A068092 -id:A068092 - cp's OEIS frontend

A095864 Largest n-digit triangular number.

6, 91, 990, 9870, 99681, 998991, 9997156, 99991011, 999961560, 9999878910, 99999957291, 999999911791, 9999997965180, 99999998250180, 999999997764120, 9999999895728690, 99999999552805215, 999999998765257141

Offset: 1

Ray Chandler, Jun 28 2004

Triangular number with index given by A095863.

Cf. A068092, A068093, A068094, A095863, A095865, A095866.

Join[{6},(#(#+1))/2&/@Table[Floor[(Sqrt[8*10^n+1]-1)/2],{n,2,30}]] (* Harvey P. Dale, Jul 27 2012 *)
triInverse[n_] := Floor[(Sqrt[1 + 8*n] - 1)/2]; tri[n_] := n*(n+1)/2; Table[tri[triInverse[10^n - 1]], {n, 18}] (* T. D. Noe, Jul 27 2012 *)

a(n) = A068093(n+1) - A068092(n+1).

A095866 Largest n-digit number - largest n-digit triangular number.

Original entry on oeis.org

3, 8, 9, 129, 318, 1008, 2843, 8988, 38439, 121089, 42708, 88208, 2034819, 1749819, 2235879, 104271309, 447194784, 1234742858, 2234186964, 3266133123, 22172578524, 114481278033, 204796672749, 841526085621, 355055477499

Offset: 1

Ray Chandler, Jun 28 2004

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A068092, A068093, A068094, A095863, A095864, A095865.

lndt[n_]:=Module[{c=Floor[(Sqrt[8*10^n+1]-1)/2]},(c(c+1))/2]; Join[{3}, Table[ 10^n-1-lndt[n],{n,2,30}]] (* Harvey P. Dale, May 16 2017 *)

a(n) = 10^n - 1 - A095864(n).

A068093 Smallest n-digit triangular number.

Original entry on oeis.org

1, 10, 105, 1035, 10011, 100128, 1000405, 10001628, 100005153, 1000006281, 10000020331, 100000404505, 1000001326005, 10000002437316, 100000012392316, 1000000042485480, 10000000037150046, 100000000000018810, 1000000000179470703, 10000000002237948990

Offset: 1

Amarnath Murthy, Feb 19 2002

Cf. A068092, A068094, A095863, A095864, A095865, A095866.

triInverse[n_] := Floor[(Sqrt[1 + 8*n] - 1)/2]; tri[n_] := n*(n+1)/2; Table[tri[1 + triInverse[10^(n-1) - 1]], {n, 20}] (* T. D. Noe, Jul 27 2012 *)

Triangular number with index given by A068092.

More terms from Sascha Kurz, Mar 06 2002

A068094 Number of n-digit triangular numbers.

Original entry on oeis.org

3, 10, 31, 96, 306, 967, 3058, 9670, 30579, 96700, 305793, 967000, 3057922, 9670000, 30579224, 96699996, 305792239, 966999967, 3057922393, 9669999669, 30579223926, 96699996687, 305792239263, 966999966873, 3057922392627, 9669999668731, 30579223926265

Offset: 1

Amarnath Murthy, Feb 19 2002

Cf. A068092, A068093, A095863, A095864, A095865, A095866.

triInverse[n_] := Floor[(Sqrt[1 + 8*n] - 1)/2]; Differences[Table[triInverse[10^(n - 1) - 1], {n, 30}]] (* T. D. Noe, Jul 27 2012 *)

Differences of successive terms of A068092.

More terms from Sascha Kurz, Mar 06 2002

A095863 Index of largest triangular number with n digits.

Original entry on oeis.org

3, 13, 44, 140, 446, 1413, 4471, 14141, 44720, 141420, 447213, 1414213, 4472135, 14142135, 44721359, 141421355, 447213594, 1414213561, 4472135954, 14142135623, 44721359549, 141421356236, 447213595499, 1414213562372

Offset: 1

Ray Chandler, Jun 28 2004

Partial sums of A068094.

Cf. A068092, A068093, A068094, A095864, A095865, A095866.

a(n) = A068092(n+1) - 1.

A095865 Smallest n-digit triangular number - smallest n-digit number.

Original entry on oeis.org

0, 0, 5, 35, 11, 128, 405, 1628, 5153, 6281, 20331, 404505, 1326005, 2437316, 12392316, 42485480, 37150046, 18810, 179470703, 2237948990, 10876002500, 22548781025, 26940078203, 242416922750, 572687476751, 4117080477500

Offset: 1

Ray Chandler, Jun 28 2004

Cf. A068092, A068093, A068094, A095863, A095864, A095866.

a(n) = A068093(n) - 10^(n-1).

A182402 Total number of digits in n-th row of a triangle formed by the positive integers.

Original entry on oeis.org

1, 2, 3, 5, 10, 12, 14, 16, 18, 20, 22, 24, 26, 34, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 171, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232

Offset: 1

Dave Durgin, Jun 19 2012

Sequence is nonlinear at each decade transition; for example, row-5 transitions from single-digit (7) to double-digit (10) where sequence jumps (3) to (5); row-14 transitions from 2-digit (92) to 3-digit (105) where sequence jumps from (26) to (34).

The rows of nonlinearity are given by A068092. - Jon Perry, May 26 2013

			1; .................... (row 1 contains 1 digit)
2,   3; ............... (row 2 contains 2 digits)
4,   5,  6; ........... (row 3 contains 3 digits)
7,   8,  9, 10; ....... (row 4 contains 5 digits)
11, 12, 13, 14, 15; ... (row 5 contains 10 digits)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A055642, A226029 (first differences).

Cf. A068092.

a182402 n = a182402_list !! (n-1)
a182402_list = map (sum . map a055642) $ t 1 [1..] where
   t i xs = ys : t (i + 1) zs where
     (ys, zs) = splitAt i xs
-- Reinhard Zumkeller, May 26 2013

f[n_] := Length@ Flatten[ IntegerDigits[ Range[n (n - 1)/2 + 1, n (n + 1)/2]]]; Array[f, 58] (* Robert G. Wilson v, Sep 04 2013 *)

a(n) = {my(x=n*(n-1)/2+1, y=n*(n+1)/2, nx=#Str(x), ny=#Str(y), s=0); for (i=nx, ny, if (i==nx, if (i==ny, s+=(y+1-x)*i, s+=(10^i-x)*i), if (i==ny, s+=(y+1-10^(i-1))*i, s+=i*(10^(i+1)-10^i+1)););); s;} \\ Michel Marcus, Jan 26 2022

def a(n): return len("".join(str(i) for i in range(n*(n+1)//2+1, (n+1)*(n+2)//2+1)))
print([a(n) for n in range(58)]) # Michael S. Branicky, Jan 26 2022

a(n) = A058183(A000217(n)) - A058183(A000217(n-1)), n >= 2. - Omar E. Pol, Jun 25 2012

Better definition from Omar E. Pol, Jun 25 2012

A226030 Smallest m such that A226029(m) = n.

Original entry on oeis.org

1, 3, 15, 46, 4, 448, 1415, 13, 14143, 44722, 14, 447215, 45, 4472137, 14142137, 140, 141421357, 447213596, 1414213563, 4472135956, 14142135625, 44721359551, 141421356238, 447213595501, 1414213562374, 4472135955001, 14142135623732, 44721359549997, 141421356237311, 447213595499959

Offset: 1

Reinhard Zumkeller, May 26 2013

nonn

a(39) = 44. - Michel Marcus, Jan 26 2022

Let k = ceiling(sqrt(2*10^m)). Then some terms are of the form k or k + 1. - David A. Corneth, Jan 27 2022

Cf. A068092, A182402, A226029.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a226030 = (+ 1) . fromJust . (`elemIndex` a226029_list)

nb(n) = {my(x=n*(n-1)/2+1, y=n*(n+1)/2, nx=#Str(x), ny=#Str(y), s=0); for (i=nx, ny, if (i==nx, if (i==ny, s+=(y+1-x)*i, s+=(10^i-x)*i), if (i==ny, s+=(y+1-10^(i-1))*i, s+=i*(10^(i+1)-10^i+1)););); s;} \\ A182402
a(n) = my(k=1, last=nb(k), new=nb(k+1)); while (new-last !=n, k++; last=new; new=nb(k+1)); k; \\ Michel Marcus, Jan 26 2022

A226029(a(n)) = n and A226029(m) <> n for m < a(n).

a(12)-a(18) from Michel Marcus, Jan 26 2022

More terms from David A. Corneth, Jan 26 2022

A077688 a(n) = sum of all cyclic permuted concatenations of the next n numbers.

Original entry on oeis.org

1, 55, 1665, 269884, 6565656565, 1121212121211, 176767676767675, 26262626262626260, 3727272727272727269, 510101010101010101005, 67777777777777777777771, 8787878787878787878787870, 1116161616161616161616161605

Offset: 1

Amarnath Murthy, Nov 16 2002

			a(2) = 23 + 32, a(4) = 78910 + 89107 + 91078 + 10789 = 269884 = sum of concatenation of numbers in each group: (7,8,9,10), (8,9,10,7), (9,10,7,8), (10,7,8,9).

Cf. A068092, A000217.

{ len10(n) = ceil(log(n+1)/log(10)) } { A077688(n) = local(m,w,s); m=0; for(k=1+(n*(n-1))/2,(n*(n+1))/2, m=m*10^len10(k)+k ); w=10^len10(m); s=0; for(k=1+(n*(n-1))/2,(n*(n+1))/2, m=(m*10^len10(k)+k)%w; s+=m ); return(s); } \ for n not in A068092 { a(n) = local(l); l=len10(n^2/2); return((n^2+1)*n*(10^(l*n)-1)/(10^l-1)/2) } \\ Max Alekseyev, Feb 11 2005

For n not in A068092 (implying that all n numbers in the concatenation have the same length L), a(n) = ((n^2+1)*n)/2 * (10^(L*n)-1)/(10^L-1), where L = ceiling(log(n^2/2)/log(10)). - Max Alekseyev, Feb 11 2005

More terms from Max Alekseyev, Feb 11 2005

A236043 Number of triangular numbers <= 10^n.

Original entry on oeis.org

2, 5, 14, 45, 141, 447, 1414, 4472, 14142, 44721, 141421, 447214, 1414214, 4472136, 14142136, 44721360, 141421356, 447213595, 1414213562, 4472135955, 14142135624, 44721359550, 141421356237, 447213595500, 1414213562373, 4472135955000, 14142135623731

Offset: 0

Derek Orr, Jan 18 2014

Except for 5, all numbers begin with either a 4 or a 1. If strictly less than, the 5 would become a 4, satisfying this conjecture.
This is not a conjecture, it is a fact and it is the result from the square root of 2 and 20 times powers of ten. - Robert G. Wilson v, Jan 11 2015
Tanton (2012) discusses the equivalent sequence based on excluding zero from the triangular numbers, and presents the relevant formula, which, being asymptotic to floor[sqrt(2*10^n)], explains the observation in the first comment. - Chris Boyd, Jan 19 2014

			There are 4472 triangular numbers less than or equal to 10^7 so a(7) = 4472.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Tanton, Cool Math Newsletter (November 2012)

Cf. A000217, A002024, A003056. Essentially the same as A068092.

[Floor(Sqrt(2*10^n+1/4) + 1/2): n in [1..30]]; // Vincenzo Librandi, Feb 08 2014

seq(floor(sqrt(2*10^n+1/4)+1/2),n=1..30); # Robert Israel, Dec 22 2024

Table[ Floor[ Sqrt[2*10^n + 1] + 1/2], {n, 25}] (* Vincenzo Librandi, Feb 08 2014; modified by Robert G. Wilson v, Jan 11 2015 *)

a236043(n)=floor(sqrt(2*10^n+1/4)+1/2) \\ Chris Boyd, Jan 19 2014

from math import isqrt
def A236043(n): return isqrt(10**n+1<<3)+1>>1 # Chai Wah Wu, Jun 14 2025

a(n) = floor( sqrt(2*10^n + 1/4) + 1/2 ), adapted from Tanton (see Links section). - Chris Boyd, Jan 19 2014
a(n) = A068092(n + 1) for n >= 2. - R. J. Mathar, Jan 20 2014
a(n) = A003056(10^n) + 1 = A002024(10^n + 1). - Andrew Howroyd, Dec 21 2024

More terms from Jon E. Schoenfield, Feb 07 2014

a(0) prepended by Andrew Howroyd, Dec 21 2024

cp's OEIS Frontend

A095864 Largest n-digit triangular number.

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A095866 Largest n-digit number - largest n-digit triangular number.

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A068093 Smallest n-digit triangular number.

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A068094 Number of n-digit triangular numbers.

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A095863 Index of largest triangular number with n digits.

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A095865 Smallest n-digit triangular number - smallest n-digit number.

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A182402 Total number of digits in n-th row of a triangle formed by the positive integers.

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A226030 Smallest m such that A226029(m) = n.

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A077688 a(n) = sum of all cyclic permuted concatenations of the next n numbers.

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