A108872 -id:A108872 - cp's OEIS frontend

A002260 Triangle read by rows: T(n,k) = k for n >= 1, k = 1..n.

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

Offset: 1

Angele Hamel (amh(AT)maths.soton.ac.uk)

Old name: integers 1 to k followed by integers 1 to k+1 etc. (a fractal sequence).
Start counting again and again.
This is a "doubly fractal sequence" - see the Franklin T. Adams-Watters link.
The PARI functions t1, t2 can be used to read a square array T(n,k) (n >= 1, k >= 1) by antidiagonals downwards: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002
Reading this sequence as the antidiagonals of a rectangular array, row n is (n,n,n,...); this is the weight array (Cf. A144112) of the array A127779 (rectangular). - Clark Kimberling, Sep 16 2008
The upper trim of an arbitrary fractal sequence s is s, but the lower trim of s, although a fractal sequence, need not be s itself. However, the lower trim of A002260 is A002260. (The upper trim of s is what remains after the first occurrence of each term is deleted; the lower trim of s is what remains after all 0's are deleted from the sequence s-1.) - Clark Kimberling, Nov 02 2009
Eigensequence of the triangle = A001710 starting (1, 3, 12, 60, 360, ...). - Gary W. Adamson, Aug 02 2010
The triangle sums, see A180662 for their definitions, link this triangle of natural numbers with twenty-three different sequences, see the crossrefs. The mirror image of this triangle is A004736. - Johannes W. Meijer, Sep 22 2010
A002260 is the self-fission of the polynomial sequence (q(n,x)), where q(n,x) = x^n + x^(n-1) + ... + x + 1. See A193842 for the definition of fission. - Clark Kimberling, Aug 07 2011
Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A002260 is reluctant sequence of sequence 1,2,3,... (A000027). - Boris Putievskiy, Dec 12 2012
This is the maximal sequence of positive integers, such that once an integer k has occurred, the number of k's always exceeds the number of (k+1)'s for the remainder of the sequence, with the first occurrence of the integers being in order. - Franklin T. Adams-Watters, Oct 23 2013
A002260 are the k antidiagonal numerators of rationals in Cantor's proof of 1-to-1 correspondence between rationals and naturals; the denominators are k-numerator+1. - Adriano Caroli, Mar 24 2015
T(n,k) gives the distance to the largest triangular number < n. - Ctibor O. Zizka, Apr 09 2020

			First six rows:
  1
  1   2
  1   2   3
  1   2   3   4
  1   2   3   4   5
  1   2   3   4   5   6

Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168. (Introduces upper trimming, lower trimming, and signature sequences.)
M. Myers, Smarandache Crescendo Subsequences, R. H. Wilde, An Anthology in Memoriam, Bristol Banner Books, Bristol, 1998, p. 19.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

N. J. A. Sloane, Table of n, a(n) for n = 1..11325
Franklin T. Adams-Watters, Doubly Fractal Sequences
Matin Amini and Majid Jahangiri, A Novel Proof for Kimberling's Conjecture on Doubly Fractal Sequences, arXiv:1612.09481 [math.NT], 2017.
Bruno Berselli, Illustration of the initial terms
Jerry Brown et al., Problem 4619, School Science and Mathematics (USA), Vol. 97(4), 1997, pp. 221-222.
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
Clark Kimberling, Fractal sequences
Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Aaron Snook, Augmented Integer Linear Recurrences, 2012. - _N. J. A. Sloane_, Dec 19 2012
Michael Somos, Sequences used for indexing triangular or square arrays
Eric Weisstein's World of Mathematics, Smarandache Sequences
Eric Weisstein's World of Mathematics, Unit Fraction

Cf. A000217, A001710, A002262, A003056, A004736 (ordinal transform), A025581, A056534, A094727, A127779.
Cf. A140756 (alternating signs).
Triangle sums (see the comments): A000217 (Row1, Kn11); A004526 (Row2); A000096 (Kn12); A055998 (Kn13); A055999 (Kn14); A056000 (Kn15); A056115 (Kn16); A056119 (Kn17); A056121 (Kn18); A056126 (Kn19); A051942 (Kn110); A101859 (Kn111); A132754 (Kn112); A132755 (Kn113); A132756 (Kn114); A132757 (Kn115); A132758 (Kn116); A002620 (Kn21); A000290 (Kn3); A001840 (Ca2); A000326 (Ca3); A001972 (Gi2); A000384 (Gi3).
Cf. A108872.

a002260 n k = k
a002260_row n = [1..n]
a002260_tabl = iterate (\row -> map (+ 1) (0 : row)) [1]
-- Reinhard Zumkeller, Aug 04 2014, Jul 03 2012

at:=0; for n from 1 to 150 do for i from 1 to n do at:=at+1; lprint(at,i); od: od: # N. J. A. Sloane, Nov 01 2006
seq(seq(i,i=1..k),k=1..13); # Peter Luschny, Jul 06 2009

FoldList[{#1, #2} &, 1, Range[2, 13]] // Flatten (* Robert G. Wilson v, May 10 2011 *)
Flatten[Table[Range[n],{n,20}]] (* Harvey P. Dale, Jun 20 2013 *)

T(n,k):=sum((i+k)*binomial(i+k-1,i)*binomial(k,n-i-k+1)*(-1)^(n-i-k+1),i,max(0,n+1-2*k),n-k+1); /* Vladimir Kruchinin, Oct 18 2013 */

t1(n)=n-binomial(floor(1/2+sqrt(2*n)),2) /* this sequence */

A002260(n)=n-binomial((sqrtint(8*n)+1)\2,2) \\ M. F. Hasler, Mar 10 2014

from math import isqrt, comb
def A002260(n): return n-comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)),2) # Chai Wah Wu, Nov 08 2024

a(n) = 1 + A002262(n).
n-th term is n - m*(m+1)/2 + 1, where m = floor((sqrt(8*n+1) - 1) / 2).
The above formula is for offset 0; for offset 1, use a(n) = n-m*(m+1)/2 where m = floor((-1+sqrt(8*n-7))/2). - Clark Kimberling, Jun 14 2011
a(k * (k + 1) / 2 + i) = i for k >= 0 and 0 < i <= k + 1. - Reinhard Zumkeller, Aug 14 2001
a(n) = (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2. - Brian Tenneson, Oct 11 2003
a(n) = n - binomial(floor((1+sqrt(8*n))/2), 2). - Paul Barry, May 25 2004
T(n,k) = A001511(A118413(n,k)); T(n,k) = A003602(A118416(n,k)). - Reinhard Zumkeller, Apr 27 2006
a(A000217(n)) = A000217(n) - A000217(n-1), a(A000217(n-1) + 1) = 1, a(A000217(n) - 1) = A000217(n) - A000217(n-1) - 1. - Alexander R. Povolotsky, May 28 2008
a(A169581(n)) = A038566(n). - Reinhard Zumkeller, Dec 02 2009
T(n,k) = Sum_{i=1..k} i*binomial(k,i)*binomial(n-k,n-i) (regarded as triangle, see the example). - Mircea Merca, Apr 11 2012
T(n,k) = Sum_{i=max(0,n+1-2*k)..n-k+1} (i+k)*binomial(i+k-1,i)*binomial(k,n-i-k+1)*(-1)^(n-i-k+1). - Vladimir Kruchinin, Oct 18 2013
G.f.: x*y / ((1 - x) * (1 - x*y)^2) = Sum_{n,k>0} T(n,k) * x^n * y^k. - Michael Somos, Sep 17 2014
a(n) = n - S(n) where S(n) = sum of distinct terms in {a(1), a(2), ..., a(n-1)}. - David James Sycamore, Mar 10 2025

More terms from Reinhard Zumkeller, Apr 27 2006
Incorrect program removed by Franklin T. Adams-Watters, Mar 19 2010
New name from Omar E. Pol, Jul 15 2012

A016789 a(n) = 3*n + 2.

Original entry on oeis.org

2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 179

Offset: 0

N. J. A. Sloane

Except for 1, n such that Sum_{k=1..n} (k mod 3)*binomial(n,k) is a power of 2. - Benoit Cloitre, Oct 17 2002
The sequence 0,0,2,0,0,5,0,0,8,... has a(n) = n*(1 + cos(2*Pi*n/3 + Pi/3) - sqrt(3)*sin(2*Pi*n + Pi/3))/3 and o.g.f. x^2(2+x^3)/(1-x^3)^2. - Paul Barry, Jan 28 2004 [Artur Jasinski, Dec 11 2007, remarks that this should read (3*n + 2)*(1 + cos(2*Pi*(3*n + 2)/3 + Pi/3) - sqrt(3)*sin(2*Pi*(3*n + 2)/3 + Pi/3))/3.]
Except for 2, exponents e such that x^e + x + 1 is reducible. - N. J. A. Sloane, Jul 19 2005
The trajectory of these numbers under iteration of sum of cubes of digits eventually turns out to be 371 or 407 (47 is the first of the second kind). - Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009
Union of A165334 and A165335. - Reinhard Zumkeller, Sep 17 2009
a(n) is the set of numbers congruent to {2,5,8} mod 9. - Gary Detlefs, Mar 07 2010
It appears that a(n) is the set of all values of y such that y^3 = k*n + 2 for integer k. - Gary Detlefs, Mar 08 2010
These numbers do not occur in A000217 (triangular numbers). - Arkadiusz Wesolowski, Jan 08 2012
A089911(2*a(n)) = 9. - Reinhard Zumkeller, Jul 05 2013
Also indices of even Bell numbers (A000110). - Enrique Pérez Herrero, Sep 10 2013
Central terms of the triangle A108872. - Reinhard Zumkeller, Oct 01 2014
A092942(a(n)) = 1 for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n-1), n >= 1, is also the complex dimension of the manifold E(S), the set of all second-order irreducible Fuchsian differential equations defined on P^1 = C U {oo}, having singular points at most in S = {a_1, ..., a_n, a_{n+1} = oo}, a subset of P^1. See the Iwasaki et al. reference, Proposition 2.1.3., p. 149. - Wolfdieter Lang, Apr 22 2016
Except for 2, exponents for which 1 + x^(n-1) + x^n is reducible. - Ron Knott, Sep 16 2016
The reciprocal sum of 8 distinct items from this sequence can be made equal to 1, with these terms: 2, 5, 8, 14, 20, 35, 41, 1640. - Jinyuan Wang, Nov 16 2018
There are no positive integers x, y, z such that 1/a(x) = 1/a(y) + 1/a(z). - Jinyuan Wang, Dec 31 2018
As a set of positive integers, it is the set sum S + S where S is the set of numbers in A016777. - Michael Somos, May 27 2019
Interleaving of A016933 and A016969. - Leo Tavares, Nov 16 2021
Prepended with {1}, these are the denominators of the elements of the 3x+1 semigroup, the numerators being A005408 prepended with {2}. See Applegate and Lagarias link for more information. - Paolo Xausa, Nov 20 2021
This is also the maximum number of moves starting with n + 1 dots in the game of Sprouts. - Douglas Boffey, Aug 01 2022 [See the Wikipedia link. - Wolfdieter Lang, Sep 29 2022]
a(n-2) is the maximum sum of the span (or L(2,1)-labeling number) of a graph of order n and its complement. The extremal graphs are stars and their complements. For example, K_{1,2} has span 3, and K_2 has span 2. Thus a(3-1) = 5. - Allan Bickle, Apr 20 2023

			G.f. = 2 + 5*x + 8*x^2 + 11*x^3 + 14*x^4 + 17*x^5 + 20*x^6 + ... - _Michael Somos_, May 27 2019

K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 149.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Applegate and J. C. Lagarias, The 3x+1 semigroup, Journal of Number Theory, Vol. 177, Issue 1, March 2006, pp. 146-159; see also the arXiv version, arXiv:math/0411140 [math.NT], 2004-2005.
H. Balakrishnan and N. Deo, Parallel algorithm for radiocoloring a graph, Congr. Numer. 160 (2003), 193-204.
Allan Bickle, Extremal Decompositions for Nordhaus-Gaddum Theorems, Discrete Math, 346 7 (2023), 113392.
L. Euler, Observatio de summis divisorum p. 9.
L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 937
L. B. W. Jolley, Summation of Series, Dover, 1961, p. 16
Tanya Khovanova, Recursive Sequences
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")
Fabian S. Reid, The Visual Pattern in the Collatz Conjecture and Proof of No Non-Trivial Cycles, arXiv:2105.07955 [math.GM], 2021.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Leo Tavares, Illustration: Capped Triangular Frames
Wikipedia, Sprouts (game)
Index entries for linear recurrences with constant coefficients, signature (2,-1).

First differences of A005449.
Cf. A002939, A017041, A017485, A125202, A017233, A179896, A017617, A016957, A008544 (partial products), A032766, A016777, A124388, A005351.
Cf. A087370.
Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.
Cf. A000096, A000217.
Cf. A016933, A016969.

List([0..70],n->3*n+2); # Muniru A Asiru, Nov 02 2018

a016789 = (+ 2) . (* 3)  -- Reinhard Zumkeller, Jul 05 2013

[3*n+2: n in [0..80]]; // Vincenzo Librandi, Apr 14 2015

seq(3*n+2, n = 0 .. 50); # Matt C. Anderson, May 18 2017

Range[2, 500, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
LinearRecurrence[{2,-1},{2,5},70] (* Harvey P. Dale, Aug 11 2021 *)

vector(100,n,3*n-1) \\ Derek Orr, Apr 13 2015

for n in range(0,100): print(3*n+2, end=', ') # Stefano Spezia, Nov 21 2018

G.f.: (2+x)/(1-x)^2.
a(n) = 3 + a(n-1).
a(n) = 1 + A016777(n).
a(n) = A124388(n)/9.
a(n) = A125199(n+1,1). - Reinhard Zumkeller, Nov 24 2006
Sum_{n>=1} (-1)^n/a(n) = (1/3)*(Pi/sqrt(3) - log(2)). - Benoit Cloitre, Apr 05 2002
1/2 - 1/5 + 1/8 - 1/11 + ... = (1/3)*(Pi/sqrt(3) - log 2). [Jolley] - Gary W. Adamson, Dec 16 2006
Sum_{n>=0} 1/(a(2*n)*a(2*n+1)) = (Pi/sqrt(3) - log 2)/9 = 0.12451569... (see A196548). [Jolley p. 48 eq (263)]
a(n) = 2*a(n-1) - a(n-2); a(0)=2, a(1)=5. - Philippe Deléham, Nov 03 2008
a(n) = 6*n - a(n-1) + 1 with a(0)=2. - Vincenzo Librandi, Aug 25 2010
Conjecture: a(n) = n XOR A005351(n+1) XOR A005352(n+1). - Gilian Breysens, Jul 21 2017
E.g.f.: (2 + 3*x)*exp(x). - G. C. Greubel, Nov 02 2018
a(n) = A005449(n+1) - A005449(n). - Jinyuan Wang, Feb 03 2019
a(n) = -A016777(-1-n) for all n in Z. - Michael Somos, May 27 2019
a(n) = A007310(n+1) + (1 - n mod 2). - Walt Rorie-Baety, Sep 13 2021
a(n) = A000096(n+1) - A000217(n-1). See Capped Triangular Frames illustration. - Leo Tavares, Oct 05 2021

A248110 Table read by rows: n-th row contains the q successors of n, where q = A007953(n), the digit sum of n in decimal representation.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Oct 01 2014

First 9 rows coincide with triangle A108872;

T(n,1) = n + 1; T(n,A007953(n)) = n + A007953(n) = A062028(n).

			.   n |   T(n,*)                                 | A007953(n)
.  ---+------------------------------------------+-----------
.   1 |    2                                     |       1
.   2 |    3,  4                                 |       2
.   3 |    4,  5,  6                             |       3
.   4 |    5,  6,  7,  8                         |       4
.   5 |    6,  7,  8,  9, 10                     |       5
.   6 |    7,  8,  9, 10, 11, 12                 |       6
.   7 |    8,  9, 10, 11, 12, 13, 14             |       7
.   8 |    9, 10, 11, 12, 13, 14, 15, 16         |       8
.   9 |   10, 11, 12, 13, 14, 15, 16, 17, 18     |       9
.  10 |   11                                     |       1
.  11 |   12, 13                                 |       2
.  12 |   13, 14, 15                             |       3
.  13 |   14, 15, 16, 17                         |       4
.  14 |   15, 16, 17, 18, 19                     |       5
.  15 |   16, 17, 18, 19, 20, 21                 |       6
.  16 |   17, 18, 19, 20, 21, 22, 23             |       7
.  17 |   18, 19, 20, 21, 22, 23, 24, 25         |       8
.  18 |   19, 20, 21, 22, 23, 24, 25, 26, 27     |       9
.  19 |   20, 21, 22, 23, 24, 25, 26, 27, 28, 29 |      10
.  20 |   21, 22                                 |       2

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Cf. A007953 (row lengths), A062028, A108872.

a248110 n k = a248110_tabf !! (n-1) !! (k-1)
a248110_row n = a248110_tabf !! (n-1)
a248110_tabf = map (\x -> [x + 1 .. x + a007953 x]) [1 ..]

A125585 Array of constant-spaced integers read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 1, 4, 5, 4, 2, 4, 6, 7, 5, 3, 5, 7, 8, 9, 6, 1, 6, 8, 10, 10, 11, 7, 2, 5, 9, 11, 13, 12, 13, 8, 3, 6, 9, 12, 14, 16, 14, 15, 9, 4, 7, 10, 13, 15, 17, 19, 16, 17, 10, 1, 8, 11, 14, 17, 18, 20, 22, 18, 19, 11, 2, 6, 12, 15, 18, 21, 21, 23, 25, 20, 21, 12

Offset: 1

Andrew S. Plewe, Jan 04 2007

Iteratively taking sums of the values in each row starting with 1 produces the "figurate" numbers. For instance: 1, 1 + 2 = 3, 1 + 2 + 3 = 6 (the triangular numbers -- A000217) 1, 1 + 3 = 4, 1 + 3 + 5 = 9 (the square numbers -- A000290) 1, 1 + 4 = 5, 1 + 4 + 7 = 10 (the pentagonal numbers -- A000326) etc.
Iterative sums of the rows in between produce sequences related to the figurate numbers: 2, 2+4=6, 2+4+6=10 (oblong, or pronic, or heteromecic numbers -- A002378) 2, 2+5=7, 2+5+8=15 (second pentagonal numbers -- A005449) 3, 3+6=9, 3+6+9=18 (triangular matchstick numbers -- A045943) etc.
Iterative products produce the n-factorial numbers: 1, 1*3=3, 1*3*5=15 (double factorial numbers: (2*n-1)!! -- A001147) 2, 2*4=8, 2*4*6=48 (double factorial numbers: (2*n)!! -- A000165) 1, 1*4=4, 1*4*7=28, (triple factorial numbers (3*n-2)!!! -- A007559) etc.

			The array begins:
  1, 2, 3,  4,  5,  6, ...
  1, 3, 5,  7,  9, 11, ...
  2, 4, 6,  8, 10, 12, ...
  1, 4, 7, 10, 13, 16, ...
  2, 5, 8, 11, 14, 17, ...
  3, 6, 9, 12, 15, 18, ...

Alois P. Heinz, Antidiagonals n = 1..141

Cf. A000217, A000290, A000326, A002378, A005449, A045943, A001147, A000165, A007559.
Columns k=1-2 give A002260, A108872.
Main diagonal gives A380548.

A:= proc(n, k) local m;
      m:= floor((sqrt(8*n-7)-1)/2);
      n + (m+1)*(k-1-m/2)
    end:
seq(seq(A(1+d-k, k), k=1..d), d=1..12); # Alois P. Heinz, Jul 16 2012

imax = 5;
A = Table[k, {i, 1, imax}, {j, 1, i}, {k, j, j + i*imax*(imax+1)/2 - 1, i} ] // Flatten[#, 1]&;
Table[A[[n-k+1, k]], {n, 1, Length[A]}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 23 2016 *)

cp's OEIS Frontend

A002260 Triangle read by rows: T(n,k) = k for n >= 1, k = 1..n.

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A016789 a(n) = 3*n + 2.

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A248110 Table read by rows: n-th row contains the q successors of n, where q = A007953(n), the digit sum of n in decimal representation.

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A125585 Array of constant-spaced integers read by antidiagonals.

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