690348035136, 2026798072013862225, 393188378280341236516, 76276518588314186021881, 14797251417754671747008400, 2870590498525818004733607721, 556879759462590938246572889476

Offset: 1

Kelvin Voskuijl, May 05 2022

nonn

This sequence is a quartic divisibility sequence. a(n+1) divides a(m+1) whenever n divides m. This is because this sequence is based on solutions to a special case of the general Jacobi quartic form y^2 = b*x^4 - 2*c*x^2 + 1. - Thomas Scheuerle, May 06 2022

			196 is a term because 196 = 14^2 is a perfect square and 196 = (2*6 + 5*6^2 + 4*6^3 + 6^4)/12 is the 6th four-dimensional pyramidal number.

Dustin Moody, Division Polynomials for Jacobi Quartic Curves, NIST publication.

Intersection of A000290 and A002415.

Cf. A007655, A100047.

Select[Table[1/12 (2 n + 5 n^2 + 4 n^3 + n^4), {n, 0, 75000}], IntegerQ[Sqrt[#]]&]

a(n) = A007655(n)^2.
a(2*n - 2) = (a(n) - a(n-1))^2, for n > 1. - Thomas Scheuerle, May 06 2022
O.g.f.: A(x) = x^2*(1 - x^2)/(1 - 196*x + 390*x^2 - 196*x^3 + x^4). With offset 0, this is the case P1 = 196, P2 = 194, Q = 1 of a 3-parameter family of fourth-order linear divisibility sequences. See A100047 for further details. - Peter Bala, Nov 28 2022

a(11)-a(14) from Amiram Eldar, May 05 2022

a(15) from Kelvin Voskuijl, Jun 05 2022

cp's OEIS Frontend

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A161498 Expansion of x*(1-x)*(1+x)/(1-13*x+36*x^2-13*x^3+x^4).

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A192422 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

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A244895 Period 5: repeat [0, 1, 1, -1, -1].

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A161498 Expansion of x(1-x)(1+x)/(1-13x+36x^2-13*x^3+x^4).

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