A080040 -id:A080040 - cp's OEIS frontend

A189746 a(1)=5, a(2)=2, a(n) = 5a(n-1) + 2a(n-2).

5, 2, 20, 104, 560, 3008, 16160, 86816, 466400, 2505632, 13460960, 72316064, 388502240, 2087143328, 11212721120, 60237892256, 323614903520, 1738550302112, 9339981317600, 50177007192224, 269564998596320, 1448179007366048, 7780025034022880, 41796483184846496

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (5,2).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{5,2},{5,2},40]
```

a[1]:5$ a[2]:2$ a[n]:=5*a[n-1]+2*a[n-2]$ makelist(a[n], n, 1, 24); /* Bruno Berselli, May 24 2011 */

G.f.: x*(5-23*x)/(1-5*x-2*x^2). - Bruno Berselli, May 24 2011

A189747 a(1)=5, a(2)=3, a(n)=5a(n-1) + 3a(n-2).

Original entry on oeis.org

5, 3, 30, 159, 885, 4902, 27165, 150531, 834150, 4622343, 25614165, 141937854, 786531765, 4358472387, 24151957230, 133835203311, 741631888245, 4109665051158, 22773220920525, 126195099756099, 699295161542070, 3875061106978647, 21473191019519445

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (5,3).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189748, A189749.

Mathematica
```
LinearRecurrence[{5,3},{5,3},40]
```

a[1]:5$ a[2]:3$ a[n]:=5*a[n-1]+3*a[n-2]$ makelist(a[n], n, 1, 23); /* Bruno Berselli, May 24 2011 */

G.f.: x*(5-22*x)/(1-5*x-3*x^2). - Bruno Berselli, May 24 2011

A189748 a(n) = 5a(n-1) + 4a(n-2) with a(1)=5, a(2)=4.

Original entry on oeis.org

5, 4, 40, 216, 1240, 7064, 40280, 229656, 1309400, 7465624, 42565720, 242691096, 1383718360, 7889356184, 44981654360, 256465696536, 1462255100120, 8337138286744, 47534711834200, 271022112317976, 1545249408926680, 8810335493905304, 50232675105233240

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (5,4).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189749.

Mathematica
```
LinearRecurrence[{5,4},{5,4},40]
```

a[1]:5$ a[2]:4$ a[n]:=5*a[n-1]+4*a[n-2]$ makelist(a[n], n, 1, 23); /* Bruno Berselli, May 24 2011 */

G.f.: x*(5-21*x)/(1-5*x-4*x^2). - Bruno Berselli, May 24 2011

A083337 a(n) = 2a(n-1) + 2a(n-2); a(0)=0, a(1)=3.

Original entry on oeis.org

0, 3, 6, 18, 48, 132, 360, 984, 2688, 7344, 20064, 54816, 149760, 409152, 1117824, 3053952, 8343552, 22795008, 62277120, 170144256, 464842752, 1269974016, 3469633536, 9479215104, 25897697280, 70753824768, 193303044096, 528113737728, 1442833563648, 3941894602752, 10769456332800

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,2).

Equals 3 * A002605.
Cf. A026150.
Cf. A028859, A028860, A030195, A080040, A106435, A108898, A125145.

a083337 n = a083337_list !! n
a083337_list =
   0 : 3 : map (* 2) (zipWith (+) a083337_list (tail a083337_list))
-- Reinhard Zumkeller, Oct 15 2011

CoefficientList[Series[3x/(1-2x-2x^2), {x, 0, 25}], x]
s = Sqrt[3]; a[n_] := Simplify[s*((1 + s)^n - (1 - s)^n)/2]; Array[a, 30, 0] (* or *)
LinearRecurrence[{2, 2}, {0, 3}, 31] (* Robert G. Wilson v, Aug 07 2018 *)

apply( a(n)=([1,1;3,1]^n)[2,1], [0..30]) \\ or: ([2,2;1,0]^n)[2,1]*3. - M. F. Hasler, Aug 06 2018

G.f.: 3x/(1 - 2x - 2x^2).
a(n) = a(n-1) + 3*A026150(n-1). a(n)/A026150(n) converges to sqrt(3).
a(n) = lower left term of [1,1; 3,1]^n. - Gary W. Adamson, Mar 12 2008

Edited and definition completed by M. F. Hasler, Aug 06 2018

A108898 a(n+3) = 3a(n+2) - 2a(n), a(0) = -1, a(1) = 1, a(2) = 3.

Original entry on oeis.org

-1, 1, 3, 11, 31, 87, 239, 655, 1791, 4895, 13375, 36543, 99839, 272767, 745215, 2035967, 5562367, 15196671, 41518079, 113429503, 309895167, 846649343, 2313089023, 6319476735, 17265131519, 47169216511, 128868696063, 352075825151, 961889042431, 2627929735167, 7179637555199

Offset: 0

Creighton Dement, Jul 16 2005

In reference to the program code, "ibasek" corresponds to the floretion 'ik'. Sequences in this same batch are "kbase" = A005665 (Tower of Hanoi with cyclic moves only.) and "ibase" = A077846.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-2).

Cf. A005665, A077846, A028860.

Cf. A002605, A026150, A028859, A030195, A080040, A083337, A106435, A125145.

a108898 n = a108898_list !! n
a108898_list = -1 : 1 : 3 :
   zipWith (-) (map (* 3) $ drop 2 a108898_list) (map (* 2) a108898_list)
-- Reinhard Zumkeller, Oct 15 2011

seriestolist(series((-1+4*x)/((x-1)*(2*x^2+2*x-1)), x=0,31)); -or- Floretion Algebra Multiplication Program, FAMP Code: 2ibaseksumseq[A*B] with A = + 'i + 'ii' + 'ij' + 'ik' and B = + .5'i + .5'j - .5'k + .5i' - .5j' + .5k' + .5'ij' + .5'ik' - .5'ji' - .5'ki'; Sumtype is set to:sum[(Y[0], Y[1], Y[2]),mod(3)

LinearRecurrence[{3, 0, -2}, {-1, 1, 3}, 40] (* Paolo Xausa, Aug 21 2024 *)

Vec(-(1 - 4*x) / ((1 - x)*(1 - 2*x - 2*x^2)) + O(x^40)) \\ Colin Barker, Apr 29 2019

a(n) = A028860(n+2)-1.
G.f.: (-1+4*x)/((x-1)*(2*x^2+2*x-1)).
From Colin Barker, Apr 29 2019: (Start)
a(n) = (-1 + (-(1-sqrt(3))^n + (1+sqrt(3))^n)/sqrt(3)).
a(n) = 3*a(n-1) - 2*a(n-3) for n>2.
(End)

A170931 Extended Lucas L(n,i) = n(L(n,i-1) + L(n,i-2)) = a^i + b^i where d = sqrt(n(n+4)); a=(n+d)/2; b=(n-d)/2.

Original entry on oeis.org

2, 4, 24, 112, 544, 2624, 12672, 61184, 295424, 1426432, 6887424, 33255424, 160571392, 775307264, 3743514624, 18075287552, 87275208704, 421401985024, 2034708774912, 9824443039744, 47436607258624, 229044201193472

Offset: 0

Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Feb 04 2010

Sequence gives the rational part of the radii of the circles in nested circles and squares inspired by Vitruvian Man, starting with a square whose sides are of length 4 (in some units). The radius of the circle is an integer in the real quadratic number field Q(sqrt(2)), namely R(n) = A(n-1) + B(n)*sqrt(2) with A(-1)=1, for n >= 1, A(n-1) = A170931(n-1)*-1^(n-1); and B(n) = A094013(n)*-1^n. See illustrations in the links. - Kival Ngaokrajang, Feb 15 2015

			L(n,0)=2, L(n,1)=n.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Kival Ngaokrajang, Illustration of initial terms, Vitruvian Man
Index entries for linear recurrences with constant coefficients, signature (4, 4).

Cf. similar sequences with d=sqrt(n*(n+k)): A000032 (k=1, classic Lucas), A080040 (k=2), A085480 (k=3).

Cf. A094013, A174968.

I:=[2,4]; [n le 5 select I[n] else 4*Self(n-1)+4*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 21 2017

CoefficientList[Series[2 (1 - 2 x) / (1 - 4 x - 4 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 16 2015 *)
LinearRecurrence[{4,4},{2,4},30] (* Harvey P. Dale, Sep 03 2016 *)

x='x+O('x^30); Vec(2*(1-2*x)/(1 - 4*x - 4*x^2)) \\ G. C. Greubel, Dec 21 2017

From R. J. Mathar, Feb 05 2010: (Start)
a(n) = 2*A084128(n) = 4*a(n-1) + 4*a(n-2).
G.f.: 2*(1-2*x)/(1 - 4*x - 4*x^2). (End)

A191347 Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 4, 2, 1, 0, 16, 17, 10, 8, 2, 1, 0, 32, 41, 28, 32, 9, 2, 1, 0, 64, 99, 76, 128, 38, 10, 2, 1, 0, 128, 239, 208, 512, 161, 44, 11, 2, 1, 0, 256, 577, 568, 2048, 682, 196, 50, 12, 3, 1

Offset: 0

Charles L. Hohn, May 31 2011

			1, 0,  0,   0,    0,    0,     0,      0,       0,        0,        0, ...
1, 1,  2,   4,    8,   16,    32,     64,     128,      256,      512, ...
1, 1,  3,   7,   17,   41,    99,    239,     577,     1393,     3363, ...
1, 1,  4,  10,   28,   76,   208,    568,    1552,     4240,    11584, ...
1, 2,  8,  32,  128,  512,  2048,   8192,   32768,   131072,   524288, ...
1, 2,  9,  38,  161,  682,  2889,  12238,   51841,   219602,   930249, ...
1, 2, 10,  44,  196,  872,  3880,  17264,   76816,   341792,  1520800, ...
1, 2, 11,  50,  233, 1082,  5027,  23354,  108497,   504050,  2341691, ...
1, 2, 12,  56,  272, 1312,  6336,  30592,  147712,   713216,  3443712, ...
1, 3, 18, 108,  648, 3888, 23328, 139968,  839808,  5038848, 30233088, ...
1, 3, 19, 117,  721, 4443, 27379, 168717, 1039681,  6406803, 39480499, ...
1, 3, 20, 126,  796, 5028, 31760, 200616, 1267216,  8004528, 50561600, ...
1, 3, 21, 135,  873, 5643, 36477, 235791, 1524177,  9852435, 63687141, ...
1, 3, 22, 144,  952, 6288, 41536, 274368, 1812352, 11971584, 79078912, ...
1, 3, 23, 153, 1033, 6963, 46943, 316473, 2133553, 14383683, 96969863, ...
...

Row 1 is A000007, row 2 is A011782, row 3 is A001333, row 4 is A026150, row 5 is A081294, row 6 is A001077, row 7 is A084059, row 8 is A108851, row 9 is A084128, row 10 is A081341, row 11 is A005667, row 13 is A141041.
Row 3*2 is A002203, row 4*2 is A080040, row 5*2 is A155543, row 6*2 is A014448, row 8*2 is A080042, row 9*2 is A170931, row 11*2 is A085447.
Cf. A191348 which uses ceiling() in place of floor().

T(n, k) = if (n==0, k==0, my(x=sqrtint(n)); sum(i=0, (k+1)\2, binomial(k, 2*i)*x^(k-2*i)*n^i));
matrix(9,9, n, k, T(n-1,k-1)) \\ Michel Marcus, Aug 22 2019

T(n, k) = if (k==0, 1, if (k==1, sqrtint(n), T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 22 2019

For each row n>=0 let T(n,0)=1 and T(n,1)=floor(sqrt(n)), then for each column k>=2: T(n,k)=T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 22 2019

T(n, k) = Sum_{i=0..floor((k+1)/2)} binomial(k, 2*i)*floor(sqrt(n))^(k-2*i)*n^i for n > 0, with T(0, 0) = 1 and T(0, k) = 0 for k > 0. - Michel Marcus, Aug 23 2019

A238799 a(0) = 1, a(n+1) = 2a(n)^3 + 3a(n).

Original entry on oeis.org

1, 5, 265, 37220045, 103124220135120334842385, 2193370648451279691104497113491599222165108730278225579497595691360405

Offset: 0

Arkadiusz Wesolowski, Mar 05 2014

a(6) has 209 digits and is too large to include.
Except for the first term, this is a subsequence of A175180.
The squares larger than 1 are in A076445.
If we define u(0) = 1 , u(n+1) = (u(n)/3)*(u(n)^2+9) / (u(n)^2 + 1), then u(n) = a(n) / A378683(n) ; this is Halley's method to calculate sqrt(3). - Robert FERREOL, Dec 21 2024

Wikipedia, Halley's method.

Cf. A175180, A378683.

RecurrenceTable[{a[0] == 1, a[n] == 2*a[n - 1]^3 + 3*a[n - 1]}, a[n], {n, 5}]
NestList[2#^3+3#&,1,5] (* Harvey P. Dale, Mar 22 2023 *)

a=1; print1(a, ", "); for(n=1, 5, b=2*a^3+3*a; print1(b, ", "); a=b);

{ A238799(n) = my(q=Mod(x,x^2-3)); lift( (1+q)*(2+q)^((3^n-1)/2) + (1-q)*(2-q)^((3^n-1)/2) )/2; } \\ Max Alekseyev, Sep 04 2018

a(n) = sqrt(2) * sinh( 3^n * arcsinh(1/sqrt(2)) ) = (1+sqrt(3))/2 * (2+sqrt(3))^((3^n-1)/2) + (1-sqrt(3))/2 * (2-sqrt(3))^((3^n-1)/2). - Max Alekseyev, Sep 04 2018

a(n) = ((1 + sqrt(3))^(3^n) + (1 - sqrt(3))^(3^n))/2^((3^n+1)/2) = A002531(3^n) = A080040(3^n)/2^((3^n+1)/2). - Robert FERREOL, Nov 19 2024

A084157 a(n) = 8a(n-1) - 16a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.

Original entry on oeis.org

0, 1, 4, 22, 112, 556, 2704, 13000, 62080, 295312, 1401664, 6644320, 31472896, 149017792, 705395968, 3338614912, 15800258560, 74772443392, 353840161792, 1674425579008, 7923565146112, 37494981225472, 177428889407488

Offset: 0

Paul Barry, May 16 2003

Binomial transform of A084156.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-16,0,12).

Cf. A026150, A083881, A084155, A084156.

Cf. A002605, A003462, A080040.

I:=[0,1,4,22]; [n le 4 select I[n] else 8*Self(n-1) -16*Self(n-2) +12*Self(n-4): n in [1..41]]; // G. C. Greubel, Oct 11 2022

LinearRecurrence[{8,-16,0,12},{0,1,4,22},30] (* Harvey P. Dale, Feb 19 2017 *)

A083881 = BinaryRecurrenceSequence(6,-6,1,3)
A026150 = BinaryRecurrenceSequence(2,2,1,1)
def A084157(n): return (A083881(n) - A026150(n))/2
[A084157(n) for n in range(41)] # G. C. Greubel, Oct 11 2022

a(n) = (A083881(n) - A026150(n))/2.
a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4).
a(n) = ((3+sqrt(3))^n + (3-sqrt(3))^n - (1+sqrt(3))^n - (1-sqrt(3))^n)/4.
G.f.: x*(1-4*x+6*x^2)/((1-2*x-2*x^2)*(1-6*x+6*x^2)).
E.g.f.: exp(2*x)*sinh(x)*cosh(sqrt(3)*x).
From G. C. Greubel, Oct 11 2022: (Start)
a(2*n) = A003462(n)*A026150(2*n) = A003462(n)*A080040(2*n)/2.
a(2*n+1) = (1/2)*(3^(n+1)*A002605(2*n+1) - A026150(2*n+1)). (End)

A067834 Numbers k such that in the ring Z[sqrt(3)] the norm of (-1+sqrt(3))^k-1 is prime.

Original entry on oeis.org

2, 3, 7, 13, 19, 43, 61, 151, 257, 751, 859, 1453, 3767, 3889, 8171, 15959, 21499, 22679, 23297, 31277, 43609, 57037, 61961, 103087, 115931, 173647, 215959, 496073

Offset: 1

Mike Oakes, Feb 09 2002

The definition implies that k must be a prime.
The norm of (-1+sqrt(3))^k-1 is given by (-2)^k - lucasV(-2, -2, k)+1, where lucasV(-2, -2, k) is the solution of the recurrence relation v(n) = -2*v(n-1) + 2*v(n-2) with v(0)=2 and v(1)=-2.
a(29) > 517000. - Serge Batalov, Oct 24 2024

			3 is a term because (-2)^3-lucasV(-2,-2,3)+1 = -8-(-20)+1 = 13 and 13 is prime.

Mike Oakes, Posting to primenumbers list on Feb 08 2002
Mike Oakes, 4 new forms of primes, digest of 4 messages in primenumbers Yahoo group, Feb 8 - Feb 11, 2002.

Cf. A080040.

v[0] = 2; v[1] = -2; v[n_] := v[n] = -2*v[n-1] + 2*v[n-2] ; s = {}; Do[If[PrimeQ[(-2)^n - v[n] + 1], Print[n]; AppendTo[s, n]], {n, 8171}]; s (* Jean-François Alcover, Apr 18 2011 *)

isok(n)={ispseudoprime(([0, 1; 2, 2]^n*[2; 2])[1, 1] - 2^n - (-1)^n)} \\ Andrew Howroyd, Oct 24 2024

Corrected and extended by Aurelien Gibier, Oct 24 2024

a(28) from Serge Batalov, Oct 24 2024

cp's OEIS Frontend

A189746 a(1)=5, a(2)=2, a(n) = 5*a(n-1) + 2*a(n-2).

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A189747 a(1)=5, a(2)=3, a(n)=5*a(n-1) + 3*a(n-2).

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A189748 a(n) = 5*a(n-1) + 4*a(n-2) with a(1)=5, a(2)=4.

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A083337 a(n) = 2*a(n-1) + 2*a(n-2); a(0)=0, a(1)=3.

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A108898 a(n+3) = 3*a(n+2) - 2*a(n), a(0) = -1, a(1) = 1, a(2) = 3.

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A170931 Extended Lucas L(n,i) = n*(L(n,i-1) + L(n,i-2)) = a^i + b^i where d = sqrt(n*(n+4)); a=(n+d)/2; b=(n-d)/2.

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A191347 Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

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A238799 a(0) = 1, a(n+1) = 2*a(n)^3 + 3*a(n).

A189746 a(1)=5, a(2)=2, a(n) = 5a(n-1) + 2a(n-2).

A189747 a(1)=5, a(2)=3, a(n)=5a(n-1) + 3a(n-2).

A189748 a(n) = 5a(n-1) + 4a(n-2) with a(1)=5, a(2)=4.

A083337 a(n) = 2a(n-1) + 2a(n-2); a(0)=0, a(1)=3.

A108898 a(n+3) = 3a(n+2) - 2a(n), a(0) = -1, a(1) = 1, a(2) = 3.

A170931 Extended Lucas L(n,i) = n(L(n,i-1) + L(n,i-2)) = a^i + b^i where d = sqrt(n(n+4)); a=(n+d)/2; b=(n-d)/2.

A238799 a(0) = 1, a(n+1) = 2a(n)^3 + 3a(n).

A084157 a(n) = 8a(n-1) - 16a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.