A000217 -id:A000217 - cp's OEIS frontend

A258257 The number of representations of n as a minimal number of triangular numbers, A000217(n).

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

Offset: 1

Martin Renner, May 24 2015

nonn

			a(5) = 1 since 5 = 1 + 1 + 3 is the only representation as a minimal number of three triangular numbers.
a(16) = 2 since 16 = 1 + 15 = 6 + 10 has two representations as a minimal number of two triangular numbers.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A061336, A180466.

t[n_] := n (n + 1)/2; a[n_] := Block[{k = 1, t, tt = t /@ Range[ Sqrt[2*n]]}, While[{} == (r = IntegerPartitions[n, {k}, tt]), k++]; Length@r]; Array[a, 100] (* Giovanni Resta, Jun 09 2015 *)

A259413 Triangular numbers (A000217) that are the sum of eleven consecutive triangular numbers.

Original entry on oeis.org

2145, 3916, 7381, 13530, 843051, 1547920, 2926990, 5374281, 335521560, 616057651, 1164924046, 2138939715, 133536727236, 245189386585, 463636832725, 851292621696, 53147281907775, 97584759792586, 184526294489911, 338812324484700, 21152484662556621

Offset: 1

Colin Barker, Jun 26 2015

			2145 is in the sequence because T(65) = 2145 = 105 + 120 + 136 + 153 + 171 + 190 + 210 + 231 + 253 + 276 + 300 = T(14) + ... + T(24).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,398,-398,0,0,-1,1).

Cf. A000217, A001110, A129803, A131557, A257711, A257712, A257713, A259414, A259415.

LinearRecurrence[{1, 0, 0, 398, -398, 0, 0, -1, 1}, {2145, 3916, 7381, 13530, 843051, 1547920, 2926990, 5374281, 335521560}, 30] (* Vincenzo Librandi, Jun 27 2015 *)

Vec(-11*x*(6*x^8-x^7+x^5-2199*x^4+559*x^3+315*x^2+161*x+195)/((x-1)*(x^4-20*x^2+1)*(x^4+20*x^2+1)) + O(x^30))

G.f.: -11*x*(6*x^8-x^7+x^5-2199*x^4+559*x^3+315*x^2+161*x+195) / ((x-1)*(x^4-20*x^2+1)*(x^4+20*x^2+1)).

A259414 Triangular numbers (A000217) that are the sum of thirteen consecutive triangular numbers.

Original entry on oeis.org

2080, 414505, 28815436, 49317346, 3428789455, 698283666730, 48548229019381, 83089887991201, 5776831256176630, 1176469718198438755, 81794153348207147926, 139990009467226925656, 9732816854065394603605, 1982118534159467652450580, 137806953149317550935817071

Offset: 1

Colin Barker, Jun 26 2015

			2080 is in the sequence because T(64) = 2080 = 66 + 78 + 91 + 105 + 120 + 136 + 153 + 171 + 190 + 210 + 231 + 253 + 276 = T(11) + ... + T(23).

Colin Barker, Table of n, a(n) for n = 1..641
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1684802,-1684802,0,0,-1,1).

Cf. A000217, A001110, A129803, A131557, A257711, A257712, A257713, A259413, A259415.

 LinearRecurrence[{1, 0, 0, 1684802, -1684802, 0, 0, -1, 1}, {2080, 414505, 28815436, 49317346, 3428789455, 698283666730, 48548229019381, 83089887991201, 5776831256176630}, 30] (* Vincenzo Librandi, Jun 27 2015 *)

Vec(-13*x*(7*x^8 +153*x^6 +31725*x^5 -9608927*x^4 +1577070*x^3 +2184687*x^2 +31725*x +160) / ((x -1)*(x^2 -36*x -1)*(x^2 +36*x -1)*(x^4 +1298*x^2 +1)) + O(x^20))

G.f.: -13*x*(7*x^8 +153*x^6 +31725*x^5 -9608927*x^4 +1577070*x^3 +2184687*x^2 +31725*x +160) / ((x -1)*(x^2 -36*x -1)*(x^2 +36*x -1)*(x^4 +1298*x^2 +1)).

A259415 Triangular numbers (A000217) that are the sum of seventeen consecutive triangular numbers.

Original entry on oeis.org

1326, 9180, 24531, 1979055, 5325216, 39529386, 106368405, 8616365901, 23185550130, 172110498456, 463127571831, 37515654714891, 100949879501796, 749369070309030, 2016457340944761, 163343152011830505, 439535752164830646, 3262752760014579156

Offset: 1

Colin Barker, Jun 26 2015

			1326 is in the sequence because T(51) = 1326 = 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 + 91 + 105 + 120 + 136 + 153 + 171 + 190 = T(3) + ... + T(19).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,4354,-4354,0,0,-1,1).

Cf. A000217, A001110, A129803, A131557, A257711, A257712, A257713, A259413, A259414.

 LinearRecurrence[{1, 0, 0, 4354, -4354, 0, 0, -1, 1}, {1326, 9180, 24531, 1979055, 5325216, 39529386, 106368405, 8616365901, 23185550130}, 30] (* Vincenzo Librandi, Jun 27 2015 *)
Module[{nn=10^6},Select[Total/@Partition[Accumulate[Range[nn]],17,1],OddQ[ Sqrt[8#+1]]&]] (* Harvey P. Dale, Mar 19 2023 *)

Vec(-51*x*(11*x^8 +15*x^6 +154*x^5 -47593*x^4 +38324*x^3 +301*x^2 +154*x +26) / ((x -1)*(x^2 -8*x -1)*(x^2 +8*x -1)*(x^4 +66*x^2 +1)) + O(x^30))

G.f.: -51*x*(11*x^8 +15*x^6 +154*x^5 -47593*x^4 +38324*x^3 +301*x^2 +154*x +26) / ((x -1)*(x^2 -8*x -1)*(x^2 +8*x -1)*(x^4 +66*x^2 +1)).

A276598 Values of m such that m^2 + 3 is a triangular number (A000217).

Original entry on oeis.org

0, 5, 30, 175, 1020, 5945, 34650, 201955, 1177080, 6860525, 39986070, 233055895, 1358349300, 7917039905, 46143890130, 268946300875, 1567533915120, 9136257189845, 53250009223950, 310363798153855, 1808932779699180, 10543232880041225, 61450464500548170

Offset: 1

Colin Barker, Sep 07 2016

			5 is in the sequence because 5^2 + 3 = 28, which is a triangular number.

Colin Barker, Table of n, a(n) for n = 1..1000
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, Integer sequences and ellipse chains inside a hyperbola, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A000129, A000217, A230044.
Cf. A001109 (k=0), A106328 (k=1), A077241 (k=2), A276599 (k=5), A276600 (k=6), A276601 (k=9), A276602 (k=10), where k is the value added to n^2.
Cf. A328791 (the resulting triangular numbers).

[n le 2 select 5*(n-1) else 6*Self(n-1) - Self(n-2): n in [1..31]]; // G. C. Greubel, Sep 15 2021

CoefficientList[Series[5*x/(1 - 6*x + x^2), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 07 2016 *)
LinearRecurrence[{6,-1},{0,5},30] (* Harvey P. Dale, Apr 26 2019 *)
(5/2)*Fibonacci[2*Range[30] -2, 2] (* G. C. Greubel, Sep 15 2021 *)

concat(0, Vec(5*x^2/(1-6*x+x^2) + O(x^30)))

a(n)=([0,1;-1,6]^n*[-5;0])[1,1] \\ Charles R Greathouse IV, Sep 07 2016

[(5/2)*lucas_number1(2*n-2, 2, -1) for n in (1..30)] # G. C. Greubel, Sep 15 2021

a(n) = 5*A001109(n-1).
a(n) = 5*( (3 - 2*sqrt(2))*(3 + 2*sqrt(2))^n - (3 + 2*sqrt(2))*(3 - 2*sqrt(2))^n )/(4*sqrt(2)).
a(n) = 6*a(n-1) - a(n-2) for n>2.
G.f.: 5*x^2 / (1-6*x+x^2).
a(n) = (5/2)*A000129(2*n-2). - G. C. Greubel, Sep 15 2021

A349243 Indices of triangular numbers A000217 with only odd digits.

Original entry on oeis.org

1, 2, 5, 10, 13, 17, 18, 26, 34, 58, 62, 101, 109, 138, 149, 154, 177, 178, 186, 189, 250, 257, 266, 382, 554, 586, 589, 621, 622, 862, 893, 1013, 1050, 1057, 1069, 1258, 1354, 1370, 1634, 1658, 1738, 1754, 1777, 1786, 1853, 1885, 1965, 2657, 2666, 2741, 2818, 3218, 3346, 3445, 3457, 3794, 3845

Offset: 1

M. F. Hasler, Nov 20 2021

S. S. Gupta, Can You Find (CYF) no. 55, Nov 11 2021.

Cf. A000217 (triangular numbers), A014261 (numbers with only odd digits), A117960 (triangular numbers with only odd digits).

a:= proc(n) option remember; local k; for k from
      1+`if`(n=1, 0, b(n-1)) while 0=mul(irem(i, 2),
      i=convert(k*(k+1)/2, base, 10)) do od; k
    end:
seq(a(n), n=1..57);  # Alois P. Heinz, Nov 22 2021

Select[Range[4000], AllTrue[IntegerDigits[#*(# + 1)/2], OddQ] &] (* Amiram Eldar, Nov 20 2021 *)
Position[Accumulate[Range[4000]],?(AllTrue[IntegerDigits[#],OddQ]&)]//Flatten (* _Harvey P. Dale, Sep 06 2023 *)

select( {is_A349243(n)=Set(digits(n*(n+1)\2)%2)==[1]}, [1..9999])

from itertools import islice, count
def A349243(): return filter(lambda n: set(str(n*(n+1)//2)) <= {'1','3','5','7','9'}, count(0))
A349243_list = list(islice(A349243(),20)) # Chai Wah Wu, Nov 22 2021

a(n) = floor(sqrt(2*A117960(n))).

A061208 Numbers which can be expressed as sum of distinct triangular numbers (A000217).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Amarnath Murthy, Apr 21 2001

nonn

These numbers were called "almost-triangular" numbers during the Peru's Selection Test for the XII IberoAmerican Olympiad (1998). All numbers >= 34 are almost-triangular: see link. [Bernard Schott, Feb 04 2013]

			25 = 1 + 3 + 6 + 15

R. E. Woodrow, The Olympiad Corner, No. 198, Crux Mathematicorum, v25-n4(2002), 207-208, exercise 2.

Cf. A000217, A007294, A051611, A051533. Complement of A053614.

gf := product(1+x^(j*(j+1)/2), j=1..100): s := series(gf, x, 200): for i from 1 to 200 do if coeff(s, x, i) > 0 then printf(`%d,`,i) fi:od:

Corrected and extended by James Sellers, Apr 24 2001

A065113 Sum of the squares of the a(n)-th and the (a(n)+1)st triangular numbers (A000217) is a perfect square.

Original entry on oeis.org

6, 40, 238, 1392, 8118, 47320, 275806, 1607520, 9369318, 54608392, 318281038, 1855077840, 10812186006, 63018038200, 367296043198, 2140758220992, 12477253282758, 72722761475560, 423859315570606, 2470433131948080, 14398739476117878, 83922003724759192

Offset: 1

Robert G. Wilson v, Nov 12 2001

The sequence of square roots of the sum of the squares of the n-th and the (n+1)st triangular numbers is A046176.

			T6 = 21 and T7 = 28, 21^2 + 28^2 = 441 + 784 = 1225 = 35^2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT] (2008)
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A001652, A002315, A003499 (first differences), A065651.

CoefficientList[ Series[2*(x - 3)/(-1 + 7x - 7x^2 + x^3), {x, 0, 24} ], x]
LinearRecurrence[{7,-7,1},{6,40,238},41] (* Harvey P. Dale, Dec 27 2011 *)

a(n)=-1+subst(poltchebi(abs(n+1))-poltchebi(abs(n)),x,3)/2

Vec(2*x*(3-x)/((1-6*x+x^2)*(1-x)) + O(x^40)) \\ Colin Barker, Mar 05 2016

a(n) = 2*A001652(n) = -1 + A002315(n).
a(n) - a(n-1) = A003499(n).
From Michael Somos, Apr 07 2003: (Start)
G.f.: 2*x*(3-x)/((1-6*x+x^2)*(1-x)).
a(n) = 6*a(n-1) - a(n-2) + 4.
a(-1-n) = -a(n) - 2. (End)
a(1)=6, a(2)=40, a(3)=238, a(n) = 7*a(n-1)-7*a(n-2)+a(n-3). - Harvey P. Dale, Dec 27 2011
a(n)^2 + (a(n)+2)^2 = A075870(n+1)^2 = A165518(n+1). - Joerg Arndt, Feb 15 2012
a(n) = (-2-(3-2*sqrt(2))^n*(-1+sqrt(2))+(1+sqrt(2))*(3+2*sqrt(2))^n)/2. - Colin Barker, Mar 05 2016
From Klaus Purath, Sep 05 2021: (Start)
(a(n+1) - a(n) - a(n-1) + a(n-2))/8 = A005319(n), for n >= 3.
((a(n) - a(n-1))^2)/2 - 2 = A005319(n)^2 = 2*A132592(n), for n>= 2.
a(n) = A265278(2*n+1).
a(n) = A293004(2*n+1).
a(n) = A213667(2*n).
a(n) = Sum_{k=1..n} A003499(k). (End)

A080248 Stirling-like number triangle defined by sequence A000217.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 13, 10, 1, 1, 40, 73, 20, 1, 1, 121, 478, 273, 35, 1, 1, 364, 2989, 3208, 798, 56, 1, 1, 1093, 18298, 35069, 15178, 1974, 84, 1, 1, 3280, 110881, 368988, 262739, 56632, 4326, 120, 1, 1, 9841, 668566, 3800761, 4310073, 1452011, 177760

Offset: 0

Paul Barry, Feb 17 2003

Columns include A003462, A016211, A021514. The defining sequence A000217(n) = C(n+1,2) is the sequence of partial sums of the sequence (0,1,2,3,4,...) which defines the Stirling numbers of the second kind A008277.
n-th row = M^n * [1,0,0,0,...], where M = an infinite lower triangular matrix with (1, 3, 6, ...) in the main diagonal and (1, 1, 1, ...) in the subdiagonal. - Gary W. Adamson, Apr 13 2009
Row sums = A124373 starting (1, 2, 6, 25, 135, ...). - Gary W. Adamson, Jul 11 2011

			Rows are
  {1},
  {1,  1},
  {1,  4,  1},
  {1, 13, 10,  1},
  {1, 40, 73, 20, 1},
  ...
For example, 73 = 13 + 6*10, 20 = 10 + 10*1.

Vincenzo Librandi, Rows n = 0..50, flattened

Cf. A000217, A008277, A124373.

gf  := k -> 1/mul(1 - x*j*(j-1)/2, j=0..k+2):
ser := k -> series(gf(k), x, 16):
T := (n, k) -> coeff(ser(k), x, n-k):
seq(print(seq(T(n, k), k=0..n)), n=0..8); # Peter Luschny, Aug 29 2020

max = 10; t[n_, n_] = n*(n+1)/2; t[n_, k_] /; k == n-1 = 1; t[, ] = 0; m = Table[t[n, k], {n, 1, max}, {k, 1, max}]; row[n_] := MatrixPower[m, n][[All, 1]]; Table[Take[row[n], n+1], {n, 0, max-1}] // Flatten (* Jean-François Alcover, Jun 25 2013, after Gary W. Adamson *)

{T(n, k) = local(s); if( k<0 || k>n, 0, forvec(v = vector(n-k, i, [0, k]), s += prod(i=1, n-k, v[i] * (v[i] + 1) / 2), 1)); s}; /* Michael Somos, Feb 06 2004 */

Columns are generated by 1/Product_{k=1..n+1} (1 - C(k + 1, 2)*x). [In other words:
T(n, k) = [x^(n-k)] 1/Product_{j=0..k+2}(1 - x*binomial(j, 2)).]
T(n, k) = (k*(k+1)/2) * T(n-1,k) + T(n-1,k-1), T(n,n)=1. - Vladimir Kruchinin, Aug 25 2020
T(n,k) = (Sum_{i=0..k} (-1)^(k-i) * (2*i + 3) * binomial(2*k + 3,k-i) * ((i+1) * (i+2) / 2)^(n+1)) * 2^(k+1) / (2*k + 3)! for 0 <= k <= n. - Werner Schulte, Oct 29 2020
The polynomials p(n,x) = Sum_{k=0..n} T(n,k) * (k!*(k+1)!/2^k) * x^(k+2) satisfy for n >= 0 the equations p(n+1,x) = p(1,x) * p''(n,x) / 2 and p(n,-1) = 0^n when p'' is the second derivative of p. - Werner Schulte, Dec 15 2020

A108552 Integer values of (12...*k)/(1+2+...+k) = k!/T(k) = A000142(k)/A000217(k), k>=1.

Original entry on oeis.org

1, 1, 8, 180, 1120, 8064, 604800, 68428800, 830269440, 10897286400, 2324754432000, 640237370572800, 11585247657984000, 221172909834240000, 93666727314800640000, 2068161339110798131200, 47726800133326110720000, 1148978521728221184000000, 28806532937614688256000000

Offset: 1

Rick L. Shepherd, Jun 09 2005

nonn

A000142(n)/A000217(n) = n!/(n*(n+1)/2) = 2*(n-1)!/(n+1) is an integer iff n = 1 or n + 1 is composite; i.e., iff n is a term of A060462.

Cf. A060462 (corresponding k), A000142 (factorials), A000217 (triangular numbers).

select(x-> denom(x)=1, [k!/(k*(k+1)/2)$k=1..30])[];  # Alois P. Heinz, Dec 11 2020

Select[Table[(n - 1)!/((n (n - 1))/2), {n, 2, 50}], IntegerQ[#] &] (* Geoffrey Critzer, May 02 2015 *)

for(n=1,50, r=2*(n-1)!/(n+1); if(denominator(r)==1, print1(r,",")))

a(m) = 2*(A060462(m)-1)!/(A060462(m)+1) = A000142(A060462(m))/A000217(A060462(m)).

Offset corrected by Alois P. Heinz, Dec 11 2020

cp's OEIS Frontend

A258257 The number of representations of n as a minimal number of triangular numbers, A000217(n).

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A259413 Triangular numbers (A000217) that are the sum of eleven consecutive triangular numbers.

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A259414 Triangular numbers (A000217) that are the sum of thirteen consecutive triangular numbers.

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A259415 Triangular numbers (A000217) that are the sum of seventeen consecutive triangular numbers.

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A276598 Values of m such that m^2 + 3 is a triangular number (A000217).

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A349243 Indices of triangular numbers A000217 with only odd digits.

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A061208 Numbers which can be expressed as sum of distinct triangular numbers (A000217).

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A108552 Integer values of (12...*k)/(1+2+...+k) = k!/T(k) = A000142(k)/A000217(k), k>=1.