A000462 -id:A000462 - cp's OEIS frontend

A376354 Numbers that end in 0 when written in base of triangular numbers (cf. A000462).

3, 6, 9, 10, 13, 15, 18, 21, 24, 27, 28, 31, 34, 36, 39, 42, 45, 48, 51, 54, 55, 58, 61, 64, 65, 66, 69, 72, 75, 76, 78, 81, 84, 87, 88, 91, 94, 97, 100, 101, 104, 105, 108, 111, 114, 115, 118, 120, 123, 126, 129, 130, 133, 135, 136, 139, 142, 145, 146, 149

Offset: 1

Clark Kimberling, Sep 22 2024

Every positive integer is in exactly one of the following sequences: this sequence, A376355, or A376356.

Conjecture: {a(n+1) - a(n) : n >= 1} = {1, 2, 3}. (See related conjectures at A376355 and A376356.)

Cf. A000462, A376355, A376356, A376357.

a[n_, poly_] := FromDigits[FoldList[{Mod[#[[1]], #2], Quotient[#[[1]], #2]} &, {n, 0}, Reverse[Map[(poly - 2)  #  (# - 1)/2 + # &, Range[Floor[Sqrt[2  n]]]]]][[All, 2]]]
t3 = Map[a[#, 3] &, Range[200]]; (* A000462 *)
m = Mod[t3, 10]
Table[Flatten[Position[m, r]], {r, 0, 2}]
p0 = Flatten[Position[m, 0]]  (* this sequence *)
p1 = Flatten[Position[m, 1]]  (* A376355 *)
p2 = Flatten[Position[m, 2]]  (* A376356 *)
(* Peter J. C. Moses, Sep 20 2024 *)

A376355 Numbers that end in 1 when written in base of triangular numbers (cf. A000462).

Original entry on oeis.org

1, 4, 7, 11, 14, 16, 19, 22, 25, 29, 32, 35, 37, 40, 43, 46, 49, 52, 56, 59, 62, 67, 70, 73, 77, 79, 82, 85, 89, 92, 95, 98, 102, 106, 109, 112, 116, 119, 121, 124, 127, 131, 134, 137, 140, 143, 147, 150, 152, 154, 157, 160, 164, 167, 169, 172, 175, 178, 182

Offset: 1

Clark Kimberling, Sep 22 2024

Every positive integer is in exactly one of the following sequences: A376355, this sequence, or A376356.

Conjecture: {a(n+1) - a(n) : n >= 1} = {2,3,4,5,6}. (See related conjectures at A376354 and A376356.)

Cf. A000462, A376354, A376356, A376357.

a[n_, poly_] := FromDigits[FoldList[{Mod[#[[1]], #2], Quotient[#[[1]], #2]} &, {n, 0}, Reverse[Map[(poly - 2)  #  (# - 1)/2 + # &, Range[Floor[Sqrt[2  n]]]]]][[All, 2]]]
t3 = Map[a[#, 3] &, Range[200]]; (* A000462 *)
m = Mod[t3, 10]
Table[Flatten[Position[m, r]], {r, 0, 2}]
p0 = Flatten[Position[m, 0]]  (* A376354 *)
p1 = Flatten[Position[m, 1]]  (* this sequence *)
p2 = Flatten[Position[m, 2]]  (* A376356 *)
(* Peter J. C. Moses, Sep 20 2024 *)

A376356 Positions of numbers in A000462 that end in 2.

Original entry on oeis.org

2, 5, 8, 12, 17, 20, 23, 26, 30, 33, 38, 41, 44, 47, 50, 53, 57, 60, 63, 68, 71, 74, 80, 83, 86, 90, 93, 96, 99, 103, 107, 110, 113, 117, 122, 125, 128, 132, 138, 141, 144, 148, 155, 158, 161, 165, 170, 173, 176, 179, 183, 188, 192, 195, 198

Offset: 1

Clark Kimberling, Sep 25 2024

Every positive integer is in exactly one of the following sequences: A376354, A376355, or this sequence.

Conjecture: {a(n+1) - a(n) : n >= 1} = {3,4,5,6,7,8,9}. (See related conjectures at A376354 and A376355.)

Cf. A000462, A376354, A376355, A376357.

a[n_, poly_] := FromDigits[FoldList[{Mod[#[[1]], #2], Quotient[#[[1]], #2]} &, {n, 0}, Reverse[Map[(poly - 2)  #  (# - 1)/2 + # &, Range[Floor[Sqrt[2  n]]]]]][[All, 2]]]
t3 = Map[a[#, 3] &, Range[200]]; (* A000462 *)
m = Mod[t3, 10]
Table[Flatten[Position[m, r]], {r, 0, 2}]
p0 = Flatten[Position[m, 0]]  (* A376354 *)
p1 = Flatten[Position[m, 1]]  (* A376355 *)
p2 = Flatten[Position[m, 2]]  (* this sequence *)
(* Peter J. C. Moses, Sep 20 2024 *)

A003056 n appears n+1 times. Also the array A(n,k) = n+k (n >= 0, k >= 0) read by antidiagonals. Also inverse of triangular numbers.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 0

N. J. A. Sloane

Also triangle read by rows: T(n,k), n>=0, k>=0, in which n appears n+1 times in row n. - Omar E. Pol, Jul 15 2012
The PARI functions t1, t2 can be used to read a triangular array T(n,k) (n >= 0, 0 <= k <= n-1) by rows from left to right: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002
Number of terms in partition of n with greatest number of distinct terms. - Amarnath Murthy, May 20 2001
Summation table for (x+y) = (0+0),(0+1),(1+0),(0+2),(1+1),(2+0), ...
Also the number of triangular numbers less than or equal to n, not counting 0 as triangular. - Robert G. Wilson v, Oct 21 2005
Permutation of A116939: a(n) = A116939(A116941(n)), a(A116942(n)) = A116939(n). - Reinhard Zumkeller, Feb 27 2006
Maximal size of partitions of n into distinct parts, see A000009. - Reinhard Zumkeller, Jun 13 2009
Also number of digits of A000462(n). - Reinhard Zumkeller, Mar 27 2011
Also the maximum number of 1's contained in the list of hook-lengths of a partition of n. E.g., a(4)=2 because hooks of partitions of n=4 comprise {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1} where the number of 1's in each is 1,2,1,2,1. Hence the maximum is 2. - T. Amdeberhan, Jun 03 2012
Fan, Yang, and Yu (2012) prove a conjecture of Amdeberhan on the generating function of a(n). - Jonathan Sondow, Dec 17 2012
Also the number of partitions of n into distinct parts p such that max(p) - min(p) <= length(p). - Clark Kimberling, Apr 18 2014
Also the maximum number of occurrences of any single value among the previous terms. - Ivan Neretin, Sep 20 2015
Where records occur gives A000217. - Omar E. Pol, Nov 05 2015
Also number of peaks in the largest Dyck path of the symmetric representation of sigma(n), n >= 1. Cf. A237593. - Omar E. Pol, Dec 19 2016

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + ...
As triangle, the sequence starts
  0;
  1, 1;
  2, 2, 2;
  3, 3, 3, 3;
  4, 4, 4, 4, 4;
  5, 5, 5, 5, 5, 5;
  6, 6, 6, 6, 6, 6, 6;
  7, 7, 7, 7, 7, 7, 7, 7;
  8, 8, 8, 8, 8, 8, 8, 8, 8;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Anna R. B. Fan, Harold R. L. Yang, and Rebecca T. Yu, On the Maximum Number of k-Hooks of Partitions of n, arXiv:1212.3505 [math.CO], 2012.
Michael Somos, Sequences used for indexing triangular or square arrays.

a(n) = A002024(n+1)-1.
Cf. A000196, A000217, A000462, A001227, A001462, A001614, A004247 (multiplication table), A006463 (partial sums), A016655, A050600, A050602, A048645, A122797, A131507, A238005.
Partial sums of A073424.
Cf. A116939, A116941, A116942,

a003056 = floor . (/ 2) . (subtract 1) .
                  sqrt . (+ 1) . (* 8) . fromIntegral
a003056_row n = replicate (n + 1) n
a003056_tabl = map a003056_row [0..]
a003056_list = concat $ a003056_tabl
-- Reinhard Zumkeller, Aug 02 2014, Oct 17 2010

[Floor((Sqrt(1+8*n)-1)/2): n in [0..80]]; // Vincenzo Librandi, Oct 23 2011

A003056 := (n,k) -> n: # Peter Luschny, Oct 29 2011
a := [ 0 ]: for i from 1 to 15 do for j from 1 to i+1 do a := [ op(a),i ]; od: od: a;
A003056 := proc(n)
    floor((sqrt(1+8*n)-1)/2) ;
end proc: # R. J. Mathar, Jul 10 2015

f[n_] := Floor[(Sqrt[1 + 8n] - 1)/2]; Table[ f[n], {n, 0, 87}] (* Robert G. Wilson v, Oct 21 2005 *)
Table[x, {x, 0, 13}, {y, 0, x}] // Flatten
T[ n_, k_] := If[ n >= k >= 0, n, 0]; (* Michael Somos, Dec 22 2016 *)
Flatten[Table[PadRight[{},n+1,n],{n,0,12}]] (* Harvey P. Dale, Jul 03 2021 *)

A003056(n)=(sqrtint(8*n+1)-1)\2  \\ M. F. Hasler, Oct 08 2011

t1(n)=floor(-1/2+sqrt(2+2*n)) /* A003056 */

t2(n)=n-binomial(floor(1/2+sqrt(2+2*n)),2) /* A002262 */

from math import isqrt
def A003056(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) = floor((sqrt(1+8*n)-1)/2). - Antti Karttunen
a(n) = floor(-1/2 + sqrt(2*n+b)) with 1/4 <= b < 9/4 or a(n) = floor((sqrt(8*n+b)-1)/2) with 1 <= b < 9. - Michael A. Childers (childers_moof(AT)yahoo.com), Nov 11 2001
a(n) = f(n,0) with f(n,k) = k if n <= k, otherwise f(n-k-1, k+1). - Reinhard Zumkeller, May 23 2009
a(n) = 2*n + 1 - A001614(n+1) = n + 1 - A122797(n+1). - Reinhard Zumkeller, Feb 12 2012
a(n) = k if k*(k+1)/2 <= n < (k+1)*(k+2)/2. - Jonathan Sondow, Dec 17 2012
G.f.: (1-x)^(-1)*Sum_{n>=1} x^(n*(n+1)/2) = (Theta_2(0,x^(1/2)) - 2*x^(1/8))/(2*x^(1/8)*(1-x)) where Theta_2 is a Jacobi Theta function. - Robert Israel, May 21 2015
a(n) = floor((A000196(1+8*n)-1)/2). - Pontus von Brömssen, Dec 10 2018
a(n+1) = a(n-a(n)) + 1, a(0) = 0. - Rok Cestnik, Dec 29 2020
a(n) = A001227(n) + A238005(n), n >= 1. - Omar E. Pol, Sep 30 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 (cf. A016655). - Amiram Eldar, Sep 24 2023
G.f. as array: (x + y - 2*x*y)/((1 - x)^2*(1 - y)^2). - Stefano Spezia, Dec 20 2023 [corrected by Stefano Spezia, Apr 22 2024]

Definition clarified by N. J. A. Sloane, Dec 08 2020

A057945 Number of triangular numbers needed to represent n with greedy algorithm.

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 1, 2, 3, 2, 1, 2, 3, 2, 3, 1, 2, 3, 2, 3, 4, 1, 2, 3, 2, 3, 4, 2, 1, 2, 3, 2, 3, 4, 2, 3, 1, 2, 3, 2, 3, 4, 2, 3, 4, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 2, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 2, 3, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 2, 3, 4, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 2, 3, 4, 3

Offset: 0

Henry Bottomley, Oct 05 2000

nonn

a(n) = sum of digits of A000462(n). - Reinhard Zumkeller, Mar 27 2011

The length of (number of moves in) Simon Norton's game in A006019 starting with an initial heap of n if both players always take, never put. - R. J. Mathar, May 13 2016

			a(35)=3 since 35=28+6+1

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000217, A002262, A056944, A057944. See A006893 for records.

a057945 n = g n $ reverse $ takeWhile (<= n) $ tail a000217_list where
   g 0 _      = 0
   g x (t:ts) = g r ts + a where (a,r) = divMod x t
-- Reinhard Zumkeller, Mar 27 2011

A057945 := proc(n)
    local a,x;
    a := 0 ;
    x := n ;
    while x > 0 do
        x := x-A057944(x) ;
        a := a+1 ;
    end do:
    a ;
end proc: # R. J. Mathar, May 13 2016

A057944[n_] := With[{k = Floor[Sqrt[8n+1]]}, Floor[(k-1)/2]* Floor[(k+1)/2]/2];
a[n_] := Module[{k = 0, x = n}, While[x>0, x = x - A057944[x]; k++]; k];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Mar 10 2019, after R. J. Mathar *)

a(0)=0, otherwise a(n)=a(A002262(n))+1.

A372281 Numbers written in base of triangular numbers where the trailing digits are made as high as possible.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 18, 19, 27, 28, 29, 37, 38, 39, 47, 48, 49, 57, 58, 59, 67, 68, 69, 77, 78, 79, 87, 88, 89, 97, 98, 99, 187, 188, 189, 197, 198, 199, 287, 288, 289, 297, 298, 299, 387, 388, 389, 397, 398, 399, 487, 488, 489, 497, 498, 499, 587

Offset: 1

Stefano Spezia, Apr 25 2024

This sequence differs from A000462 whose leading digits are made as high as possible.

			The digits (from right to left) have values 1, 3, 6, 10, etc. (A000217), hence a(61) = 587 because 61 = 5*6 + 8*3 + 7*1.

Cf. A000217, A000462.

9*A000292(n) gives the number of terms with at most n digits.

A000217[n_]:=n(n+1)/2; a[n_]:=Module[{s=0}, num=n; digits={}; s=Ceiling[Root[9(#1^3+3#1^2+2#1)-6num &, 1]]; While[s>0, AppendTo[digits, d=Ceiling[(num-9(s-1)^2)/A000217[s]]]; num-=d*A000217[s]; s--]; FromDigits[digits]]; Array[a,61]

cp's OEIS Frontend

A376354 Numbers that end in 0 when written in base of triangular numbers (cf. A000462).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A376355 Numbers that end in 1 when written in base of triangular numbers (cf. A000462).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A376356 Positions of numbers in A000462 that end in 2.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A003056 n appears n+1 times. Also the array A(n,k) = n+k (n >= 0, k >= 0) read by antidiagonals. Also inverse of triangular numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

Extensions

A057945 Number of triangular numbers needed to represent n with greedy algorithm.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A372281 Numbers written in base of triangular numbers where the trailing digits are made as high as possible.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica