A000217 -id:A000217 - cp's OEIS frontend

A014134 Numbers that are not the sum of a square (A000290) and a triangular number (A000217).

8, 13, 18, 20, 23, 27, 33, 34, 38, 41, 43, 47, 48, 58, 60, 62, 63, 68, 69, 73, 76, 83, 86, 88, 89, 90, 93, 97, 98, 99, 108, 111, 112, 113, 118, 123, 125, 132, 133, 134, 135, 138, 139, 143, 146, 148, 151, 158, 160, 163, 164, 167, 168, 173, 174

Offset: 1

N. J. A. Sloane

n is in the sequence if for some prime p == 5 or 7 (mod 8), 8*n+1 is divisible by p but not by p^2. The first member of the sequence that does not have this property is 16078. - Robert Israel, Mar 17 2015

R. J. Mathar, Table of n, a(n) for n = 1..4646

Cf. A140867.

N:= 1000: # to generate all terms <= N
{$1 .. N} minus {seq(seq(x*(x+1)/2 + y^2, y = 0 .. floor(sqrt(N - x*(x+1)/2))),
x = 0 .. floor((sqrt(8*N+1)-1)/2))};
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Mar 17 2015

fQ[n_] := Block[{k = 0, lmt = 1 + Floor@ Sqrt@ n}, While[k < lmt && !IntegerQ@ Sqrt[ 8(n - k^2) + 1], k++]; k == lmt]; Select[ Range@ 175, fQ@# &] (* Robert G. Wilson v, Nov 29 2015 *)

is_A014134(n)=for(k=0,sqrtint(n*2),issquare(n-k*(k+1)/2)&return);1 /* M. F. Hasler, Jan 05 2009 */

A028401 The (2^n+1)-th triangular number (cf. A000217).

Original entry on oeis.org

3, 6, 15, 45, 153, 561, 2145, 8385, 33153, 131841, 525825, 2100225, 8394753, 33566721, 134242305, 536920065, 2147581953, 8590131201, 34360131585, 137439739905, 549757386753, 2199026401281, 8796099313665, 35184384671745

Offset: 2

N. J. A. Sloane

Number of types of Boolean functions of n variables under a certain group.

Also the number of ordered decompositions of 2^n into 3 nonnegative integers (e.g., 2 = 0+0+2 = 0+2+0 = 2+0+0 = 1+1+0 = 1+0+1 = 0+1+1). - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Aug 02 2007

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Daniel Poveda Parrilla, Illustration of initial terms
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
Index entries for sequences related to Boolean functions
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Equals 2*A036562(n-4) - 1, n > 3.

Cf. A000217.

Drop[#, 2] &@ CoefficientList[Series[3 x^2*(1 - 5 x + 5 x^2)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 25}], x] (* Michael De Vlieger, Jul 08 2019 *)

def A028401(n): return ((m:=1<2 else 3 # Chai Wah Wu, Jul 11 2024

From Ralf Stephan, Aug 23 2003: (Start)
a(n) = (3/8)*2^n + (1/32)*4^n + 1.
a(n) = 3*A007581(n-2) = (3/4)*A060919(n-1). (End)
a(n) = (2^n+4)*(2^n+8)/32. - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Aug 02 2007
G.f.: 3*x^2*(1-5*x+5*x^2)/((1-x)*(1-2*x)*(1-4*x)). - Colin Barker, Mar 09 2012
a(n) = a(n-1) + 3*A000217(2^(n-3)) for n > 2. - Daniel Poveda Parrilla, Dec 27 2016
E.g.f.: (32*exp(x) + 12*exp(2*x) + exp(4*x) - 45 - 60*x)/32. - Stefano Spezia, Jul 11 2024

More terms from Vladeta Jovovic, Feb 24 2000

Simpler definition from Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Aug 02 2007

A058498 Number of solutions to c(1)t(1) + ... + c(n)t(n) = 0, where c(i) = +-1 for i>1, c(1) = t(1) = 1, t(i) = triangular numbers (A000217).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 2, 0, 6, 8, 13, 0, 33, 52, 105, 0, 310, 485, 874, 0, 2974, 5240, 9488, 0, 30418, 55715, 104730, 0, 352467, 642418, 1193879, 0, 4165910, 7762907, 14493951, 0, 50621491, 95133799, 179484713, 0, 637516130, 1202062094, 2273709847, 0, 8173584069

Offset: 1

Naohiro Nomoto, Dec 20 2000

nonn

			a(8) = 2 because there are two solutions: 1 - 3 + 6 + 10 + 15 - 21 + 28 - 36 = 1 - 3 - 6 + 10 - 15 + 21 + 28 - 36 = 0.

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..633 (first 280 terms from Alois P. Heinz)

Cf. A000217.

b:= proc(n, i) option remember; local m; m:= (2+(3+i)*i)*i/6;
      `if`(n>m, 0, `if`(n=m, 1,
      b(abs(n-i*(i+1)/2), i-1) +b(n+i*(i+1)/2, i-1)))
    end:
a:= n-> `if`(irem(n, 4)=1, 0, b(n*(n+1)/2, n-1)):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 31 2011

b[n_, i_] := b[n, i] = With[{m = (2+(3+i)*i)*i/6}, If[n>m, 0, If[n == m, 1, b[Abs[n - i*(i+1)/2], i-1] + b[n + i*(i+1)/2, i-1]]]]; a[n_] := If[Mod[n, 4] == 1, 0, b[n*(n+1)/2, n-1]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 30 2017, after Alois P. Heinz *)

a(n) = [x^(n*(n+1)/2)] Product_{k=1..n-1} (x^(k*(k+1)/2) + 1/x^(k*(k+1)/2)). - Ilya Gutkovskiy, Feb 01 2024

More terms from Sascha Kurz, Oct 13 2001

More terms from Alois P. Heinz, Oct 31 2011

A113231 Ascending descending base exponent transform of triangular numbers (A000217).

Original entry on oeis.org

1, 4, 34, 956, 106721, 75818480, 490656737694, 22960404169011552, 7141530219670856270919, 20319415706020976355219258316, 1104797870481014132439711155738991604

Offset: 1

Jonathan Vos Post, Jan 07 2006

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. Since the parity of the triangular numbers cycles odd, odd, even, even; the parity of this sequence cycles odd, even, even, even. The smallest prime in this sequence is a(5) = 127601. What is the next prime? What is the first triangular value?

			a(1) = 1 because T(1)^T(1) = 1^1 = 1.
a(2) = 4 because T(1)^T(2) + T(2)^T(1) = 1^3 + 3^1 = 4.
a(3) = 34 = 1^6 + 3^3 + 6^1.
a(4) = 956 = 1^10 + 3^6 + 6^3 + 10^1.
a(5) = 106721 = 1^15 + 3^10 + 6^6 + 10^3 + 15^1.
a(6) = 75818480 = 1^21 + 3^15 + 6^10 + 10^6 + 15^3 + 21^1.
a(7) = 490656737694 = 1^28 + 3^21 + 6^15 + 10^10 + 15^6 + 21^3 + 28^1.
a(8) = 22960404169011552 = 1^36 + 3^28 + 6^21 + 10^15 + 15^10 + 21^6 + 28^3 + 36^1.
a(9) = 7141530219670856270919 = 1^45 + 3^36 + 6^28 + 10^21 + 15^15 + 21^10 + 28^6 + 36^3 + 45^1.

G. C. Greubel, Table of n, a(n) for n = 1..40

Cf. A000217, A005408, A113122, A113153, A113154.

A000217[n_] := Binomial[n + 1, 2]; Table[Sum[A000217[k]^(A000217[n - k + 1]), {k, 1, n}], {n, 1, 10}] (* G. C. Greubel, May 18 2017 *)

for(n=1,10, print1(sum(k=1,n, (binomial(k+1,2))^(binomial(n-k+2,2))), ", ")) \\ G. C. Greubel, May 18 2017

a(n) = Sum_{i=1..n} (T(i))^(T(n-i+1)), where T(n) are the triangle numbers.
a(n) = Sum_{i=1..n} ((i*(i+1)/2))^((n-i+1)*(n-i+2)/2).
a(n) = Sum_{i=1..n} (A000217(i))^(A000217(n-i+1)).
log(a(n)) ~ n^2 * (-1 + 2*LambertW(2^(-3/2)*exp(1/2)*n))^3 / (8*LambertW(2^(-3/2)*exp(1/2)*n)^2). - Vaclav Kotesovec, Jun 07 2025

A165517 Indices of the least triangular numbers (A000217) for which three consecutive triangular numbers sum to a perfect square (A000290).

Original entry on oeis.org

0, 5, 14, 63, 152, 637, 1518, 6319, 15040, 62565, 148894, 619343, 1473912, 6130877, 14590238, 60689439, 144428480, 600763525, 1429694574, 5946945823, 14152517272, 58868694717, 140095478158, 582740001359, 1386802264320

Offset: 1

Ant King, Sep 25 2009, Oct 01 2009

Those perfect squares that can be expressed as the sum of three consecutive triangular numbers correspond to integer solutions of the equation T(k)+T(k+1)+T(k+2)=s^2, or equivalently to 3k^2 + 9k + 8 = 2s^2. Hence solutions occur whenever (3k^2 + 9k + 8)/2 is a perfect square, or equivalently when s>=2 and sqrt(24s^2 - 15) is congruent to 3 mod 6. This sequence returns the index of the smallest of the 3 triangular numbers, the values of s^2 are given in A165516 and, with the exception of the first term, the values of s are in A129445.

			The fourth perfect square that can be expressed as the sum of three consecutive triangular numbers is 6241 = T(63) + T(64) + T(65). Hence a(4)=63.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Tom Beldon and Tony Gardiner, Triangular Numbers and Perfect Squares, The Mathematical Gazette, Vol. 86, No. 507, (2002), pp. 423-431.
Index entries for linear recurrences with constant coefficients, signature (1,10,-10,-1,1).

Cf. A000290, A129445, A000217, A165516.

I:=[0, 5, 14, 63, 152]; [n le 5 select I[n] else Self(n-1) + 10*Self(n-2) - 10*Self(n-3) - Self(n-4) + Self(n-5): n in [1..50]]; // G. C. Greubel, Oct 21 2018

TriangularNumber[ n_ ]:=1/2 n (n+1);Select[ Range[ 0,10^7 ], IntegerQ[ Sqrt[ TriangularNumber[ # ]+TriangularNumber[ #+1 ]+TriangularNumber[ #+2 ] ] ] & ]
CoefficientList[Series[x*(x^3 + x^2 - 9*x - 5)/((x - 1)*(x^4 - 10*x^2 + 1)), {x,0,50}], x] (* or *) LinearRecurrence[{1,10,-10,-1,1}, {0, 5, 14, 63, 152}, 50] (* G. C. Greubel, Feb 17 2017 *)

x='x+O('x^50); concat([0], Vec(x*(x^3 + x^2 - 9*x - 5)/((x - 1)*(x^4 - 10*x^2 + 1)))) \\ G. C. Greubel, Feb 17 2017

a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - a(n-4) + a(n-5).
G.f.: x(x^3 + x^2 - 9x - 5)/((x-1)(x^4 - 10x^2 + 1)).
a(n) = 10*a(n-2) - a(n-4) + 12. - Zak Seidov, Sep 25 2009

a(1) = 0 added by N. J. A. Sloane, Sep 28 2009, at the suggestion of Alexander R. Povolotsky

More terms from Zak Seidov, Sep 25 2009

A232096 a(n) = largest m such that m! divides 1+2+...+n; a(n) = A055881(A000217(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 18 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080

A042963 gives the positions of ones and A014601 the positions of larger terms.

Cf. also A232097, A076733, A232094, A232095, A232098.

(define (A232096 n) (A055881 (A000217 n)))

a(n) = A055881(A000217(n)).

a(n) = A231719(A226061(n+1)). [Not a practical way to compute this sequence, but follows from the definitions]

A333516 Irregular triangle read by rows in which row n lists the first A000217(n) terms of A002260, n >= 1.

Original entry on oeis.org

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

Offset: 1

Andrew Slattery, Mar 25 2020

a(n) equals the difference between n and the largest number less than n that can be expressed as the sum of the i-th triangular number and the j-th tetrahedral number for integers i < j.

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 2, 1, 2, 3;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6;
  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;
  ...

Row sums give A000292.
Right border gives A000027.
Cf. A000217, A002260, A124171.

T:= n-> seq([$1..i][], i=1..n):
seq(T(n), n=1..7);  # Alois P. Heinz, Apr 10 2020

from math import comb, isqrt
from sympy import integer_nthroot
def A333516(n): return (r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(n>comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2)+1 # Chai Wah Wu, Nov 10 2024

a(n) = A002260(A124171(n)).

A383834 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000217(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

1, 7, 31, 97, 241, 511, 967, 1681, 2737, 4231, 6271, 8977, 12481, 16927, 22471, 29281, 37537, 47431, 59167, 72961, 89041, 107647, 129031, 153457, 181201, 212551, 247807, 287281, 331297, 380191, 434311, 494017, 559681, 631687, 710431, 796321, 889777, 991231, 1101127, 1219921, 1348081

Offset: 0

Miguel-Ángel Pérez García-Ortega, May 11 2025

			For n=1, the short leg is A002061(1) = 3 and the long leg is A212135(2) = 4 so the sum of the legs is then a(1) = 3 + 4 = 7.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A000217, A383833, A002061, A006007, A058919, A336535.

a=Table[(n(n+1))/2,{n,0,40}];Apply[Join,Map[{2#^2+4#+1}&,a]]

a(n) = 2*(A000217(n))^2 + 4*A000217(n) + 1.

a(n) = 6*A006007(n) + 1

A190405 Decimal expansion of Sum_{k>=1} (1/2)^T(k), where T=A000217 (triangular numbers); based on column 1 of the natural number array, A000027.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, May 10 2011

See A190404.
Binary expansion is .1010010001... (A023531). - Rick L. Shepherd, Jan 05 2014
From Amiram Eldar, Dec 07 2020: (Start)
This constant is not a quadratic irrational (Duverney, 1995).
The Engel expansion of this constant are the powers of 2 (A000079) above 1. (End)

			0.64163256065515386629...

Danny Rorabaugh, Table of n, a(n) for n = 0..10000
Daniel Duverney, Sommes de deux carrés et irrationalité de valeurs de fonctions thêta, Comptes rendus de l'Académie des sciences, Série 1, Mathématique, Vol. 320, No. 9 (1995), pp. 1041-1044.

A190404: (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T = A000217 (triangular numbers).
A190405: Sum_{k>=1} (1/2)^T(k), where T = A000217 (triangular numbers).
A190406: Sum_{k>=1} (1/2)^S(k-1), where S = A001844 (centered square numbers).
A190407: Sum_{k>=1} (1/2)^V(k), where V = A058331 (1 + 2*k^2).
Cf. A000079.

RealDigits[EllipticTheta[2, 0, 1/Sqrt[2]]/2^(7/8) - 1, 10, 120] // First (* Jean-François Alcover, Feb 12 2013 *)
RealDigits[Total[(1/2)^Accumulate[Range[50]]],10,120][[1]] (* Harvey P. Dale, Oct 18 2013 *)
(* See also A190404 *)

th2(x)=2*x^.25 + 2*suminf(n=1,x^(n+1/2)^2)
th2(sqrt(.5))/2^(7/8)-1 \\ Charles R Greathouse IV, Jun 06 2016

def A190405(b):  # Generate the constant with b bits of precision
    return N(sum([(1/2)^(j*(j+1)/2) for j in range(1,b)]),b)
A190405(409) # Danny Rorabaugh, Mar 25 2015

A276600 Values of m such that m^2 + 6 is a triangular number (A000217).

Original entry on oeis.org

0, 2, 3, 7, 15, 20, 42, 88, 117, 245, 513, 682, 1428, 2990, 3975, 8323, 17427, 23168, 48510, 101572, 135033, 282737, 592005, 787030, 1647912, 3450458, 4587147, 9604735, 20110743, 26735852, 55980498, 117214000, 155827965, 326278253, 683173257, 908231938

Offset: 1

Colin Barker, Sep 07 2016

2*a(n+2) gives the y members of all positive solutions (x(n), y(n)), proper and improper, of the Pell equation x^2 - 2*y^2 = 7^2, n >= 0. The corresponding x members are x(n) = A106525(n). - Wolfdieter Lang, Sep 29 2016

			7 is in the sequence because 7^2 + 6 = 55, which is a triangular number.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A000217, A230044.
Cf. A001109 (k=0), A106328 (k=1), A077241 (k=2), A276598 (k=3), A276599 (k=5), A276601 (k=9), A276602 (k=10), where k is the value added to n^2.
Cf. A275794, A275796, A106525.

I:=[0,2,3,7,15,20]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..41]]; // G. C. Greubel, Sep 15 2021

LinearRecurrence[{0,0,6,0,0,-1}, {0,2,3,7,15,20}, 41] (* G. C. Greubel, Sep 15 2021 *)

concat(0, Vec(x^2*(2+3*x+7*x^2+3*x^3+2*x^4)/(1-6*x^3+x^6) + O(x^40)))

def A276600_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2*(2+3*x+7*x^2+3*x^3+2*x^4)/(1-6*x^3+x^6) ).list()
a=A276600_list(41); a[1:] # G. C. Greubel, Sep 15 2021

a(n) = 6*a(n-3) - a(n-6) for n>6.
G.f.: x^2*(2 + 3*x + 7*x^2 + 3*x^3 + 2*x^4)/(1 - 6*x^3 + x^6).
From Wolfdieter Lang, Sep 29 2016: (Start)
Trisection:
a(2+3*n) = 15*S(n-1,6) - 2*S(n-2,6) = A275794(n),
a(3+3*n) = 20*S(n-1,6) - 3*S(n-2,6) = A275796(n),
a(4+3*n) = 7*(6*S(n-1,6) - S(n-2,6)) = 7*A001109(n+1) for n >= 0, with the Chebyshev polynomials S(n, 6) = A001109(n+1), n >= -1, with S(-2, 6) = -1.
(End)

cp's OEIS Frontend

A014134 Numbers that are not the sum of a square (A000290) and a triangular number (A000217).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A028401 The (2^n+1)-th triangular number (cf. A000217).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A058498 Number of solutions to c(1)t(1) + ... + c(n)t(n) = 0, where c(i) = +-1 for i>1, c(1) = t(1) = 1, t(i) = triangular numbers (A000217).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A113231 Ascending descending base exponent transform of triangular numbers (A000217).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A165517 Indices of the least triangular numbers (A000217) for which three consecutive triangular numbers sum to a perfect square (A000290).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A232096 a(n) = largest m such that m! divides 1+2+...+n; a(n) = A055881(A000217(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A333516 Irregular triangle read by rows in which row n lists the first A000217(n) terms of A002260, n >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple