A002605 -id:A002605 - cp's OEIS frontend

A073397 Eighth convolution of A002605(n) (generalized (2,2)-Fibonacci), n>=0, with itself.

1, 18, 198, 1680, 12060, 76824, 446952, 2420352, 12363120, 60151520, 280833696, 1265442048, 5528697408, 23507763840, 97575960960, 396398370816, 1579498956288, 6184543546368, 23833455191040, 90522348871680, 339263015528448, 1255995653197824, 4597442198728704

Offset: 0

Wolfdieter Lang, Aug 02 2002

For a(n) in terms of U(n+1) and U(n), with U(n) = A002605(n), see A073387 and the row polynomials of triangles A073405 and A073406.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-126,384,-144,-2016,3360,4608,-12384,-8512, 24768,18432,-26880,-32256,4608,24576,16128,4608,512).

Ninth (m=8) column of triangle A073387.

Cf. A002605, A073387, A073394, A073405, A073406.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^9 )); // G. C. Greubel, Oct 06 2022

CoefficientList[Series[1/(1-2*x-2*x^2)^9, {x,0,30}], x] (* G. C. Greubel, Oct 06 2022 *)

def A073397_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2)^9 ).list()
A073397_list(30) # G. C. Greubel, Oct 06 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A002605(k) and c(k) = A073394(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+8, 8)*binomial(n-k, k)*2^(n-k).
G.f.: 1/(1-2*x*(1+x))^9.

A073406 Coefficient triangle of polynomials (rising powers) related to convolutions of A002605(n), n>=0, (generalized (2,2)-Fibonacci). Companion triangle is A073405.

Original entry on oeis.org

2, 36, 12, 1056, 672, 96, 43968, 40416, 10752, 864, 2396160, 2815488, 1051776, 156672, 8064, 161879040, 226492416, 105981696, 22125312, 2121984, 76032, 13044326400, 20766633984, 11446769664, 2995605504

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are q(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...

The k-th convolution of U0(n) := A002605(n), n>= 0, ((2,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073387(n+k,k) = 2*(p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*U0(n))/(k!*12^k)), k=1,2,..., where the companion polynomials p(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073405(k,m).

			k=2: U2(n)=2*((36+12*n)*(n+1)*U0(n+1)+(36+12*n)*(n+2)*U0(n))/(2!*12^2), cf. A073389.
2; 36,12; 1056,672,96; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

W. Lang, First 7 rows.

Cf. A002605, A073387, A073403-A073405.

Recursion for row polynomials defined in the comments: p(k, n)= 2*((n+2)*p(k-1, n+1)+2*(n+2*(k+1))*p(k-1, n)+(n+3)*q(k-1, n+1)); q(k, n)= 4*((n+1)*p(k-1, n+1)+(n+2*(k+1))*q(k-1, n)), k >= 1. [Corrected by Sean A. Irvine, Nov 25 2024]

A073398 Ninth convolution of A002605(n) (generalized (2,2)-Fibonacci), n>=0, with itself.

Original entry on oeis.org

1, 20, 240, 2200, 16940, 115104, 711040, 4072640, 21930480, 112157760, 549010176, 2587777920, 11802273600, 52287866880, 225756241920, 952486588416, 3935984616960, 15961485957120, 63628396339200, 249702113464320, 965924035135488, 3687247950397440

Offset: 0

Wolfdieter Lang, Aug 02 2002

For a(n) in terms of U(n+1) and U(n), with U(n) = A002605(n), see A073387 and the row polynomials of triangles A073405 and A073406.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (20,-160,600,-660,-2496,7680,1920,-28320,7040, 66560,-14080,-113280,-15360,122880,79872,-42240,-76800,-40960,-10240,-1024).

Tenth (m=9) column of triangle A073387.

Cf. A002605, A073397, A073405, A073406.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^10 )); // G. C. Greubel, Oct 06 2022

CoefficientList[Series[1/(1-2*x-2*x^2)^10, {x,0,30}], x] (* G. C. Greubel, Oct 06 2022 *)

def A073398_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2)^10 ).list()
A073398_list(30) # G. C. Greubel, Oct 06 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A002605(k) and c(k) = A073397(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+9, 9)*binomial(n-k, k)*2^(n-k).
G.f.: 1/(1-2*x*(1+x))^10.

A073404 Coefficient triangle of polynomials (falling powers) related to convolutions of A002605(n), n>=0, (generalized (2,2)-Fibonacci). Companion triangle is A073403.

Original entry on oeis.org

2, 12, 36, 96, 672, 1056, 864, 10752, 40416, 43968, 8064, 156672, 1051776, 2815488, 2396160, 76032, 2121984, 22125312, 105981696, 226492416, 161879040, 718848, 27205632, 404656128, 2995605504

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are q(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,..

The k-th convolution of U0(n) := A002605(n), n>= 0, ((2,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073387(n+k,k) = 2*(p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*U0(n))/(k!*(2^2+4*2)^k), k=1,2,..., where the companion polynomials p(k,n) := sum(b(k,m)*n^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A073403(k,m).

			k=2: U2(n)=(2*(36+12*n)*(n+1)*U0(n+1)+2*(36+12*n)*(n+2)*U0(n))/(2!*12^2), cf. A073389.
1; 12,36; 96,672,1056; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

W. Lang, First 7 rows.

Cf. A002605, A073387, A073403, A073405.

Recursion for row polynomials defined in the comments: see A073405.

A105607 Sylvester cyclotomic numbers for A002605.

Original entry on oeis.org

1, 2, 6, 8, 44, 10, 328, 56, 408, 76, 18272, 52, 136384, 568, 3856, 3104, 7598336, 424, 56714752, 2896, 215104, 31648, 3159738368, 3088, 536013824, 236224, 71910912, 161344, 1313964867584, 2320, 9807567290368, 9634304, 667730944, 13160704, 37860806656, 172864, 4078438577864704, 98232832, 37201186816, 9584896

Offset: 1

Paul Barry, Apr 15 2005

nonn

Primitive parts of A002605.

Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number.

Cf. A002605, A105608, A105609.

f[n_] := FullSimplify[ Expand[ Times @@ ((1+Sqrt[3])-(1-Sqrt[3])*Exp[2Pi*I*Select[Range[n-1], GCD[ #, n] == 1 &]/n])]]; Table[ f[n], {n, 1, 32}] (* Robert G. Wilson v, Aug 02 2005 *)

a(n) = A002605(n)/A105606(n); a(n) = Product_{k=1..n-1, gcd(n, k)=1} (1+sqrt(3))-(1-sqrt(3))*exp(2*Pi*i*k/n), i = sqrt(-1). - Robert G. Wilson v, Aug 02 2005

More terms from David Wasserman, May 06 2008

A105608 Sylvester dividends for A002605.

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 16, 6, 88, 1, 960, 1, 656, 264, 896, 1, 48960, 1, 53504, 1968, 36544, 1, 2795520, 44, 272768, 2448, 2980864, 1, 1547335680, 1, 2781184, 109632, 15196672, 14432, 8635760640, 1, 113429504, 818304, 8677064704, 1, 4808968273920, 1, 9252356096, 415337472, 6319476736, 1, 26795484119040, 328, 3584860454912

Offset: 1

Paul Barry, Apr 15 2005

Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number.

Cf. A105607, A105609.

f[n_] := FullSimplify[ Expand[ Times @@ ((1+Sqrt[3])-(1-Sqrt[3])*Exp[2Pi*I*Select[Range[n-1], GCD[ #, n] > 1 &]/n])]]; Table[ f[n], {n, 1, 32}] (* Robert G. Wilson v, Aug 02 2005 *)

a(n) = A002605(n)/A105607(n); a(n) = Product_{k=1..n-1, gcd(n, k)>1} (1+sqrt(3))-(1-sqrt(3))*exp(2*Pi*i*k/n), i = sqrt(-1).

More terms from David Wasserman, May 06 2008

A175289 Pisano period of A002605 modulo n.

Original entry on oeis.org

1, 1, 3, 1, 24, 3, 48, 1, 9, 24, 10, 3, 12, 48, 24, 1, 144, 9, 180, 24, 48, 10, 22, 3, 120, 12, 27, 48, 840, 24, 320, 1, 30, 144, 48, 9, 36, 180, 12, 24, 280, 48, 308, 10, 72, 22, 46, 3, 336, 120, 144, 12, 936, 27, 120, 48, 180, 840, 29, 24, 60, 320, 144, 1, 24, 30, 1122, 144

Offset: 1

R. J. Mathar, Mar 24 2010

nonn

a(79)=6240. [John W. Layman, Aug 10 2010]

			Reading 0, 1, 2, 6, 16, 44, 120, 328, 896, 2448,.. modulo 12 gives 0, 1, 2, 6, 4, 8, 0, 4, 8, 0, 4, 8 ,.. with period length a(n=12)= 3.

Michael De Vlieger, Table of n, a(n) for n = 1..200

Cf. A001175, A046738, A175181, A175182, A175183, A175184, A175185, A175286.

a={1};For[n=2,n<=80,n++,{x={{0,1}}; t={1,1}; While[ !MemberQ[x,t], {xl = x[[ -1]]; AppendTo[x,t]; t={Mod[2*(t[[1]]+xl[[1]]),n], Mod[2*(t[[2]] + xl[[2]]),n]};}]; p = Flatten[Position[x,t]][[1]]; AppendTo[a, Length[x] - p+1];}]; Print[a]; (* John W. Layman, Aug 10 2010 *)

Terms beyond a(28)=48 from John W. Layman, Aug 10 2010

A127259 Sequence arising from the factorization of F(n)=A002605 and L(n)=A080040 F(0)=0, F(1)=1, F(n)=2F(n-1)+2F(n-2), L(0)=2, L(1)=2, L(n)=2L(n-1)+2L(n-2).

Original entry on oeis.org

2, 1, 10, 8, 76, 6, 568, 56, 424, 44, 31648, 52, 236224, 328, 2320, 3104, 13160704, 408, 98232832, 2896, 129088, 18272, 5472827392, 3088, 537496576, 136384, 71911936, 161344, 2275853910016, 3856, 16987204845568

Offset: 1

Miklos Kristof, Mar 26 2007

nonn

			F(12)=a(1)*a(2)*a(3)*a(4)*a(6)*a(12)=2*1*10*8*6*52=49920
F(9)=a(2)*a(6)*a(18)= 1*6*408=2448
L(12)=a(8)*a(24)=56*3088=172928
L(21)=a(1)*a(3)*a(7)*a(21)=2*10*568*129088=1466439680

Cf. A002605, A080040.

with(numtheory): a[1]:=2:a[2]:=1:for n from 3 to 60 do a[n]:=round(evalf((sqrt(3)-1)^degree(cyclotomic(n,x),x)*cyclotomic(n,2+sqrt(3)),30)) od: seq(a[n],n=1..60);

(sqrt(3)-1)^degree(cyclotomic(n,x),x)*cyclotomic(n,2+sqrt(3)) L(n)=2*F(n-1)+F(n+1) F(2n)=Product(d|2n) a(d), F(2n+1)=Product(d|2n+1) a(2d). L(2n+1)=Product(d|2n+1, a(d)), for k>0: L(2^k*(2n+1))=Product(d|2n+1, a(2^(k+1)*d)). for odd prime p, a(p)=L(p)/2, a(2p)=f(p) a(1)=2, a(2)=1; a(2^(k+1))=L(2^k);

A155084 A Catalan transform of [x^n](1/(1-2x-2x^2)) (A002605).

Original entry on oeis.org

1, 2, 8, 32, 132, 552, 2328, 9872, 42020, 179336, 766888, 3284272, 14081224, 60426576, 259490736, 1114965792, 4792924356, 20611174920, 88662405768, 381494338032, 1641837542232, 7067257125744, 30425523536592

Offset: 0

Paul Barry, Jan 19 2009

Hankel transform is 4^n.

Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. [_N. J. A. Sloane_, Oct 08 2012]

Cf. A000108, A002605, A101850, A039599, A108411.

G.f.: 1/(1-2x*c(x)-2(x*c(x))^2), where c(x) is the g.f. of A000108.
G.f.: 1/(1-2x-4x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-..... (continued fraction).
a(n) = Sum_{k=0..n} (k/(2n-k))*binomial(2n-k, n-k)*A002605(k), a(0) = 1.
a(n) = Sum_{0<=k<=n} A039599(n,k)*A108411(k). [Philippe Deléham, Nov 15 2009]
Apparently 3*n*a(n) +6*(3-4*n)*a(n-1) +4*(11*n-18)*a(n-2) +8*(2*n-3)*a(n-3)=0. - R. J. Mathar, Oct 25 2012

A156710 Triangle read by rows, A123191 * (A002605 * (A002605 * 0^(n-k))).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 6, 6, 1, 3, 6, 18, 16, 1, 3, 6, 18, 48, 44, 1, 3, 6, 18, 48, 132, 120, 136, 18, 48, 132, 360, 328, 1, 3, 6, 18, 48, 132, 360, 984, 896, 1, 3, 6, 18, 48, 132, 360, 984, 2688, 2448, 1, 3, 6, 18, 48, 132, 360, 984, 2688, 7344, 6688

Offset: 0

Gary W. Adamson, Feb 14 2009

Row sums = A002605: (1, 2, 6, 16, 44, 120,...)

As a property of eigentriangles, sum of row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
1, 3, 2;
1, 3, 6, 6;
1, 3, 6, 18, 16;
1, 3, 6, 18, 48, 44;
1, 3, 6, 18, 48, 132, 120;
1, 3, 6, 18, 48, 132, 360, 328;
1, 3, 6, 18, 48, 132, 360, 984, 896;
1, 3, 6, 18, 48, 132, 360, 984, 2688, 2448;
1, 3, 6, 18, 48, 132, 360, 984, 2688, 7344, 6688;
...

Cf. A002605, A123191.

Triangle read by rows, A123191 * (A002605 * (A002605 * 0^(n-k))). A123191 is
unsigned, (A002605 * 0^(n-k))= an infinite lower triangular matrix with
A002605 as the main diagonal prefaced with a 1: (1, 1, 2, 6, 16, 44,...)
and the rest zeros.

cp's OEIS Frontend

A073397 Eighth convolution of A002605(n) (generalized (2,2)-Fibonacci), n>=0, with itself.

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A073406 Coefficient triangle of polynomials (rising powers) related to convolutions of A002605(n), n>=0, (generalized (2,2)-Fibonacci). Companion triangle is A073405.

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A073398 Ninth convolution of A002605(n) (generalized (2,2)-Fibonacci), n>=0, with itself.

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Mathematica

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A073404 Coefficient triangle of polynomials (falling powers) related to convolutions of A002605(n), n>=0, (generalized (2,2)-Fibonacci). Companion triangle is A073403.

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A105607 Sylvester cyclotomic numbers for A002605.

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A105608 Sylvester dividends for A002605.

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A175289 Pisano period of A002605 modulo n.

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A127259 Sequence arising from the factorization of F(n)=A002605 and L(n)=A080040 F(0)=0, F(1)=1, F(n)=2*F(n-1)+2*F(n-2), L(0)=2, L(1)=2, L(n)=2*L(n-1)+2*L(n-2).

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A127259 Sequence arising from the factorization of F(n)=A002605 and L(n)=A080040 F(0)=0, F(1)=1, F(n)=2F(n-1)+2F(n-2), L(0)=2, L(1)=2, L(n)=2L(n-1)+2L(n-2).