Displaying 1-10 of 22 results found.
3, 8, 11, 16, 19, 21, 24, 29, 32, 37, 42, 45, 50, 53, 55, 58, 63, 66, 71, 74, 76, 79, 84, 87, 92, 97, 100, 105, 108, 110, 113, 118, 121, 126, 129, 131, 134, 139, 142, 144, 147, 152, 155, 160, 163, 165, 168, 173, 176, 181, 186, 189, 194, 197, 199, 202, 207
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MAPLE
option remember;
if n =1 then
3;
else
for a from procname(n-1)+1 do
return a;
end if;
end do:
end if;
end proc:
MATHEMATICA
a3[n_] := Floor[n (Sqrt[5] + 3)/2];
a5[n_] := Floor[n (Sqrt[5] + 5)/2];
PROG
(PARI) A001950(n) = floor(n*(sqrt(5)+3)/2); A003231(n) = floor(n*(sqrt(5)+5)/2);
(Haskell)
a003234 n = a003234_list !! (n-1)
a003234_list = [x | x <- [1..],
a003231 (a001950 x) == a001950 (a003231 x) - 1]
(Python)
from math import isqrt
from itertools import count, islice
def A003234_gen(startvalue=1): # generator of terms >= startvalue return filter(lambda n:((m:=(n+isqrt(5*n**2)>>1)+n)+isqrt(5*m**2)>>1)+(m<<1)+1==((k:=(n+isqrt(5*n**2)>>1)+(n<<1))+isqrt(5*k**2)>>1)+k, count(max(1, startvalue)))
1, 2, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 51, 52, 54, 56, 57, 59, 60, 61, 62, 64, 65, 67, 68, 69, 70, 72, 73, 75, 77, 78, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MATHEMATICA
a3221[n_] := Floor[n(5 + Sqrt[5])/2];
a1950[n_] := Floor[n(1 + Sqrt[5])^2/4];
PROG
(PARI) A001950(n) = floor(n*(sqrt(5)+3)/2); A003231(n) = floor(n*(sqrt(5)+5)/2);
(Haskell)
a003233 n = a003233_list !! (n-1)
a003233_list = [x | x <- [1..],
a003231 (a001950 x) == a001950 (a003231 x)]
(Python)
from math import isqrt
from itertools import count, islice
def A003233_gen(startvalue=1): # generator of terms >= startvalue return filter(lambda n:((m:=(n+isqrt(5*n**2)>>1)+n)+isqrt(5*m**2)>>1)+(m<<1)==((k:=(n+isqrt(5*n**2)>>1)+(n<<1))+isqrt(5*k**2)>>1)+k, count(max(1, startvalue)))
EXTENSIONS
Definition from Michel Marcus moved from comment to name by Eric M. Schmidt, Aug 17 2014
Floor(r*n), where r = (5 - sqrt(5))/2; the Beatty complement of A003231.
+20
3
1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 51, 52, 53, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 69, 70, 71, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 87, 88, 89, 91
COMMENTS
Let r = (5 - sqrt(5))/2 and s = (5 + sqrt(5))/2. Then 1/r + 1/s = 1, so that A249115 and A003231 are a pair of complementary Beatty sequences. Let tau = (1 + sqrt(5))/2, the golden ratio. Let R = {h*tau, h >= 1} and S = {k*(tau - 1), k >= 1}. Then A249115(n) is the position of n*(tau - 1) in the ordered union of R and S.
MATHEMATICA
Table[Floor[(5 - Sqrt[5])/2*n], {n, 1, 200}]
First differences of the Beatty sequence A003231 for 2 + tau, where tau = golden ratio = (1 + sqrt(5))/2.
+20
3
3, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4
FORMULA
a(n) = floor(n*r) - floor(n*r - r), where r = 2 + tau, n >= 1.
MATHEMATICA
z = 500; r = 2+GoldenRatio; b = Table[Floor[k*r], {k, 0, z}]; (* A003231 *)
2, 3, 5, 7, 8, 10, 12, 13, 15, 16, 18, 20, 21, 23, 24, 26, 28, 29, 31, 33, 34, 36, 37, 39, 41, 42, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 62, 63, 65, 67, 68, 70, 71, 73, 75, 76, 78, 80, 81, 83, 84, 86, 88, 89, 91, 92, 94, 96, 97, 99, 101, 102, 104, 105
COMMENTS
This is the function named phi in the Carlitz-Scoville-Vaughan link. - Eric M. Schmidt, Aug 14 2014
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
4, 11, 15, 22, 26, 29, 33, 40, 44, 51, 58, 62, 69, 73, 76, 80, 87, 91, 98, 102, 105, 109, 116, 120, 127, 134, 138, 145, 149, 152, 156, 163, 167, 174, 178, 181, 185, 192, 196, 199, 203, 210, 214, 221, 225, 228, 232, 239, 243, 250, 257, 261, 268, 272, 275, 279
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = ceiling((1/phi^2) * A003231( A003234(n))), where phi is the golden ratio. a(n) = 5*k - 1 - A003231(k), where k = A003234(n). [Cf. Theorems 4.1(ii) and 5.9(vii) in Carlitz.] Conjecture: a(n) = floor((3-phi)* A003234(n)). (End)
3, 6, 9, 12, 15, 18, 21, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 61, 64, 67, 70, 73, 76, 79, 82, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 122, 125, 128, 131, 134, 137, 140, 143, 145, 148, 151, 154, 157, 159, 162, 165, 168, 171
COMMENTS
This is the function named lambda in [Carlitz]. - Eric M. Schmidt, Aug 14 2014
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = 3n - j(n), where j(n) is the maximum number such that j(n) <= A003249(n). [Carlitz, Theorem 7.15.] - Eric M. Schmidt, Aug 17 2014
EXTENSIONS
Sequence corrected and extended by, and definition from Eric M. Schmidt, Aug 17 2014
1, 2, 3, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 77, 78, 79, 81, 82, 84, 85, 86, 88, 89, 90, 92
COMMENTS
This is the function named K in [Carlitz].
PROG
(PARI) a31(n) = (5*n+sqrtint(5*n^2))\2; \\ A003231a50(n) = (sqrtint(n^2*5)+n*3)\2; \\ A001950a(n) = my(m=1, N=a31(n)); while(a50(m) < N, m++); m-1; \\ Michel Marcus, Nov 14 2023
Sums-complement of the Beatty sequence for sqrt(5).
+10
19
1, 10, 19, 28, 37, 48, 57, 66, 75, 86, 95, 104, 113, 124, 133, 142, 151, 162, 171, 180, 189, 198, 209, 218, 227, 236, 247, 256, 265, 274, 285, 294, 303, 312, 323, 332, 341, 350, 359, 370, 379, 388, 397, 408, 417, 426, 435, 446, 455, 464, 473, 484, 493, 502
COMMENTS
The sums-complement of a sequence s(1), s(2), ... of positive integers is introduced here as the set of numbers c(1), c(2), ... such that no c(n) is a sum s(j)+s(j+1)+...+s(k) for any j and k satisfying 1 <= j <= k. If this set is not empty, the term "sums-complement" also applies to the (possibly finite) sequence of numbers c(n) arranged in increasing order. In particular, the difference sequence D(r) of a Beatty sequence B(r) of an irrational number r > 2 has an infinite sums-complement, abbreviated as SC(r) in the following table:
r B(r) D(r) SC(r)
----------------------------------------------------
Simpler description: this sequence represents those positive integers that CANNOT be expressed as a difference of two elements of A022839. There is a 20-state Fibonacci automaton for the terms of this sequence (see a276871.pdf). It takes as input the Zeckendorf representation of n and accepts iff n is a member of A276871. (End)
EXAMPLE
The Beatty sequence for sqrt(5) is A022839 = (0,2,4,6,8,11,13,15,...), with difference sequence s = A081427 = (2,2,2,2,3,2,2,2,3,2,...). The sums s(j)+s(j+1)+...+s(k) include (2,3,4,5,6,7,8,9,11,12,...), with complement (1,10,19,28,37,...).
MATHEMATICA
z = 500; r = Sqrt[5]; b = Table[Floor[k*r], {k, 0, z}]; (* A022839 *) c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]; (* A276871 *)
1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74
COMMENTS
This is the function named u in [Carlitz].
First differs from A187947 at a(46)=51.
MATHEMATICA
nmax = 80;
A001950[n_] := Floor[n*GoldenRatio^2];
A003231[n_] := 2*n + Floor[n*GoldenRatio];
PROG
(Haskell)
a242094 n = a242094_list !! (n-1)
a242094_list = c [1..] a003249_list where
c (v:vs) ws'@(w:ws) = if v == w then c vs ws else v : c vs ws'
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