A060143 -id:A060143 - cp's OEIS frontend

A183137 a(n) = [1/s] + [2/s] + ... + [n/s], where s = (golden ratio)^2 = (3+sqrt(5))/2 and [] = floor.

0, 0, 1, 2, 3, 5, 7, 10, 13, 16, 20, 24, 28, 33, 38, 44, 50, 56, 63, 70, 78, 86, 94, 103, 112, 121, 131, 141, 152, 163, 174, 186, 198, 210, 223, 236, 250, 264, 278, 293, 308, 324, 340, 356, 373, 390, 407, 425, 443, 462, 481, 500, 520, 540, 561, 582, 603, 625

Offset: 1

Clark Kimberling, Dec 26 2010

nonn

A183136(n) + a(n) = A000217(n+1) (the triangular numbers).

			a(7) = 7 = 0+0+1+1+1+2+2.

Johan Kok, Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs, arXiv:2507.16500 [math.CO], 2025. See p. 4.

Cf. A001622, A104457, A183136, A000217, A005206, A060143, A060144.

Accumulate[With[{c=GoldenRatio^2},Floor[Range[60]/c]]]  (* Harvey P. Dale, Apr 20 2011 *)

a(n+1) = a(n) + n - A005206(n). - John Furey, Jun 03 2015

A379663 a(n) is the number of integer-sided triangles whose sides are in geometric progression with smallest side n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 4, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 5, 4, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 4, 2, 1, 1, 1, 3, 6, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Felix Huber, Jan 07 2025

nonn

The integer sides of the triangles are n, n*r, n*r^2 with rational r >= 1. From the triangle inequality n + n*r >= n*r^2 follows r <= (1 + sqrt(5))/2 (golden ratio). Therefore 1 <= r = c/d < (1 + sqrt(5))/2, where c and d are coprimes and d^2 divides n.

			The a(18) = 2 integer-sided triangles whose sides form a geometric sequence are [18, 18, 18] with r = 1, [18, 24, 32] with r = 4/3.
The a(25) = 4 integer-sided triangles whose sides form a geometric sequence are [25, 25, 25] with r = 1, [25, 30, 36] with r = 6/5, [25, 35, 49] with r = 7/5, [25, 40, 64] with r = 8/5.
The a(36) = 4 integer-sided triangles whose sides form a geometric sequence are [36, 36, 36] with r = 1, [36, 54, 81] with r = 3/2, [36, 48, 64] with r = 4/3, [36, 42, 49] with r = 7/6.
See also the linked Maple program "Triangles for a given n".

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Triangles for a given n
Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Geometric Progression
Wikipedia, Triangle Inequality

Cf. A000188, A000201, A008833, A027750, A046951, A057918, A060143, A366398, A370599, A376900, A379033, A379340, A379341, A379705.

A379663:=n->floor(2*expand(NumberTheory:-LargestNthPower(n,2))/(1+sqrt(5)))+1;
seq(A379663(n),n=1..88);

a(n) = A060143(A000188(n)) + 1.

A134299 Maximal length of a sequence such that v(0)=n, v(k+2) = v(k)-v(k+1), v(k) >= 0.

Original entry on oeis.org

4, 5, 6, 5, 7, 6, 5, 8, 6, 7, 6, 6, 9, 6, 7, 8, 6, 7, 6, 7, 10, 6, 7, 8, 7, 9, 6, 7, 8, 7, 7, 8, 7, 11, 7, 7, 8, 7, 9, 8, 7, 10, 7, 7, 8, 7, 9, 8, 7, 8, 7, 9, 8, 7, 12, 8, 7, 8, 7, 9, 8, 7, 10, 8, 9, 8, 7, 11, 8, 7, 8, 8, 9, 8, 7, 10, 8, 9, 8, 8, 9, 8, 7, 10, 8, 9, 8, 8, 13, 8, 9, 8, 8, 9, 8, 8, 10, 8, 9, 8

Offset: 1

M. F. Hasler, Oct 18 2007

nonn

This is also the maximal index at which n can occur in a Fibonacci-like sequence u(k+2) = u(k)+u(k+1) of nonnegative numbers.

A sequence of this length is obtained for v(0) = n, v(1) = A019446(n) = ceiling(n/tau) or A060143(n) = floor(n/tau).

			a(2007)=11 since there is no such sequence longer than v = (2007, 1240, 767, 473, 294, 179, 115, 64, 51, 13, 38).

Eric Angelini, Max Alekseyev and M. F. Hasler, Longest Lucas sequence from start to n, postings to Sequence Fans mailing list, Oct 18, 2007.

Cf. A101803, A019446, A060143.

A134299( goal, mi=0, mx=0, new=0 ) = { for( j=mi,goal, a=[goal,new=j]; while( mi<=new=a[ #a-1]-new, a=concat(a,new)); if( #a>mx, mx=#a)); mx }

A192002 Counting sequence for Wythoff AB-numbers smaller than n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 21, 21, 21

Offset: 1

Wolfdieter Lang, Jun 28 2011

a(n) is the number of Wythoff AB-numbers from A003623 which are less than n.
a(n) is also the number of Wythoff A-pairs (two consecutive numbers which are both Wythoff A-numbers) not exceeding n.
a(n) is also the number of Wythoff BA-numbers (including 2 = B(A(1)) which however has Wythoff representation 0 for B(1)) not exceeding n-2. From the identity B(A(n)) = A(B(n)) - 1.
For the Wythoff representation of numbers see A189921 and A135817.

			a(9) = 2 = A(10) + A(9) - (3*9+1) = 16 + 14 - 28.
a(9) = 2 = z(9) - z(8) - 9 = 6 + 5 - 9.
There are a(9)=2 AB-numbers <9, namely 3=A(B(1)) and 8=A(B(B(1))).
There are a(9)=2 A-pairs <=9, namely 3,4 and 8,9.
There are a(9)=2 BA-numbers <=7, namely 2 (see the comment above) and 7 = B(A(B(1))).

Martin Griffiths, A formula for an infinite family of Fibonacci-word sequences, Fib. Q., 56 (2018), 75-80.

Cf. A000201 (Wythoff A-numbers), A001950 (Wythoff B-numbers), A003623 (Wythoff AB-numbers), A123740.

from math import isqrt
def A192002(n): return (n+isqrt(m:=5*n**2)>>1)+(n+1+isqrt(m+10*n+5)>>1)-3*n-1 # Chai Wah Wu, Aug 10 2022

a(n) = Sum_{j=0..n-1} A123740(j), with A123740(0)=0.
a(n) = A(n+1) + A(n) - (3*n+1), with the Wythoff A-numbers A000201.
a(n) = z(n) + z(n-1) - n, with z(n) = A005206(n) = A060143(n+1) which counts A-numbers <=n.
Note that no floor function definitions are necessary.
A(n) (which is as Beatty sequence also floor(n*phi), with phi=(1+sqrt(5))/2) can be defined from the rabbit sequence A005614(n-1), n>=1, which results from a substitution rule, via z(n) by A(n):= z(n-1) + n, B(n):= A(n) + n.
a(n) = floor(n/phi) - floor((1+n)/(1+phi)). - Frank Ruskey, Nov 30 2011

A384947 Positive integers m for which A183136(m) != f(m), where f(m) = floor( (m(m+1)/2)/phi - m/2 + 1/(2phi) ) and phi = (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

15, 18, 36, 39, 41, 47, 49, 52, 91, 94, 96, 102, 103, 104, 107, 109, 123, 125, 128, 130, 136, 138, 141, 235, 238, 240, 246, 247, 248, 251, 252, 253, 267, 268, 269, 272, 273, 274, 277, 280, 281, 282, 285, 287, 303, 306, 322, 324, 327, 328

Offset: 1

Hoang Xuan Thanh, Jun 13 2025

nonn

f(m) is an approximation to A183136(m) = Sum_{k=1..m} floor(k/phi) based on assuming the floor in each term decreases it by 1/2 from what is otherwise a triangular sum; and further offset + 1/(2*phi) in f(m) chosen to improve the accuracy of this approximation.

The actual values of frac(k/phi) can differ from 1/2 each by a net amount which is enough to make m a term of this sequence.

			41 is term, because A183136(41) = 512 != 511 = floor(((41^2+1)*phi - 41) / (2*phi^2)).

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Hoang Xuan Thanh)

Cf. A183136, A001622, A060143.

PositionIndex[MapIndexed[# != Floor[PolygonalNumber[#2[[1]]]/GoldenRatio - #2[[1]]/2 + 1/(2*GoldenRatio)] &, Accumulate[Floor[Range[500]/GoldenRatio]]]][True] (* Paolo Xausa, Jun 20 2025 *)

cp's OEIS Frontend

A183137 a(n) = [1/s] + [2/s] + ... + [n/s], where s = (golden ratio)^2 = (3+sqrt(5))/2 and [] = floor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A379663 a(n) is the number of integer-sided triangles whose sides are in geometric progression with smallest side n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A134299 Maximal length of a sequence such that v(0)=n, v(k+2) = v(k)-v(k+1), v(k) >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A192002 Counting sequence for Wythoff AB-numbers smaller than n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

A384947 Positive integers m for which A183136(m) != f(m), where f(m) = floor( (m*(m+1)/2)/phi - m/2 + 1/(2*phi) ) and phi = (1+sqrt(5))/2 is the golden ratio.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A384947 Positive integers m for which A183136(m) != f(m), where f(m) = floor( (m(m+1)/2)/phi - m/2 + 1/(2phi) ) and phi = (1+sqrt(5))/2 is the golden ratio.