A014217 -id:A014217 - cp's OEIS frontend

A152737 a(n) = floor(n^phi) where phi is the golden ratio.

0, 1, 3, 5, 9, 13, 18, 23, 28, 34, 41, 48, 55, 63, 71, 79, 88, 97, 107, 117, 127, 137, 148, 159, 171, 182, 194, 207, 219, 232, 245, 258, 272, 286, 300, 315, 329, 344, 359, 375, 391, 406, 423, 439, 456, 473, 490, 507, 525, 542, 561, 579, 597, 616, 635, 654, 673

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 12 2008

Georg Fischer, Table of n, a(n) for n = 0..9999 [first 1615 terms from Paolo P. Lava]

Cf. A001622, A014217.

phi:=(1+Sqrt(5))/2; [Floor(n^phi): n in [0..50]]; // G. C. Greubel, Sep 01 2018

Mathematica
```
a[n_]:=Floor[n^GoldenRatio];
```

a(n)=floor(n^((1+sqrt(5))/2)) \\ Charles R Greathouse IV, Jul 29 2011

Offset changed to 0 by Georg Fischer, Oct 19 2024

A169613 Triangular array: T(n,k)=floor(F(n)/F(n-k)), k=1,2,...,n-2; n>=3, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

2, 1, 3, 1, 2, 5, 1, 2, 4, 8, 1, 2, 4, 6, 13, 1, 2, 4, 7, 10, 21, 1, 2, 4, 6, 11, 17, 34, 1, 2, 4, 6, 11, 18, 27, 55, 1, 2, 4, 6, 11, 17, 29, 44, 89, 1, 2, 4, 6, 11, 18, 28, 48, 72, 144, 1, 2, 4, 6, 11, 17, 29, 46, 77, 116, 233, 1, 2, 4, 6, 11, 17, 29, 47, 75, 125, 188, 377, 1, 2, 4, 6, 11

Offset: 3

Clark Kimberling, Dec 03 2009

Combinatorial limit of row n is essentially A014217.

			The first 6 rows:
2
1 3
1 2 5
1 2 4 8
1 2 4 6 13
1 2 4 7 10 21

Cf. A000045, A014217, A169614, A169615, A169616.

T[n_, k_] := Floor[Fibonacci[n]/Fibonacci[n-k]]; Table[T[n, k], {n, 3, 15}, {k, 1, n-2}] // Flatten (* Jean-François Alcover, Jul 16 2017 *)

from sympy import fibonacci as F, floor
def T(n, k): return floor(F(n)/F(n - k))
for n in range(3, 16): print([T(n, k) for k in range(1, n - 1)]) # Indranil Ghosh, Jul 17 2017

Offset corrected by Jean-François Alcover, Jul 16 2017

A142710 a(n) = A142585(n) + A142586(n).

Original entry on oeis.org

2, 2, 6, 14, 38, 112, 276, 814, 1998, 5702, 14226, 39404, 99908, 270922, 695106, 1859134, 4807518, 12748472, 33128916, 87394454, 227792678, 599050102, 1564242906, 4106054164, 10733283588, 28143585362, 73614464826, 192899714414, 504751433798, 1322156172352

Offset: 0

Paul Curtz, Sep 25 2008

Sum of the binomial and inverse binomial transforms of A014217.

Starting at a(1), the last digits form a period-4 sequence 2, 6, 4, 8.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7,-12,-11,16,-4).

Cf. A000032, A014217, A142585, A142586, A147586.

[n eq 0 select 2 else (-1)^n*Lucas(n) +Lucas(2*n) -(1+(-1)^n)*2^(n-1): n in [0..50]]; // G. C. Greubel, Oct 26 2022

Join[{2},LinearRecurrence[{2,7,-12,-11,16,-4},{2,6,14,38,112,276},30]] (* Harvey P. Dale, Nov 25 2013 *)

def A142710(n): return (-1)^n*lucas_number2(n,1,-1) + lucas_number2(2*n,1,-1) - (1 + (-1)^n)*2^(n-1) -int(n==0)
[A142710(n) for n in range(51)] # G. C. Greubel, Oct 26 2022

a(n) = +2*a(n-1) +7*a(n-2) -12*a(n-3) -11*a(n-4) +16*a(n-5) -4*a(n-6), n>6. - R. J. Mathar, Jun 14 2010
G.f.: 2*(1-x-6*x^2+6*x^3+7*x^4-2*x^6)/((1-2*x)*(1+2*x)*(1+x-x^2)*(1-3*x+x^2)). - Colin Barker, Aug 13 2012
a(n) = (-1)^n*LucasL(n) + LucasL(2*n) - (1 + (-1)^n)*2^(n-1) - [n=0]. - G. C. Greubel, Oct 26 2022

Offset set to zero and extended - R. J. Mathar, Jun 14 2010

A350092 a(n) = floor(x^n) where x = 1 + sqrt(5)/2 = A176055.

Original entry on oeis.org

1, 2, 4, 9, 20, 42, 90, 191, 405, 857, 1816, 3848, 8150, 17263, 36564, 77445, 164031, 347423, 735855, 1558567, 3301098, 6991839, 14808952, 31365865, 66433969, 140709405, 298027302, 631231956, 1336970739, 2831749467, 5997741619, 12703420605, 26906276616

Offset: 0

Michel Lagneau, Dec 14 2021

a(n+1)/a(n) tends to A176055 when n tends towards infinity.

Cf. A176055, A058066 (x*n), A014217 (phi^n).

Maple
```
seq(floor((1+sqrt(5)/2)^n), n=0..32);
```

a[n_] := Floor[(GoldenRatio + 1/2)^n]; Array[a, 33, 0] (* Amiram Eldar, Dec 14 2021 *)

from sympy import floor, sqrt
def A350092(n): return floor((1+sqrt(5)/2)**n) # Chai Wah Wu, Dec 17 2021

A057146 The sequence 2, floor(a), floor(a^2), floor(a^3), ..., with a = 1+sqrt(5).

Original entry on oeis.org

2, 3, 10, 33, 109, 354, 1148, 3716, 12026, 38918, 125943, 407562, 1318899, 4268047, 13811692, 44695576, 144637922, 468058148, 1514667986, 4901568568, 15861809082, 51329892437, 166107021206, 537533612162, 1739495309150, 5629125066952, 18216231370504

Offset: 0

N. J. A. Sloane, Sep 12 2000, May 14 2007

nonn

G. Harman, One hundred years of normal numbers

Cf. A014217.

[2] cat [Floor((1+Sqrt(5))^n): n in [1..30]]; // Vincenzo Librandi, Feb 20 2016

Join[{2}, Table[Floor[(Sqrt[5] + 1)^n], {n, 1, 30}]] (* Artur Jasinski, Nov 22 2006 *)

a(n) = if (n==0, 2, floor((1+sqrt(5))^n)); \\ Michel Marcus, Feb 19 2016

a(n) = floor( (1+sqrt(5))^n ) for n>0, a(0)=2. - Artur Jasinski, Nov 22 2006

A228560 Curvature (rounded down) of the circle inscribed in the n-th golden triangle arranged in a spiral form.

Original entry on oeis.org

2, 4, 7, 11, 18, 30, 49, 79, 129, 209, 338, 547, 886, 1434, 2320, 3754, 6075, 9830, 15905, 25735, 41641, 67376, 109017, 176394, 285412, 461806, 747218, 1209024

Offset: 1

Kival Ngaokrajang, Aug 25 2013

Starting with a golden triangle whose base is of length 1 and whose sides are of length phi = (1+sqrt(5))/2, create the next golden triangle at the base of the previous triangle, i.e., sides length = 1 and base length = phi-1, and so on. a(n) is the floor of the curvature (inverse of the radius) of the circle inscribed in the n-th triangle.
The golden triangles created by this process are the same as the golden triangles inscribed in a logarithmic spiral.
The logarithmic spiral can be approximated by circular arcs of radii 1, phi-1, (phi-1)^2, ... which are the sides of bisected golden gnomons and center located at their related apex. The sequence whose n-th term is the curvature (rounded down) of the n-th such circular arc is A014217. See illustration in link.

Kival Ngaokrajang, Illustration of initial terms.
Wikipedia, Golden triangle.

Cf. A001521 (for 45-45-90 triangles), A065565 (for 3:4:5 triangles), A014217.

A140471 Floored n-th power of Viswanath's constant.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 22, 25, 28, 32, 36, 41, 46, 52, 59, 67, 76, 86, 98, 111, 125, 142, 161, 182, 206, 233, 264, 299, 339, 384, 434, 492, 557, 630, 713, 808, 914, 1035, 1172, 1326, 1502, 1700, 1924

Offset: 0

Alonso del Arte, Jun 28 2008

nonn

For sufficiently large terms of a random Fibonacci sequence, the powers of Viswanath's constant approximate the absolute value of the terms in such a sequence (with a few notable exceptions).

			a(7) = 2 because V^7 is approximately 2.381734947432 and floored that is 2.

Eric Weisstein, Random Fibonacci sequence at MathWorld

Cf. A014217, floored n-th power of the golden ratio; A000149, floored n-th power of e; A001672, floored n-th power of Pi.

V = 1.1319882487943; Table[Floor[V^n], {n, 0, 49}]

a(n) = floor(v^n), where v = 1.1319882487943 as given by A078416.

More terms from Alois P. Heinz, Mar 08 2020

A277752 a(n) = Sum_{k=0..n} (-1)^k*floor(phi^k), where phi is the golden ratio (A001622).

Original entry on oeis.org

1, 0, 2, -2, 4, -7, 10, -19, 27, -49, 73, -126, 195, -326, 516, -848, 1358, -2213, 3564, -5785, 9341, -15135, 24467, -39612, 64069, -103692, 167750, -271454, 439192, -710659, 1149838, -1860511, 3010335, -4870861, 7881181, -12752058, 20633223, -33385298, 54018504, -87403820, 141422306

Offset: 0

Ilya Gutkovskiy, Oct 31 2016

Alternating sum of A014217.

Eric Weisstein's World of Mathematics, Golden Ratio
Index entries for linear recurrences with constant coefficients, signature (0,3,-1,-2,1).

Cf. A000032, A000045, A001622, A014217, A020956.

Accumulate[Table[(-1)^n Floor[GoldenRatio^n], {n, 0, 40}]]
LinearRecurrence[{0, 3, -1, -2, 1}, {1, 0, 2, -2, 4}, 41]

G.f.: (1 - x^2 - x^3)/((1 - x)^2*(1 + 2*x - x^3)).
a(n) = 3*a(n-2) - a(n-3) - 2*a(n-4) + a(n-5).
a(n) = Sum_{k=0..n} (-1)^k*floor(Fibonacci(2k+3)/Fibonacci(k+3)).
a(n) = Sum_{k=0..n} (-1)^k*(L(k) - (1 + (-1)^k)/2), where L(k) is the Lucas numbers beginning at 2 (A000032).
a(n) = 2^(-n-2)*(9*2^n - 2^(n+1)*n - (-2)^n - 2*(1 + sqrt(5))*(sqrt(5) - 1)^n + 2*(sqrt(5) - 1)*(-1-sqrt(5))^n).
a(n) ~ (-1)^n*phi^(n-1).
a(n) = (-1)^n*Lucas(n-1) - (1/4)*(2*n -9 +(-1)^n). - G. C. Greubel, Oct 31 2016

A301653 Expansion of x(1 + 2x)/((1 - x)(1 + x)(1 - x - x^2)).

Original entry on oeis.org

0, 1, 3, 5, 10, 16, 28, 45, 75, 121, 198, 320, 520, 841, 1363, 2205, 3570, 5776, 9348, 15125, 24475, 39601, 64078, 103680, 167760, 271441, 439203, 710645, 1149850, 1860496, 3010348, 4870845, 7881195, 12752041, 20633238, 33385280, 54018520, 87403801, 141422323, 228826125, 370248450

Offset: 0

Ilya Gutkovskiy, Mar 25 2018

Apparently (for n > 0), numbers that have a unique partition into a sum of distinct Lucas numbers (A000204).

Eric Weisstein's World of Mathematics, Lucas Number
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A000071, A000204, A001622, A003263, A014217, A054770, A294203.

CoefficientList[Series[x (1 + 2 x)/((1 - x) (1 + x) (1 - x - x^2)) , {x, 0, 40}], x]
LinearRecurrence[{1, 2, -1, -1}, {0, 1, 3, 5}, 41]
Table[LucasL[n + 1] - (3 - (-1)^n)/2, {n, 0, 40}]
Table[Floor[GoldenRatio^(n + 1)] - 1, {n, 0, 40}]

a(n) = fibonacci(n) + fibonacci(n+2) + ((-1)^n - 3)/2; \\ Altug Alkan, Mar 25 2018

G.f.: x*(1 + 2*x)/((1 - x)*(1 + x)*(1 - x - x^2)).
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4).
a(n) = Lucas(n+1) - (3 - (-1)^n)/2.
a(n) = floor(phi^(n+1)) - 1, where phi = (1 + sqrt(5))/2 is the golden ratio (A001622).
a(n) = Sum_{k>=0} A051601(n-k,k) (conjectured). - Greg Dresden, May 18 2023

cp's OEIS Frontend

A152737 a(n) = floor(n^phi) where phi is the golden ratio.

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A169613 Triangular array: T(n,k)=floor(F(n)/F(n-k)), k=1,2,...,n-2; n>=3, where F=A000045 (Fibonacci numbers).

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A142710 a(n) = A142585(n) + A142586(n).

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A350092 a(n) = floor(x^n) where x = 1 + sqrt(5)/2 = A176055.

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A057146 The sequence 2, floor(a), floor(a^2), floor(a^3), ..., with a = 1+sqrt(5).

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Magma

Mathematica

PARI

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A228560 Curvature (rounded down) of the circle inscribed in the n-th golden triangle arranged in a spiral form.

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A140471 Floored n-th power of Viswanath's constant.

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Comments

Examples

Links

Crossrefs

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Mathematica

Formula

Extensions

A277752 a(n) = Sum_{k=0..n} (-1)^k*floor(phi^k), where phi is the golden ratio (A001622).

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A301653 Expansion of x(1 + 2x)/((1 - x)(1 + x)(1 - x - x^2)).