A002024 -id:A002024 - cp's OEIS frontend

A072649 n occurs Fibonacci(n) times (cf. A000045).

1, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Antti Karttunen, Jun 02 2002

Number of digits in Zeckendorf-binary representation of n. E.g., the Zeckendorf representation of 12 is 8+3+1, which in binary notation is 10101, which consists of 5 digits. - Clark Kimberling, Jun 05 2004
First position where value n occurs is A000045(n+1), i.e., a(A000045(n)) = n-1, for n >= 2 and a(A000045(n)-1) = n-2, for n >= 3.
This is the number of distinct Fibonacci numbers greater than 0 which are less than or equal to n. - Robert G. Wilson v, Dec 10 2006
The smallest nondecreasing sequence a(n) such that a(Fibonacci(n-1)) = n. - Tanya Khovanova, Jun 20 2007

			1, 1, then F(2) 2's, then F(3) 3's, then F(4) 4's, ..., then F(k) k's, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See page 4.
Popular Computing (Calabasas, CA), A Coding Exercise (from a suggestion by R. W. Hamming), Vol. 5 (No. 54, Sep 1977), p. PC55-18.

Cf. A000045, A095791, A095792.
Cf. A001622 (golden ratio phi), A073010.
Used to construct A003714. Cf. also A002024, A072643, A072648, A072650.
Cf. A131234.
Partial sums: A256966, A256967.

a072649 n = a072649_list !! (n-1)
a072649_list = f 1 where
   f n = (replicate (fromInteger $ a000045 n) n) ++ f (n+1)
-- Reinhard Zumkeller, Jul 04 2011

A072649 := proc(n)
    local j;
    for j from ilog[(1+sqrt(5))/2](n) while combinat[fibonacci](j+1)<=n do
    end do;
    j-1
end proc:
seq(A072649(n), n=1..120);  # Alois P. Heinz, Mar 18 2013

Table[Table[n, {Fibonacci[n]}], {n, 10}] // Flatten (* Robert G. Wilson v, Jan 14 2007 *)
a[n_] := Module[{j}, For[j = Floor@Log[GoldenRatio, n], Fibonacci[j+1] <= n, j++]; j-1];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Nov 17 2022, after Alois P. Heinz *)

a(n) = -1+floor(log(((n+0.2)*sqrt(5)))/log((1+sqrt(5))/2))

a(n)=local(m); if(n<1,0,m=0; until(fibonacci(m)>n,m++); m-2)

from sympy import fibonacci
def a(n):
    if n<1: return 0
    m=0
    while fibonacci(m)<=n: m+=1
    return m-2
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 09 2017

def A072649(n):
    a, b, c = 0, 1, -2
    while a <= n:
        a, b = b, a+b
        c += 1
    return c # Chai Wah Wu, Nov 04 2024
(MIT/GNU Scheme) (define (A072649 n) (let ((b (A072648 n))) (+ -1 b (floor->exact (/ n (A000045 (1+ b))))))) ;; (The implementation below is better)

(define (A072649 n) (if (<= n 3) n (let loop ((k 5)) (if (> (A000045 k) n) (- k 2) (loop (+ 1 k)))))) ;; (Use this with the memoized implementation of A000045 given under that entry. No floating point arithmetic is involved). - Antti Karttunen, Oct 06 2017

G.f.: (Sum_{n>1} x^Fibonacci(n))/(1-x). - Michael Somos, Apr 25 2003
From Hieronymus Fischer, May 02 2007: (Start)
a(n) = floor(log_phi((sqrt(5)*n + sqrt(5*n^2+4))/2)) - 1, where phi is A001622.
a(n) = floor(arcsinh(sqrt(5)*n/2)/log(phi)) - 1.
a(n) = A108852(n) - 2. (End)
a(n) = -1 + floor( log_phi( (n+0.2)*sqrt(5) ) ), where log_phi(x) is the logarithm to the base (1+sqrt(5))/2. - Ralf Stephan, May 14 2007
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Feb 18 2024

Typo fixed by Charles R Greathouse IV, Oct 28 2009

A000194 n appears 2n times, for n >= 1; also nearest integer to square root of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

N. J. A. Sloane

Define the oblong root obrt(x) to be the (larger) solution of y * (y+1) = x; i.e., obrt(x) = sqrt(x+1/4) - 1/2. So obrt(x) is an integer iff x is an oblong number (A002378). Then a(n) = ceiling(obrt(n)). - Franklin T. Adams-Watters, Jun 24 2015
From Wolfdieter Lang, Mar 12 2019: (Start)
The general Pell equation is related to the non-reduced form F(n) = Xvec^T A(n) Xvec = x^2 - D(n)*y^2 with D(n) = A000037(n) (D not a square), Xvec = (x,y)^T (T for transposed) and A(n) = matrix[[1,0], [0,-D(n)]]. The discriminant of F(n) = [1, 0, -D(n)] is 4*D(n).
The first reduced form appears after two applications of an equivalence transformation A' = R^T A R obtained with R = R(t) = matrix([0, -1], [1, t]), namely first with t = 0, leading to the still not reduced form [-D, 0, 1], and then with t = ceiling(f(4*D(n))/2 - 1), where f(4*D(n)) = ceiling(2*sqrt(D(n))). This can be shown to be a(n), which is also D(n) - n, for n >= 1 (see a formula below).
This leads to the reduced form FR(n) = [1, 2*a(n), -(D(n) - a(n)^2)] = [1, 2*a(n), -(n - a(n)*(a(n) - 1))]. Example: n = 5, a(5) = 2: D(5) = 7 and FR(5) = [1, 4, -3].  (End)

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + 3*x^10 + ...

Titu Andreescu, Dorin Andrica, and Zuming Feng, 104 Number Theory Problems, Birkhäuser, 2006, 59-60.
B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 78, Entry 24.

T. D. Noe, Table of n, a(n) for n = 0..1000
Jonathan M. Borwein and others, Nearest Integer Zeta Functions, solution to Problem 10212, The American Mathematical Monthly, Vol. 101, No. 6 (1994), pp. 579-580.
G. Gutin, Problem 913 (BCC20.5), Mediated digraphs, in Research Problems from the 20th British Combinatorial Conference, Discrete Math., 308 (2008), 621-630.
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 2002), 559-564.
Michael Somos, Sequences used for indexing triangular or square arrays.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000037, A002024, A002061, A002378, A053187, A105209, A259351.
Partial sums of A005369.
Cf. A000196 (floor(sqrt(n))), A003059 (ceiling(sqrt(n))).

a000194 n = a000194_list !! (n-1)
a000194_list = concat $ zipWith ($) (map replicate [2,4..]) [1..]
-- Reinhard Zumkeller, Mar 18 2011

Digits := 100; f := n->round(evalf(sqrt(n))); [ seq(f(n), n=0..100) ];
# More efficient:
a := n -> isqrt(n): seq(a(n), n=0..98); # Peter Luschny, Mar 13 2019

A000194[n_] := Floor[(1 + Sqrt[4 n - 3])/2]; (* Enrique Pérez Herrero, Apr 14 2010 *)
Flatten[Table[PadRight[{}, 2 n, n], {n, 10}]] (* Harvey P. Dale, Nov 16 2011 *)
CoefficientList[Series[x QPochhammer[-x^2, x^4] QPochhammer[x^8, x^8]/(1 - x), {x, 0, 50}], x] (* Eric W. Weisstein, Jan 10 2024 *)

{a(n) = ceil( sqrtint(4*n) / 2)}; /* Michael Somos, Feb 11 2004 */

a(n)=(sqrtint(4*n) + 1)\2 \\ Charles R Greathouse IV, Jun 08 2020

apply( {A000194(n)=sqrtint(4*n)\/2}, [0..99]) \\ M. F. Hasler, Jun 22 2024

from math import isqrt
def A000194(n): return (m:=isqrt(n))+int(n-m*(m+1)>=1) # Chai Wah Wu, Jul 30 2022

a(n) = A000037(n) - n.
G.f.: x * f(x^2, x^6)/(1-x) where f(,) is Ramanujan's two-variable theta function. - Michael Somos, May 31 2000
a(n) = a(n - 2*a(n - a(n-1))) + 1. - Benoit Cloitre, Oct 27 2002
a(n+1) = a(n) + A005369(n).
a(n) = floor((1/2)*(1 + sqrt(4*n - 3))). - Zak Seidov, Jan 18 2006
a(n) = A000037(n) - n. - Jaroslav Krizek, Jun 14 2009
a(n) = floor(A027434(n)/2). - Gregory R. Bryant, Apr 17 2013
From Mikael Aaltonen, Jan 17 2015: (Start)
a(n) = floor(sqrt(n) + 1/2).
a(n) = sqrt(A053187(n)). (End)
a(0) = 0, and a(n) = k for k from the closed interval [k^2 - k + 1, k*(k+1)] = [A002061(k), A002378(k)], for k >= 1. See A053187. - Wolfdieter Lang, Mar 12 2019
a(n) = floor(2*sqrt(n)) - floor(sqrt(n)). - Ridouane Oudra, Jun 08 2020
Sum_{n>=1} 1/a(n)^s = 2*zeta(s-1), for s > 2 (Borwein, 1994). - Amiram Eldar, Oct 31 2020

Additional comments from Michael Somos, May 31 2000
Edited by M. F. Hasler, Mar 01 2014
Initial 0 added by N. J. A. Sloane, Nov 13 2017

A204016 Symmetric matrix based on f(i,j) = max(j mod i, i mod j), by antidiagonals.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 2, 0, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 0, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 0, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 0, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 7

Offset: 1

Clark Kimberling, Jan 10 2012

A204016 represents the matrix M given by f(i,j) = max{(j mod i), (i mod j)} for i >= 1 and j >= 1.  See A204017 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
Guide to symmetric matrices M based on functions f(i,j) and characteristic polynomial sequences (c.p.s.) with interlaced zeros:
f(i,j)..........................M.........c.p.s.
C(i+j,j)........................A007318...A045912
min(i,j)........................A003983...A202672
max(i,j)........................A051125...A203989
(i+j)*min(i,j)..................A203990...A203991
|i-j|...........................A049581...A203993
max(i-j+1,j-i+1)................A143182...A203992
min(i-j+1,j-i+1)................A203994...A203995
min(i(j+1),j(i+1))..............A203996...A203997
max(i(j+1)-1,j(i+1)-1)..........A203998...A203999
min(i(j+1)-1,j(i+1)-1)..........A204000...A204001
min(2i+j,i+2j)..................A204002...A204003
max(2i+j-2,i+2j-2)..............A204004...A204005
min(2i+j-2,i+2j-2)..............A204006...A204007
max(3i+j-3,i+3j-3)..............A204008...A204011
min(3i+j-3,i+3j-3)..............A204012...A204013
min(3i-2,3j-2)..................A204028...A204029
1+min(j mod i, i mod j).........A204014...A204015
max(j mod i, i mod j)...........A204016...A204017
1+max(j mod i, i mod j).........A204018...A204019
min(i^2,j^2)....................A106314...A204020
min(2i-1, 2j-1).................A157454...A204021
max(2i-1, 2j-1).................A204022...A204023
min(i(i+1)/2,j(j+1)/2)..........A106255...A204024
gcd(i,j)........................A003989...A204025
gcd(i+1,j+1)....................A204030...A204111
min(F(i+1),F(j+1)),F=A000045....A204026...A204027
gcd(F(i+1),F(j+1)),F=A000045....A204112...A204113
gcd(L(i),L(j)),L=A000032........A204114...A204115
gcd(2^i-1,2^j-2)................A204116...A204117
gcd(prime(i),prime(j))..........A204118...A204119
gcd(prime(i+1),prime(j+1))......A204120...A204121
gcd(2^(i-1),2^(j-1))............A144464...A204122
max(floor(i/j),floor(j/i))......A204123...A204124
min(ceiling(i/j),ceiling(j/i))..A204143...A204144
Delannoy matrix.................A008288...A204135
max(2i-j,2j-i)..................A204154...A204155
-1+max(3i-j,3j-i)...............A204156...A204157
max(3i-2j,3j-2i)................A204158...A204159
floor((i+1)/2)..................A204164...A204165
ceiling((i+1)/2)................A204166...A204167
i+j.............................A003057...A204168
i+j-1...........................A002024...A204169
i*j.............................A003991...A204170
..abbreviation below:  AOE means "all 1's except"
AOE f(i,i)=i....................A204125...A204126
AOE f(i,i)=A000045(i+1).........A204127...A204128
AOE f(i,i)=A000032(i)...........A204129...A204130
AOE f(i,i)=2i-1.................A204131...A204132
AOE f(i,i)=2^(i-1)..............A204133...A204134
AOE f(i,i)=3i-2.................A204160...A204161
AOE f(i,i)=floor((i+1)/2).......A204162...A204163
...
Other pairs (M, c.p.s.): (A204171, A204172) to (A204183, A204184)
See A202695 for a guide to choices of symmetric matrix M for which the zeros of the characteristic polynomials are all positive.

			Northwest corner:
  0 1 1 1 1 1 1 1
  0 1 2 2 2 2 2 2
  1 2 0 3 3 3 3 3
  1 2 3 0 4 4 4 4
  1 2 3 4 0 5 5 5
  1 2 3 4 5 0 6 6
  1 2 3 4 5 6 0 7

Cf. A204017, A202453.

f[i_, j_] := Max[Mod[i, j], Mod[j, i]];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]  (* A204016 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]               (* A204017 *)
TableForm[Table[c[n], {n, 1, 10}]]

A003057 n appears n - 1 times.

Original entry on oeis.org

2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14

Offset: 2

N. J. A. Sloane

The PARI functions t1, t2 can be used to read a triangular array T(n,k) (n >= 2, 1 <= k <= n - 1) by rows from left to right: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002
Smallest integer such that n-1 <= C(a(n),2). - Frank Ruskey, Nov 06 2007
a(n) = inverse (frequency distribution) sequence of A161680. - Jaroslav Krizek, Jun 19 2009
Taken as a triangle t(n, m) with offset 1, i.e., n >= m >= 1, this gives all positive integer limits r = r (a = m, b = A063929(n, m)) of the (a,b)-Fibonacci ratio F(a,b;k+1)/F(a,b;k) for k -> infinity. See the Jan 11 2015 comment on A063929. - Wolfdieter Lang, Jan 12 2015
Square array, T(n,k) = n + k + 2, n > = 0 and k >= 0, read by antidiagonals. Northwest corner:
  2, 3, 4, 5, ...
  3, 4, 5, 6, ...
  4, 5, 6, 7, ...
  5, 6, 7, 8, ...
  ... - Franck Maminirina Ramaharo, Nov 21 2018
a(n) is the pair chromatic number of an empty graph with n vertices.  (The pair chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of colors is repeated.) - Allan Bickle, Dec 26 2021

			(a,b)-Fibonacci ratio limits r(a,b) (see a comment above): as a triangle with offset 1 one has t(3, m) = 4 for m = 1, 2, 3. This gives the limits r(a = m,b = A063929(3, m)), i.e., r(1,12) = r(2,8) = r(3,4) = 4 (and the limit 4 appears only for these three (a,b) values). - _Wolfdieter Lang_, Jan 12 2015

Vincenzo Librandi, Table of n, a(n) for n = 2..10000
Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013), 171-190.
Michael Somos, Sequences used for indexing triangular or square arrays

Cf. A002024, A002260.

[Round(Sqrt(2*(n-1)))+1: n in [2..60]]; // Vincenzo Librandi, Jun 23 2011

seq(n$(n-1),n=2..15); # Robert Israel, Jan 12 2015

Flatten[Table[PadRight[{},n-1,n],{n,15}]] (* Harvey P. Dale, Feb 26 2012 *)

t1(n)=floor(3/2+sqrt(2*n-2)) /* A003057 */

t2(n)=n-1-binomial(floor(1/2+sqrt(2*n-2)),2) /* A002260(n-2) */

from math import isqrt
def A003057(n): return (k:=isqrt(m:=n-1<<1))+int((m<<2)>(k<<2)*(k+1)+1)+1 # Chai Wah Wu, Jul 26 2022

a(n) = A002260(n) + A004736(n).
a(n) = A002024(n-1) + 1 = floor(sqrt(2*(n - 1)) + 1/2) + 1 = round(sqrt(2*(n - 1))) + 1. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 21 2003
a(n) = ceiling((sqrt(8*n - 7) + 1)/2). - Reinhard Zumkeller, Aug 28 2001, modified by Frank Ruskey, Nov 06 2007, restored by M. F. Hasler, Jan 13 2015
a(n) = A080036(n-1) - (n - 1) for n >= 2. - Jaroslav Krizek, Jun 19 2009
G.f.: (2*x^2 + Sum_{n>=2} x^(n*(n - 1)/2 + 2))/(1 - x) = (x^2 + x^(15/8)*theta_2(0,sqrt(x))/2)/(1 - x) where theta_2 is the second Jacobi theta function. - Robert Israel, Jan 12 2015

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 21 2003

A362614 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k modes.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 1, 0, 5, 2, 0, 7, 3, 1, 0, 11, 3, 1, 0, 16, 4, 2, 0, 21, 6, 3, 0, 29, 8, 4, 1, 0, 43, 7, 5, 1, 0, 54, 13, 8, 2, 0, 78, 12, 8, 3, 0, 102, 17, 11, 5, 0, 131, 26, 12, 6, 1, 0, 175, 29, 17, 9, 1, 0, 233, 33, 18, 11, 2, 0, 295, 47, 25

Offset: 0

Gus Wiseman, May 04 2023

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			Triangle begins:
   1
   0   1
   0   2
   0   2   1
   0   4   1
   0   5   2
   0   7   3   1
   0  11   3   1
   0  16   4   2
   0  21   6   3
   0  29   8   4   1
   0  43   7   5   1
   0  54  13   8   2
   0  78  12   8   3
   0 102  17  11   5
   0 131  26  12   6   1
   0 175  29  17   9   1
Row n = 8 counts the following partitions:
  (8)         (53)    (431)
  (44)        (62)    (521)
  (332)       (71)
  (422)       (3311)
  (611)
  (2222)
  (3221)
  (4211)
  (5111)
  (22211)
  (32111)
  (41111)
  (221111)
  (311111)
  (2111111)
  (11111111)

Alois P. Heinz, Rows n = 0..800, flattened

Row sums are A000041.
Row lengths are A002024.
Removing columns 0 and 1 and taking sums gives A362607, ranks A362605.
Column k = 1 is A362608, ranks A356862.
This statistic (mode-count) is ranked by A362611.
For co-modes we have A362615, ranked by A362613.
A008284 counts partitions by length.
A096144 counts partitions by number of minima, A026794 by maxima.
A238342 counts compositions by number of minima, A238341 by maxima.
A275870 counts collapsible partitions.
Cf. A002865, A003056, A098859, A240219, A359893, A360071, A362609, A362610, A362612, A372542.

msi[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],Length[msi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]

Sum_{k=0..A003056(n)} k * T(n,k) = A372542. - Alois P. Heinz, May 05 2024

A050873 Triangular array T read by rows: T(n,k) = gcd(n,k).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 2, 1, 4, 1, 1, 1, 1, 5, 1, 2, 3, 2, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Clark Kimberling

The function T(n,k) = T(k,n) is defined for all integer k,n but only the values for 1 <= k <= n as a triangular array are listed here.
For each divisor d of n, the number of d's in row n is phi(n/d). Furthermore, if {a_1, a_2, ..., a_phi(n/d)} is the set of positive integers <= n/d that are relatively prime to n/d then T(n,a_i * d) = d. - Geoffrey Critzer, Feb 22 2015
Starting with any row n and working downwards, consider the infinite rectangular array with k = 1..n. A repeating pattern occurs every A003418(n) rows. For example, n=3: A003418(3) = 6. The 6-row pattern starting with row 3 is {1,1,3}, {1,2,1}, {1,1,1}, {1,2,3}, {1,1,1}, {1,2,1}, and this pattern repeats every 6 rows, i.e., starting with rows {9,15,21,27,...}. - Bob Selcoe and Jamie Morken, Aug 02 2017

			Rows:
  1;
  1, 2;
  1, 1, 3;
  1, 2, 1, 4;
  1, 1, 1, 1, 5;
  1, 2, 3, 2, 1, 6; ...

T. D. Noe, Rows n=1..100, flattened
Marcelo Polezzi, A Geometrical Method for Finding an Explicit Formula for the Greatest Common Divisor, The American Mathematical Monthly, Vol. 104, No. 5 (May, 1997), pp. 445-446.
Eric Weisstein's World of Mathematics, Greatest Common Divisor
Wikipedia, Greatest Common Divisor

Cf. A003989.
Cf. A002262, A054531, A226314.
Cf. A018804 (row sums), A245717.
Cf. A132442 (sums of divisors).
Cf. A003418.

a050873 = gcd
a050873_row n = a050873_tabl !! (n-1)
a050873_tabl = zipWith (map . gcd ) [1..] a002260_tabl
-- Reinhard Zumkeller, Dec 12 2015, Aug 13 2013, Jun 10 2013

ColumnForm[Table[GCD[n, k], {k, 12}, {n, k}], Center] (* Alonso del Arte, Jan 14 2011 *)

{T(n, k) = gcd(n, k)} /* Michael Somos, Jul 18 2011 */

a(n) = gcd(A002260(n), A002024(n)); A054521(n) = A000007(a(n)). - Reinhard Zumkeller, Dec 02 2009
T(n,k) = A075362(n,k)/A051173(n,k), 1 <= k <= n. - Reinhard Zumkeller, Apr 25 2011
T(n, k) = T(k, n) = T(-n, k) = T(n, -k) = T(n, n+k) = T(n+k, k). - Michael Somos, Jul 18 2011
T(n,k) = A051173(n,k) / A051537(n,k). - Reinhard Zumkeller, Jul 07 2013

A014132 Complement of triangular numbers (A000217); also array T(n,k) = ((n+k)^2 + n-k)/2, n, k > 0, read by antidiagonals.

Original entry on oeis.org

2, 4, 5, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79

Offset: 1

N. J. A. Sloane

Numbers that are not triangular (nontriangular numbers).
Also definable as follows: a(1)=2; for n>1, a(n) is smallest integer greater than a(n-1) such that the condition "n and a(a(n)) have opposite parities" can always be satisfied. - Benoit Cloitre and Matthew Vandermast, Mar 10 2003
Record values in A256188 that are greater than 1. - Reinhard Zumkeller, Mar 26 2015
From Daniel Forgues, Apr 10 2015: (Start)
With n >= 1, k >= 1:
  t(n+k) - k, 1 <= k <= n+k-1, n >= 1;
  t(n+k-1) + n, 1 <= n <= n+k-1, k >= 1;
where t(n+k) = t(n+k-1) + (n+k) is the (n+k)-th triangular number, while the number of compositions of n+k into 2 parts is C(n+k-1, 2-1) = n+k-1, the number of nontriangular numbers between t(n+k-1) and t(n+k), just right!
Related to Hilbert's Infinite Hotel:
0) All rooms, numbered through the positive integers, are full;
1) An infinite number of trains, each containing an infinite number of passengers, arrives: i.e., a 2-D lattice of pairs of positive integers;
2) Move occupant of room m, m >= 1, to room t(m) = m*(m+1)/2, where t(m) is the m-th triangular number;
3) Assign n-th passenger from k-th train to room t(n+k-1) + n, 1 <= n <= n+k-1, k >= 1;
4) Everybody has his or her own room, no room is empty, for m >= 1.
If situation 1 happens again, repeat steps 2 and 3, you're back to 4.
(End)
1711 + 2*a(n)*(58 + a(n)) is prime for n<=21. The terms that do not have this property start 29,32,34,43,47,58,59,60,62,63,65,68,70,73,...  - Benedict W. J. Irwin, Nov 22 2016
Also numbers k with the property that in the symmetric representation of sigma(k) both Dyck paths have a central peak or both Dyck paths have a central valley. (Cf. A237593.) - Omar E. Pol, Aug 28 2018

			From _Boris Putievskiy_, Jan 14 2013: (Start)
Start of the sequence as a table (read by antidiagonals, right to left), where the k-th row corresponds to the k-th column of the triangle (shown thereafter):
   2,  4,  7, 11, 16, 22, 29, ...
   5,  8, 12, 17, 23, 30, 38, ...
   9, 13, 18, 24, 31, 39, 48, ...
  14, 19, 25, 32, 40, 49, 59, ...
  20, 26, 33, 41, 50, 60, 71, ...
  27, 34, 42, 51, 61, 72, 84, ...
  35, 43, 52, 62, 73, 85, 98, ...
  (...)
Start of the sequence as a triangle (read by rows), where the i elements of the i-th row are t(i) + 1 up to t(i+1) - 1, i >= 1:
   2;
   4,  5;
   7,  8,  9;
  11, 12, 13, 14;
  16, 17, 18, 19, 20;
  22, 23, 24, 25, 26, 27;
  29, 30, 31, 32, 33, 34, 35;
  (...)
Row number i contains i numbers, where t(i) = i*(i+1)/2:
  t(i) + 1, t(i) + 2, ..., t(i) + i = t(i+1) - 1
(End) [Edited by _Daniel Forgues_, Apr 11 2015]

T. D. Noe, Table of n, a(n) for n = 1..1000
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence arXiv:math/0305308 [math.NT], 2003.
Bakir Farhi, An explicit formula generating the non-Fibonacci numbers, arXiv:1105.1127 [math.NT], May 05 2011. See Example 5 p. 456.
J. Lambek and L. Moser, Inverse and complementary sequences of natural numbers, Amer. Math. Monthly, 61 (1954), 454-458.
Cristinel Mortici, Remarks on Complementary Sequences, Fibonacci Quart. 48 (2010), no. 4, 343-347.
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A000217, A006002, A035214, A080036, A002024, A007401, A003057, A114327, A002260, A004736, A118011, A237593.
Cf. A000124 (left edge: quasi-triangular numbers), A000096 (right edge: almost-triangular numbers), A006002 (row sums), A001105 (central terms).
Cf. A242401 (subsequence).
Cf. A004202, A256188.
Cf. A145397 (the non-tetrahedral numbers).

a014132 n = n + round (sqrt $ 2 * fromInteger n)
a014132_list = filter ((== 0) . a010054) [0..]
-- Reinhard Zumkeller, Dec 12 2012

IsTriangular:=func< n | exists{ k: k in [1..Isqrt(2*n)] | n eq (k*(k+1) div 2)} >; [ n: n in [1..90] | not IsTriangular(n) ]; // Klaus Brockhaus, Jan 04 2011

f[n_] := n + Round[Sqrt[2n]]; Array[f, 71] (* or *)
Complement[ Range[83], Array[ #(# + 1)/2 &, 13]] (* Robert G. Wilson v, Oct 21 2005 *)
DeleteCases[Range[80],?(OddQ[Sqrt[8#+1]]&)] (* _Harvey P. Dale, Jul 24 2021 *)

PARI
```
a(n)=if(n<1,0,n+(sqrtint(8*n-7)+1)\2)
```

isok(n) = !ispolygonal(n,3); \\ Michel Marcus, Mar 01 2016

from math import isqrt
def A014132(n): return n+(isqrt((n<<3)-7)+1>>1) # Chai Wah Wu, Jun 17 2024

a(n) = n + round(sqrt(2*n)).
a(a(n)) = n + 2*floor(1/2 + sqrt(2n)) + 1.
a(n) = a(n-1) + A035214(n), a(1)=2.
a(n) = A080036(n) - 1.
a(n) = n + A002024(n). - Vincenzo Librandi, Jul 08 2010
A010054(a(n)) = 0. - Reinhard Zumkeller, Dec 10 2012
From Boris Putievskiy, Jan 14 2013: (Start)
a(n) = A007401(n)+1.
a(n) = A003057(n)^2 - A114327(n).
a(n) = ((t+2)^2 + i - j)/2, where
i = n-t*(t+1)/2,
j = (t*t+3*t+4)/2-n,
t = floor((-1+sqrt(8*n-7))/2). (End)
A248952(a(n)) < 0. - Reinhard Zumkeller, Oct 20 2014
a(n) = A256188(A004202(n)). - Reinhard Zumkeller, Mar 26 2015
From Robert Israel, Apr 20 2015 (Start):
a(n) = A118011(n) - n.
G.f.: x/(1-x)^2 + x/(1-x) * Sum(j>=0, x^(j*(j+1)/2)) = x/(1-x)^2 + x^(7/8)/(2-2*x) * Theta2(0,sqrt(x)), where Theta2 is a Jacobi theta function. (End)
G.f. as array: x*y*(2 - 2*y + x^2*y + y^2 - x*(1 + y))/((1 - x)^3*(1 - y)^3). - Stefano Spezia, Apr 22 2024

Following Alford Arnold's comment: keyword tabl and correspondent crossrefs added by Reinhard Zumkeller, Dec 12 2012

I restored the original definition. - N. J. A. Sloane, Jan 27 2019

A053141 a(0)=0, a(1)=2 then a(n) = a(n-2) + 2sqrt(8a(n-1)^2 + 8*a(n-1) + 1).

Original entry on oeis.org

0, 2, 14, 84, 492, 2870, 16730, 97512, 568344, 3312554, 19306982, 112529340, 655869060, 3822685022, 22280241074, 129858761424, 756872327472, 4411375203410, 25711378892990, 149856898154532, 873430010034204, 5090723162050694, 29670908962269962, 172934730611569080

Offset: 0

Wolfdieter Lang

Solution to b(b+1) = 2a(a+1) in natural numbers including 0; a = a(n), b = b(n) = A001652(n).
The solution of a special case of a binomial problem of H. Finner and K. Strassburger (strass(AT)godot.dfi.uni-duesseldorf.de).
Also the indices of triangular numbers that are half other triangular numbers [a of T(a) such that 2T(a)=T(b)]. The T(a)'s are in A075528, the T(b)'s are in A029549 and the b's are in A001652. - Bruce Corrigan (scentman(AT)myfamily.com), Oct 30 2002
Sequences A053141 (this entry), A016278, A077259, A077288 and A077398 are part of an infinite series of sequences. Each depends upon the polynomial p(n) = 4k*n^2 + 4k*n + 1, when 4k is not a perfect square. Equivalently, they each depend on the equation k*t(x)=t(z) where t(n) is the triangular number formula n(n+1)/2. The dependencies are these: they are the sequences of positive integers n such that p(n) is a perfect square and there exists a positive integer m such that k*t(n)=t(m). A053141 is for k=2, A016278 is for k=3, A077259 is for k=5. - Robert Phillips (bobanne(AT)bellsouth.net), Oct 11 2007, Nov 27 2007
Jason Holt observes that a pair drawn from a drawer with A053141(n)+1 red socks and A001652(n) - A053141(n) blue socks will as likely as not be matching reds: (A053141+1)*A053141/((A001652+1)*A001652) = 1/2, n>0. - Bill Gosper, Feb 07 2010
The values x(n)=A001652(n), y(n)=A046090(n) and z(n)=A001653(n) form a nearly isosceles Pythagorean triple since y(n)=x(n)+1 and x(n)^2 + y(n)^2 = z(n)^2; e.g., for n=2, 20^2 + 21^2 = 29^2. In a similar fashion, if we define b(n)=A011900(n) and c(n)=A001652(n), a(n), b(n) and c(n) form a nearly isosceles anti-Pythagorean triple since b(n)=a(n)+1 and a(n)^2 + b(n)^2 = c(n)^2 + c(n) + 1; i.e., the value a(n)^2 + b(n)^2 lies almost exactly between two perfect squares; e.g., 2^2 + 3^2 = 13 = 4^2 - 3 = 3^2 + 4; 14^2 + 15^2 = 421 = 21^2 - 20 = 20^2 + 21. - Charlie Marion, Jun 12 2009
Behera & Panda call these the balancers and A001109 are the balancing numbers. - Michel Marcus, Nov 07 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Counting families of generalized balancing numbers, The Australasian Journal of Combinatorics (2020) Vol. 77, Part 3, 318-325.
A. Behera and G. K. Panda, On the Square Roots of Triangular Numbers, Fib. Quart., 37 (1999), pp. 98-105.
Martin V. Bonsangue, Gerald E. Gannon and Laura J. Pheifer, Misinterpretations can sometimes be a good thing, Math. Teacher, vol. 95, No. 6 (2002) pp. 446-449.
P. Catarino, H. Campos, and P. Vasco, On some identities for balancing and cobalancing numbers, Annales Mathematicae et Informaticae, 45 (2015) pp. 11-24.
Refik Keskin and Olcay Karaatli, Some New Properties of Balancing Numbers and Square Triangular Numbers, Journal of Integer Sequences, Vol. 15 (2012), Article #12.1.4.
aBa Mbirika, Janee Schrader, and Jürgen Spilker, Pell and Associated Pell Braid Sequences as GCDs of Sums of k Consecutive Pell, Balancing, and Related Numbers, J. Int. Seq. (2023) Vol. 26, Art. 23.6.4.
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.
G. K. Panda, Sequence balancing and cobalancing numbers, Fib. Q., Vol. 45, No. 3 (2007), 265-271. See p. 266.
Michael Penn, (co) balancing numbers, YouTube video, 2022.
Robert Phillips, Polynomials of the form 1+4ke+4ke^2, 2008.
Robert Phillips, A triangular number result, 2009.
Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 2021.
Vladimir Pletser, Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2102.12392 [math.GM], 2021.
Vladimir Pletser, Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations, arXiv:2102.13494 [math.NT], 2021.
Vladimir Pletser, Congruence Properties of Indices of Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2103.03019 [math.GM], 2021.
Vladimir Pletser, Searching for multiple of triangular numbers being triangular numbers, 2021.
Vladimir Pletser, Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers, 2021.
Burkard Polster, Nice merging together, Mathologer video (2015).
B. Polster and M. Ross, Marching in squares, arXiv preprint arXiv:1503.04658 [math.HO], 2015.
A. Tekcan, M. Tayat, and M. E. Ozbek, The diophantine equation 8x^2-y^2+8x(1+t)+(2t+1)^2=0 and t-balancing numbers, ISRN Combinatorics, Volume 2014, Article ID 897834, 5 pages.
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A000129, A001108, A001109, A001652, A001653.
Cf. A011900, A029549, A053142, A075528, A103200.
Partial sums of A001542.
Cf. A000194, A001541, A002024, A016278, A046090, A055997, A077259, A077288, A077398, A084068.

a053141 n = a053141_list !! n
a053141_list = 0 : 2 : map (+ 2)
   (zipWith (-) (map (* 6) (tail a053141_list)) a053141_list)
-- Reinhard Zumkeller, Jan 10 2012

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!(2*x/((1-x)*(1-6*x+x^2)))); // G. C. Greubel, Jul 15 2018

A053141 := proc(n)
    option remember;
    if n <= 1 then
        op(n+1,[0,2]) ;
    else
        6*procname(n-1)-procname(n-2)+2 ;
    end if;
end proc: # R. J. Mathar, Feb 05 2016

Join[{a=0,b=1}, Table[c=6*b-a+1; a=b; b=c, {n,60}]]*2 (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
a[n_] := Floor[1/8*(2+Sqrt[2])*(3+2*Sqrt[2])^n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 28 2013 *)
Table[(Fibonacci[2n + 1, 2] - 1)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)

concat(0,Vec(2/(1-x)/(1-6*x+x^2)+O(x^30))) \\ Charles R Greathouse IV, May 14 2012

{x=1+sqrt(2); y=1-sqrt(2); P(n) = (x^n - y^n)/(x-y)};
a(n) = round((P(2*n+1) - 1)/2);
for(n=0, 30, print1(a(n), ", ")) \\ G. C. Greubel, Jul 15 2018

[(lucas_number1(2*n+1, 2, -1)-1)/2 for n in range(30)] # G. C. Greubel, Apr 27 2020

a(n) = (A001653(n)-1)/2 = 2*A053142(n) = A011900(n)-1. [Corrected by Pontus von Brömssen, Sep 11 2024]
a(n) = 6*a(n-1) - a(n-2) + 2, a(0) = 0, a(1) = 2.
G.f.: 2*x/((1-x)*(1-6*x+x^2)).
Let c(n) = A001109(n). Then a(n+1) = a(n)+2*c(n+1), a(0)=0. This gives a generating function (same as existing g.f.) leading to a closed form: a(n) = (1/8)*(-4+(2+sqrt(2))*(3+2*sqrt(2))^n + (2-sqrt(2))*(3-2*sqrt(2))^n). - Bruce Corrigan (scentman(AT)myfamily.com), Oct 30 2002
a(n) = 2*Sum_{k = 0..n} A001109(k). - Mario Catalani (mario.catalani(AT)unito.it), Mar 22 2003
For n>=1, a(n) = 2*Sum_{k=0..n-1} (n-k)*A001653(k). - Charlie Marion, Jul 01 2003
For n and j >= 1, A001109(j+1)*A001652(n) - A001109(j)*A001652(n-1) + a(j) = A001652(n+j). - Charlie Marion, Jul 07 2003
From Antonio Alberto Olivares, Jan 13 2004: (Start)
a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3).
a(n) = -(1/2) - (1-sqrt(2))/(4*sqrt(2))*(3-2*sqrt(2))^n + (1+sqrt(2))/(4*sqrt(2))*(3+2*sqrt(2))^n. (End)
a(n) = sqrt(2)*cosh((2*n+1)*log(1+sqrt(2)))/4 - 1/2 = (sqrt(1+4*A029549)-1)/2. - Bill Gosper, Feb 07 2010 [typo corrected by Vaclav Kotesovec, Feb 05 2016]
a(n+1) + A055997(n+1) = A001541(n+1) + A001109(n+1). - Creighton Dement, Sep 16 2004
From Charlie Marion, Oct 18 2004: (Start)
For n>k, a(n-k-1) = A001541(n)*A001653(k)-A011900(n+k); e.g., 2 = 99*5 - 493.
For n<=k, a(k-n) = A001541(n)*A001653(k) - A011900(n+k); e.g., 2 = 3*29 - 85 + 2. (End)
a(n) = A084068(n)*A084068(n+1). - Kenneth J Ramsey, Aug 16 2007
Let G(n,m) = (2*m+1)*a(n)+ m and H(n,m) = (2*m+1)*b(n)+m where b(n) is from the sequence A001652 and let T(a) = a*(a+1)/2. Then T(G(n,m)) + T(m) = 2*T(H(n,m)). - Kenneth J Ramsey, Aug 16 2007
Let S(n) equal the average of two adjacent terms of G(n,m) as defined immediately above and B(n) be one half the difference of the same adjacent terms. Then for T(i) = triangular number i*(i+1)/2, T(S(n)) - T(m) = B(n)^2 (setting m = 0 gives the square triangular numbers). - Kenneth J Ramsey, Aug 16 2007
a(n) = A001108(n+1) - A001109(n+1). - Dylan Hamilton, Nov 25 2010
a(n) = (a(n-1)*(a(n-1) - 2))/a(n-2) for n > 2. - Vladimir Pletser, Apr 08 2020
a(n) = (ChebyshevU(n, 3) - ChebyshevU(n-1, 3) - 1)/2 = (Pell(2*n+1) - 1)/2. - G. C. Greubel, Apr 27 2020
E.g.f.: (exp(3*x)*(2*cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)) - 2*exp(x))/4. - Stefano Spezia, Mar 16 2024
a(n) = A000194(A029549(n)) = A002024(A075528(n)). - Pontus von Brömssen, Sep 11 2024

Name corrected by Zak Seidov, Apr 11 2011

A362615 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k co-modes.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 1, 0, 5, 2, 0, 7, 3, 1, 0, 10, 4, 1, 0, 13, 7, 2, 0, 16, 11, 3, 0, 23, 14, 4, 1, 0, 30, 19, 6, 1, 0, 35, 29, 11, 2, 0, 50, 34, 14, 3, 0, 61, 46, 23, 5, 0, 73, 69, 27, 6, 1, 0, 95, 81, 44, 10, 1, 0, 123, 105, 53, 14, 2

Offset: 0

Gus Wiseman, May 04 2023

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

			Triangle begins:
   1
   0   1
   0   2
   0   2   1
   0   4   1
   0   5   2
   0   7   3   1
   0  10   4   1
   0  13   7   2
   0  16  11   3
   0  23  14   4   1
   0  30  19   6   1
   0  35  29  11   2
   0  50  34  14   3
   0  61  46  23   5
   0  73  69  27   6   1
   0  95  81  44  10   1
Row n = 8 counts the following partitions:
  (8)         (53)     (431)
  (44)        (62)     (521)
  (332)       (71)
  (422)       (3221)
  (611)       (3311)
  (2222)      (4211)
  (5111)      (32111)
  (22211)
  (41111)
  (221111)
  (311111)
  (2111111)
  (11111111)

Alois P. Heinz, Rows n = 0..800, flattened

Row sums are A000041.
Row lengths are A002024.
Removing columns 0 and 1 and taking sums gives A362609, ranks A362606.
Column k = 1 is A362610, ranks A359178.
This statistic (co-mode count) is ranked by A362613.
For mode instead of co-mode we have A362614, ranked by A362611.
A008284 counts partitions by length.
A096144 counts partitions by number of minima, A026794 by maxima.
A238342 counts compositions by number of minima, A238341 by maxima.
A275870 counts collapsible partitions.
Cf. A003056, A098859, A325347, A359893, A362607, A362608, A362612, A372632.

comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],Length[comsi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]

Sum_{k=0..A003056(n)} k * T(n,k) = A372632(n). - Alois P. Heinz, May 07 2024

A038567 Denominators in canonical bijection from positive integers to positive rationals <= 1.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16

Offset: 0

N. J. A. Sloane

n occurs phi(n) times (cf. A000010).
Least k such that phi(1) + phi(2) + phi(3) + ... + phi(k) >= n. - Benoit Cloitre, Sep 17 2002
Sum of numerator and denominator of fractions arranged by Cantor's ordering (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, ...) with equivalent fractions removed. - Ron R. King, Mar 07 2009 [This applies to a(1, 2, ...) without initial term a(0) = 1 which could correspond to 0/1. - Editor's Note.]
Care has to be taken in considering the offset which may be 0 or 1 in related sequences (see crossrefs), e.g., A038568 & A038569 also have offset 0, in A038566 offset has been changed to 1. - M. F. Hasler, Oct 18 2021

			Arrange fractions by increasing denominator then by increasing numerator: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, ...: this is A038566/A038567.

S. Cook, Problem 511: An Enumeration Problem, Journal of Recreational Mathematics, Vol. 9:2 (1976-77), 137. Solution by the Problem Editor, JRM, Vol. 10:2 (1977-78), 122-123.
Hans Lauwerier, Fractals, Princeton University Press, 1991, p. 23.

David Wasserman, Table of n, a(n) for n = 0..100000
Paul Yiu, Recreational Mathematics, 24.3.1 Appendix: Two enumerations of the rational numbers in (0,1), page 633. [Wayback Machine link]
Index entries for "core" sequences.
Index entries for sequences related to enumerating the rationals.
Index entries for sequences related to Stern's sequences.

Cf. A020652, A020653, A038566 - A038569, A093602, A182972, A182973 - A182976.
A054427 gives mapping to Stern-Brocot tree.
Cf. A037162.

import Data.List (genericTake)
a038567 n = a038567_list !! n
a038567_list = concatMap (\x -> genericTake (a000010 x) $ repeat x) [1..]
-- Reinhard Zumkeller, Dec 16 2013, Jul 29 2012

with (numtheory): A038567 := proc (n) local sum, k; sum := 1: k := 2: while (sum < n) do: sum := sum + phi(k): k := k + 1: od: RETURN (k-1): end: # Ulrich Schimke (ulrschimke(AT)aol.com)

a[n_] := (k = 0; While[ Total[ EulerPhi[ Range[k]]] <= n, k++]; k); Table[ a[n], {n, 0, 77}] (* Jean-François Alcover, Dec 08 2011, after Pari *)
Flatten[Table[Table[n,{EulerPhi[n]}],{n,20}]] (* Harvey P. Dale, Mar 12 2013 *)

a(n)=if(n<0,0,s=1; while(sum(i=1,s,eulerphi(i))

from sympy import totient
def a(n):
    s=1
    while sum(totient(i) for i in range(1, s + 1))Indranil Ghosh, May 23 2017

from functools import lru_cache
@lru_cache(maxsize=None)
def A002088(n): # based on second formula in A018805
    if n == 0:
        return 0
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*((A002088(k1)<<1)-1)
        j, k1 = j2, n//j2
    return n*(n-1)-c+j>>1
def A038567(n):
    kmin, kmax = 0, 1
    while A002088(kmax) <= n:
        kmax <<= 1
    kmin = kmax>>1
    while True:
        kmid = kmax+kmin>>1
        if A002088(kmid) > n:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Jun 10 2025

From Henry Bottomley, Dec 18 2000: (Start)
a(n) = A020652(n) + A020653(n) for all n > 0, e.g., a(1) = 2 = 1 + 1 = A020652(1) + A020653(1). [Corrected and edited by M. F. Hasler, Dec 10 2021]
n = a(A015614(n)) = a(A002088(n)) - 1 = a(A002088(n-1)). (End)
a(n) = A002024(A169581(n)). - Reinhard Zumkeller, Dec 02 2009
a(A002088(n)) = n for n > 1. - Reinhard Zumkeller, Jul 29 2012
a(n) = A071912(2*n+1). - Reinhard Zumkeller, Dec 16 2013
a(n) ~ c * sqrt(n), where c = Pi/sqrt(3) = 1.813799... (A093602). - Amiram Eldar, Dec 27 2024

More terms from Erich Friedman

cp's OEIS Frontend

A072649 n occurs Fibonacci(n) times (cf. A000045).

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A000194 n appears 2n times, for n >= 1; also nearest integer to square root of n.

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A204016 Symmetric matrix based on f(i,j) = max(j mod i, i mod j), by antidiagonals.

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A003057 n appears n - 1 times.

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A362614 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k modes.

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A050873 Triangular array T read by rows: T(n,k) = gcd(n,k).

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A053141 a(0)=0, a(1)=2 then a(n) = a(n-2) + 2sqrt(8a(n-1)^2 + 8*a(n-1) + 1).