A001333 -id:A001333 - cp's OEIS frontend

A085292 Product of Lucas (A000204) and a Pell companion series (A001333).

1, 9, 28, 119, 451, 1782, 6931, 27119, 105868, 413649, 1615681, 6311522, 24654241, 96306849, 376200748, 1469546399, 5740457491, 22423834422, 87593763331, 342165736199, 1336595027068, 5221113899769, 20395130698081, 79669083012482

Offset: 1

Gary W. Adamson, Jun 24 2003

Index entries for linear recurrences with constant coefficients, signature (2, 7, 2, -1).

Cf. A085293, A000204, A001333, A002203.

L[0] = 2; L[1] = 1; L[n] = L[n - 1] + L[n - 2]; P[0] = P[1] = 1; P[n_] := P[n] = 2P[n - 1] + P[n - 2]; Table[ L[n]P[n], {n, 1, 24}]
With[{nn=30},Rest[LinearRecurrence[{2,1},{1,1},nn]LucasL[Range[0,nn-1]]]] (* Harvey P. Dale, Apr 20 2012 *)
LinearRecurrence[{2, 7, 2, -1},{1, 9, 28, 119},24] (* Ray Chandler, Aug 03 2015 *)

a(n) = A000204(n) * A001333(n).
a(n) = 2*a(n-1)+7*a(n-2)+2*a(n-3)-a(n-4). G.f.: -x*(2*x^3-3*x^2-7*x-1) / (x^4-2*x^3-7*x^2-2*x+1). - Colin Barker, Oct 15 2013
2* A085292(n) = A085293(n).

Edited and extended by Robert G. Wilson v, Jun 24 2003

A111108 a(n) = A001333(n) - (-2)^(n-1), n > 0.

Original entry on oeis.org

0, 5, 3, 25, 25, 131, 175, 705, 1137, 3875, 7095, 21649, 43225, 122435, 259423, 698625, 1541985, 4011971, 9107175, 23143825, 53559817, 133933475, 314086735, 776787009, 1838300625, 4512108515, 10745077143, 26237143825, 62749602745

Offset: 1

Creighton Dement, Oct 14 2005

Conjecture: for odd primes p, p divides a(p). Note that (a(n)) and A001333 have different offsets.

The conjecture follows from the formula A001333(n) = ((1-sqrt(2))^n + (1+sqrt(2))^n)/2. - Max Alekseyev, Oct 16 2005

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,2).

Cf. A001333, A000204.

LinearRecurrence[{0,5,2},{0,5,3},30] (* Harvey P. Dale, May 03 2022 *)

concat(0, Vec(x^2*(5 + 3*x) / ((1 + 2*x)*(1 - 2*x - x^2)) + O(x^35))) \\ Colin Barker, May 01 2019

From Colin Barker, Apr 30 2019: (Start)
G.f.: x^2*(5 + 3*x) / ((1 + 2*x)*(1 - 2*x - x^2)).
a(n) = 5*a(n-2) + 2*a(n-3) for n>3.
(End)

A131607 Pell companion numbers A001333 without last digit.

Original entry on oeis.org

1, 4, 9, 23, 57, 139, 336, 811, 1960, 4732, 11424, 27580, 66585, 160752, 388089, 936931, 2261953, 5460839, 13183632, 31828103, 76839840, 185507784, 447855408, 1081218600, 2610292609, 6301803820, 15213900249, 36729604319, 88673108889, 214075822099, 516824753088

Offset: 4

Paul Curtz, Oct 02 2007

Andrew Howroyd, Table of n, a(n) for n = 4..1000

Cf. A001333, A131727.

Table[Floor[(((1 - Sqrt[2])^n + (1 + Sqrt[2])^n)/2)/10], {n, 4, 29}] (* Metin Sariyar, Jan 03 2020 *)

a(n)={polcoef((1 - x) / (1 - 2*x - x^2) + O(x*x^n), n)\10} \\ Andrew Howroyd, Jan 02 2020

a(n) = floor(A001333(n) / 10). - Andrew Howroyd, Jan 02 2020
Conjectures from Colin Barker, Jan 03 2020: (Start)
G.f.: x^4*(1 + x - 2*x^2 + x^3 + x^4) / ((1 - x)*(1 + x^2)*(1 - 2*x - x^2)*(1 - x^2 + x^4)).
a(n) = 3*a(n-1) - a(n-2) - a(n-3) - a(n-6) + 3*a(n-7) - a(n-8) - a(n-9) for n>12.
(End)

Offset changed and terms a(24) and beyond from Andrew Howroyd, Jan 02 2020

A062133 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A001333(n+1), n >= 0 (associated Pell numbers).

Original entry on oeis.org

0, 1, 2, 20, 36, 16, 456, 944, 672, 160, 14304, 33760, 28800, 10880, 1536, 575040, 1466752, 1413120, 666880, 157440, 14848, 27659520, 74774784, 79278080, 43330560, 13153280, 2128896, 143360, 1548126720

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials pPL1(n,x) := Sum_{m=0..n} a(n,m)*x^m, and pPL2(n,x) := Sum_{m=0..n} A062134(n,m)*x^m appear in the k-fold convolution of the associated Pell numbers PL(n) := A001333(n+1), n >= 0, as follows: PL(k; n) := A054458(n+k,k) = (2*pPL1(k,n)*PL(n+1)+pPL2(k,n)*PL(n))/(k!*8^k), k >= 0.

			Triangle begins:
  {0};
  {1,2};
  {20,36,16};
  {456,944,672,160};
  ...
pPL1(2,n) = 4*(5+9*n+4*n^2) = 4*(1+n)*(5+4*n).
pPL2(2,n) = 8*(1+3*n+2*n^2) = 8*(1+n)*(1+2*n).
PL(2; n) = A054460(n) = (1+n)*((5+4*n)*PL(n+1)+(1+2*n)*PL(n))/16.

Cf. A062134(n, m) (companion triangle), A054458(n, m) (convolution triangle).

A131721 Overlay of Pell companion numbers: a(n) = A001333(n) + A001333(n-6).

Original entry on oeis.org

1, 1, 3, 7, 17, 41, 100, 240, 580, 1400, 3380, 8160, 19700, 47560, 114820, 277200, 669220, 1615640, 3900500, 9416640, 22733780, 54884200, 132502180, 319888560, 772279300, 1864447160, 4501173620, 10866794400, 26234762420, 63336319240

Offset: 0

Paul Curtz, Sep 15 2007

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

Cf. A131710.

LinearRecurrence[{2, 1}, {1, 1, 3, 7, 17, 41, 100, 240}, 50] (* Paolo Xausa, Mar 01 2024 *)

a(n) = A001333(n), n<6. a(n) = A001333(n)+A001333(n-6) = 20*A000129(n-3), n>=6.
O.g.f.: (x-1)(x^2+1)(x^4-x^2+1)/(-1+2x+x^2). - R. J. Mathar, Jul 16 2008
For n>5, a(n) = 100*A000129(n+1) -240*A000129(n). - R. J. Mathar, Feb 05 2024

Edited and extended by R. J. Mathar, Jul 16 2008

A138683 a(1)=1 and, for n>1, a(n) is the smallest integer greater than a(n-1) such that a(n) and a(k) do not sum to a term of A001333 (Numerators of continued fraction convergents to sqrt(2)).

Original entry on oeis.org

2, 3, 6, 7, 8, 12, 13, 16, 17, 18, 19, 20, 26, 27, 30, 31, 32, 36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 60, 61, 64, 65, 66, 70, 71, 74, 75, 76, 77, 78, 84, 85, 88, 89, 90, 94, 95, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115

Offset: 1

John W. Layman, Mar 26 2008

nonn

The graph of the first differences (A138684) of this sequence is fractal-like.

Cf. A001333, A005652, A114889, A138684.

A143966 Eigentriangle with row sums = A001333 starting (1, 3, 7, 17, 41, 99, ...).

Original entry on oeis.org

1, 2, 1, 2, 2, 3, 2, 2, 6, 7, 2, 2, 6, 14, 17, 2, 2, 6, 14, 34, 41, 2, 2, 6, 14, 34, 82, 99, 2, 2, 6, 14, 34, 82, 198, 239, 2, 2, 6, 14, 34, 82, 198, 478, 577

Offset: 0

Gary W. Adamson, Sep 06 2008

Right border = A001333.

Row sums = shifted A001333: (1, 3, 7, 17, 41, 99, ...) = INVERT transform of (1, 2, 2, 2, ...). Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle:
  1;
  2, 1;
  2, 2, 3;
  2, 2, 6,  7;
  2, 2, 6, 14, 17;
  2, 2, 6, 14, 34, 41;
  2, 2, 6, 14, 34, 82, 99;
  2, 2, 6, 14, 34, 82, 198, 239;
  ...

Cf. A001333.

n-th row of the triangle = n terms of twice A001333: (1, 1, 3, 7, 17, 41, 99, ...) followed by A001333(n).

A158843 G.f.: A(x) = exp( Sum_{n>=1} A001333(n)^n * 2^n*x^n/n ).

Original entry on oeis.org

1, 2, 20, 952, 336112, 742166496, 10043945021760, 814531629739559808, 393150002983518264270592, 1123538097532735360702239462912, 18948231465474675384343860006353603584, 1881331085022567366434813565917484763975526400

Offset: 0

Paul D. Hanna, Apr 09 2009

nonn

Compare to g.f.: exp( Sum_{n>=1} 2*A001333(n)*x^n/n ) = 1/(1-2*x-x^2), which is the g.f. of the Pell numbers A000129 (with offset), where A001333(n) = A000129(n+1) - A000129(n).

			G.f.: A(x) = 1 + 2*x + 20*x^2 + 952*x^3 + 336112*x^4 + 742166496*x^5 +...
log(A(x)) = 2*x + 6^2*x^2/2 + 14^3*x^3/3 + 34^4*x^4/4 + 82^5*x^5/5 +...
log(G(x)) = 2*x + 6*x^2/2 + 14*x^3/3 + 34*x^4/4 + 82*x^5/5 +...
G(x) = 1 + 2*x + 5*x^2 + 12*x^3 + 29*x^4 + 70*x^5 + 169*x^6 +... (A000129).

Cf. A165937, A000129, A001333.

{a(n)=local(LD=Vec(2*(1+x)/(1-2*x-x^2 +x*O(x^n)))); polcoeff(exp(sum(m=1,n,LD[m]^m*x^m/m)+x*O(x^n)),n)}

A160756 Triangle read by rows, infinite lower triangular Toeplitz matrix with A078008 in every column convolved with A001333.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 2, 2, 0, 3, 6, 2, 2, 0, 7, 10, 6, 2, 6, 0, 17, 22, 10, 6, 6, 14, 0, 41, 42, 22, 10, 18, 14, 34, 0, 99, 86, 42, 22, 30, 42, 34, 82, 0, 239, 170, 86, 42, 66, 70, 102, 82, 198, 0, 577

Offset: 0

Gary W. Adamson, May 25 2009

Row sums = A001333: (1, 1, 3, 7, 17, 41,...). Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
0, 1;
2, 0, 1;
2, 2, 0, 3;
6, 2, 2, 0, 7;
10, 6, 2, 6, 0, 17;
22, 10, 6, 6, 14, 0, 41;
42, 22, 10, 18, 14, 34, 0, 99;
86, 42, 22, 30, 42, 34, 82, 0, 239;
170, 86, 42, 66, 70, 102, 82, 198, 0, 577;
...
Example: row 4 = (6, 2, 2, 0, 7) = (6, 2, 2, 0, 1) * (1, 1, 1, 3, 7).

Cf. A078008, A001333

Let M = an infinite lower triangular Toeplitz matrix with A078008 (1, 0, 2, 2, 6, 10, 22, 42, 86, 170,...). Let Q = the eigensequence of that triangle prefaced with a 1: (1, 1, 1, 3, 7, 17,...) where A001333 = (1, 1, 3, 7, 17,...). The triangle = M * Q.

A176481 Triangle, read by rows, defined by T(n, k) = b(n) - b(k) - b(n-k) + 2, where b(n) = A001333(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 11, 13, 11, 1, 1, 25, 33, 33, 25, 1, 1, 59, 81, 87, 81, 59, 1, 1, 141, 197, 217, 217, 197, 141, 1, 1, 339, 477, 531, 545, 531, 477, 339, 1, 1, 817, 1153, 1289, 1337, 1337, 1289, 1153, 817, 1, 1, 1971, 2785, 3119, 3249, 3283, 3249, 3119, 2785, 1971, 1

Offset: 0

Roger L. Bagula, Apr 18 2010

Row sums are: {1, 2, 5, 12, 37, 118, 369, 1112, 3241, 9194, 25533, ...}.

			Triangle begins as:
  1;
  1,    1;
  1,    3,    1;
  1,    5,    5,    1;
  1,   11,   13,   11,    1;
  1,   25,   33,   33,   25,    1;
  1,   59,   81,   87,   81,   59,    1;
  1,  141,  197,  217,  217,  197,  141,    1;
  1,  339,  477,  531,  545,  531,  477,  339,    1;
  1,  817, 1153, 1289, 1337, 1337, 1289, 1153,  817,    1;
  1, 1971, 2785, 3119, 3249, 3283, 3249, 3119, 2785, 1971, 1;

G. C. Greubel, Rows n=0..100 of triangle, flattened
B. Adamczewski, Ch. Frougny, A. Siegel and W. Steiner, Rational numbers with purely periodic beta-expansion, arXiv:0907.0206 [math.NT], 2009-2010; Bull. Lond. Math. Soc. 42:3 (2010), pp. 538-552.

Cf. A001333.

b:= func< n| Round(((1+Sqrt(2))^n + (1-Sqrt(2))^n)/2) >; [[b(n)-b(k)-b(n-k)+2: k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 06 2019

b[n_]:= LucasL[n, 2]/2; T[n_, k_]:= b[n] -b[k] -b[n-k] +2;
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 06 2019 *)

{b(n) = round(((1+sqrt(2))^n + (1-sqrt(2))^n)/2)};
{T(n,k) = b(n) -b(k) -b(n-k) +2};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, May 06 2019

def b(m): return lucas_number2(m,2,-1)/2
def T(n, k): return b(n) - b(k) - b(n-k) + 2
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 06 2019

Let b(n) = ((1+sqrt(2))^n + (1-sqrt(2))^n)/2 = A001333(n), then T(n, k) = b(n) - b(k) - b(n-k) + 2.

Edited by G. C. Greubel, May 06 2019

cp's OEIS Frontend

A085292 Product of Lucas (A000204) and a Pell companion series (A001333).

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A111108 a(n) = A001333(n) - (-2)^(n-1), n > 0.

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A131607 Pell companion numbers A001333 without last digit.

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A062133 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A001333(n+1), n >= 0 (associated Pell numbers).

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A131721 Overlay of Pell companion numbers: a(n) = A001333(n) + A001333(n-6).

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A138683 a(1)=1 and, for n>1, a(n) is the smallest integer greater than a(n-1) such that a(n) and a(k) do not sum to a term of A001333 (Numerators of continued fraction convergents to sqrt(2)).

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A143966 Eigentriangle with row sums = A001333 starting (1, 3, 7, 17, 41, 99, ...).

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A158843 G.f.: A(x) = exp( Sum_{n>=1} A001333(n)^n * 2^n*x^n/n ).

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PARI

A160756 Triangle read by rows, infinite lower triangular Toeplitz matrix with A078008 in every column convolved with A001333.

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