A076708 -id:A076708 - cp's OEIS frontend

A012132 Numbers z such that x(x+1) + y(y+1) = z*(z+1) is solvable in positive integers x,y.

3, 6, 8, 10, 11, 13, 15, 16, 18, 20, 21, 23, 26, 27, 28, 31, 33, 36, 37, 38, 40, 41, 43, 44, 45, 46, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 71, 73, 74, 75, 76, 77, 78, 80, 81, 83, 86, 88, 89, 91, 92, 93

Offset: 1

Sander van Rijnswou (sander(AT)win.tue.nl)

nonn

Theorem (Sierpinski, 1963): n is a term iff n^2+(n+1)^2 is a composite number. - N. J. A. Sloane, Feb 29 2020
For n > 1, A047219 is a subset of this sequence. This is because n^2 + (n+1)^2 is divisible by 5 if n is (1 or 3) mod 5 (also see A027861). - Dmitry Kamenetsky, Sep 02 2008
From Hermann Stamm-Wilbrandt, Sep 10 2014: (Start)
For n > 0, A212160 is a subset of this sequence (n^2 + (n+1)^2 is divisible by 13 if n == (2 or 10) (mod 13)).
For n >= 0, A212161 is a subset of this sequence (n^2 + (n+1)^2 is divisible by 17 if n == (6 or 10) (mod 17)).
The above are for divisibility by 5, 13, 17; notation (1,3,5), (2,10,13), (6,10,17). Divisibility by p for a and p-a-1; notation (a,p-a-1,p). These are the next tuples: (8,20,29), (15,21,37), (4,36,41), (11,41,53), ... . The corresponding sequences are a subset of this sequence (8,20,37,49,66,78,... for (8,20,29)). These sequences have no entries in the OEIS yet. For any prime of the form 4*k+1 there is exactly one of these tuples/sequences.
For n > 1, A000217 (triangular numbers) is a subset of this sequence (3,6,10,15,...); z=A000217(n), y=z-1, x=n.
For n > 0, A001652 is a subset of this sequence; z=A001652(n), x=y=A053141(n).
For n > 1, A001108(=A115598) is a subset of this sequence; z=A001108(n), x=A076708(n), y=x+1.
For n > 0, A124124(2*n+1)(=A098790(2*n)) is a subset of this sequence (6,37,218,...); z=A124124(2*n+1), x=a(n)-1, y=a(n)+1, a(m) = 6*a(m-1) - a(m-2) + 2, a(0)=0, a(1)=4.
(End)

Aviezri S. Fraenkel, Diophantine equations involving generalized triangular and tetrahedral numbers, pp. 99-114 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
H. Finner and K. Strassburger, Increasing sample sizes do not necessarily increase the power of UMPU-tests for 2 X 2-tables, Metrika, 54, 77-91, (2001).
Heiko Harborth, Fermat-like binomial equations, Applications of Fibonacci numbers, Proc. 2nd Int. Conf., San Jose/Ca., August 1986, 1-5 (1988).
W. Sierpinski, On triangular numbers which are sums of two smaller triangular numbers, (Polish), Wiadom. Mat. (2) 7 (1963), 27-28. See MR0182602.

Complement of A027861. - Michael Somos, Jun 08 2000
Cf. A000217, A047219, A027861, A166080.
Cf. also A212160, A212161, A001652, A001108, A115598, A124124, A098790.

Select[Range[100], !PrimeQ[#^2 + (#+1)^2]& ] (* Jean-François Alcover, Jan 17 2013, after Michael Somos *)

More terms and references from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 09 2000

A239969 Least positive k such that triangular(n) + triangular(n+k) is a triangular number (A000217), or -1 if no such k exists.

Original entry on oeis.org

2, 5, 1, 3, 20, 2, 4, 16, 3, 5, 31, 4, 6, 119, 5, 7, 16, 6, 8, 103, 7, 9, 2, 8, 10, 26, 9, 11, 464, 10, 12, 1, 11, 13, 313, 12, 5, 58, 13, 15, 37, 14, 3, 493, 15, 17, 31, 16, 18, 47, 17, 2, 79, 9, 20, 796, 19, 21, 883, 20, 22, 89, 4, 23, 58, 22, 24, 100, 23, 25, 1276

Offset: 3

Alex Ratushnyak, Mar 30 2014

nonn

In other words, smallest solution k>0 to 4*k^2 + 8*(k + 1)*n + 8*n^2 + 4*k + 1 = m^2. - Ralf Stephan, Apr 01 2014

			a(3) = 2 because triangular(3)+triangular(3+2)=21 is a triangular number.
a(5) = 1 because triangular(5)+triangular(5+1)=36 is a triangular number.
In other words, k=a(3)=2 is the smallest positive solution to 4*k^2 + 28*k + 97 = m^2, and k=a(5)=1 is the smallest positive solution to 4*k^2 + 44*k + 241 = m^2.

Reinhard Zumkeller, Table of n, a(n) for n = 3..1000

Cf. A000217, A082183, A076708, A076049, A239970.

Cf. A010054.

a239969 n = head [k | k <- [1..],
                      a010054 (a000217 n + a000217 (n + k)) == 1]
-- Reinhard Zumkeller, Apr 03 2014

triangular(n) = n*(n+1)/2;
is_triangular(n) = issquare(8*n+1);
s=[]; for(n=3, 100, k=1; while(!is_triangular(triangular(n)+triangular(n+k)), k++); s=concat(s, k)); s \\ Colin Barker, Mar 31 2014

First PROG corrected by Colin Barker, Apr 04 2014

A262140 The first of nine consecutive positive integers the sum of the squares of which is equal to the sum of the squares of eight consecutive positive integers.

Original entry on oeis.org

20, 136, 812, 4752, 27716, 161560, 941660, 5488416, 31988852, 186444712, 1086679436, 6333631920, 36915112100, 215157040696, 1254027132092, 7309005751872, 42600007379156, 248291038523080, 1447146223759340, 8434586304032976, 49160371600438532

Offset: 1

Colin Barker, Sep 12 2015

For the first of the corresponding eight consecutive positive integers, see A262139.

			20 is in the sequence because 20^2 + ... + 28^2 = 5244 = 22^2 + ... + 29^2.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A076708, A262139.

Vec(4*x*(x-5)/((x-1)*(x^2-6*x+1)) + O(x^40))

a(n) = 4*A076708(n+1).
a(n) = 7*a(n-1)-7*a(n-2)+a(n-3) for n>3.
G.f.: 4*x*(x-5) / ((x-1)*(x^2-6*x+1)).
E.g.f.: exp(x)*(exp(2*x)*(4*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)) - 4). - Stefano Spezia, Aug 08 2025

A251924 Numbers n such that the sum of the triangular numbers T(n) and T(n+1) is equal to a hexagonal number H(m) for some m.

Original entry on oeis.org

0, 34, 1188, 40390, 1372104, 46611178, 1583407980, 53789260174, 1827251437968, 62072759630770, 2108646576008244, 71631910824649558, 2433376321462076760, 82663163018885960314, 2808114166320660573948, 95393218491883573553950, 3240561314557720840260384

Offset: 1

Colin Barker, Dec 11 2014

Also nonnegative integers x in the solutions to 2*x^2-4*y^2+4*x+2*y+2 = 0, the corresponding values of y being A008844.

First bisection of A076708. [Bruno Berselli, Dec 11 2014]

			34 is in the sequence because T(34)+T(35) = 595+630 = 1225 = H(25).

Colin Barker, Table of n, a(n) for n = 1..653
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A000217, A000384, A008844, A076708.

LinearRecurrence[{35,-35,1},{0,34,1188},20] (* Harvey P. Dale, Feb 04 2019 *)

concat(0, Vec(2*x^2*(x-17)/((x-1)*(x^2-34*x+1)) + O(x^100)))

a(n) = 35*a(n-1)-35*a(n-2)+a(n-3).
G.f.: 2*x^2*(x-17) / ((x-1)*(x^2-34*x+1)).
a(n) = (-8-(4+3*sqrt(2))*(17+12*sqrt(2))^(-n)+(-4+3*sqrt(2))*(17+12*sqrt(2))^n)/8. - Colin Barker, Mar 02 2016

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A012132 Numbers z such that x*(x+1) + y*(y+1) = z*(z+1) is solvable in positive integers x,y.

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A239969 Least positive k such that triangular(n) + triangular(n+k) is a triangular number (A000217), or -1 if no such k exists.

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A262140 The first of nine consecutive positive integers the sum of the squares of which is equal to the sum of the squares of eight consecutive positive integers.

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A251924 Numbers n such that the sum of the triangular numbers T(n) and T(n+1) is equal to a hexagonal number H(m) for some m.

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A012132 Numbers z such that x(x+1) + y(y+1) = z*(z+1) is solvable in positive integers x,y.