Ralf Steiner - cp's OEIS frontend

A349682 a(n) = A000292(6*n + 1) where A000292 are the tetrahedral numbers.

1, 84, 455, 1330, 2925, 5456, 9139, 14190, 20825, 29260, 39711, 52394, 67525, 85320, 105995, 129766, 156849, 187460, 221815, 260130, 302621, 349504, 400995, 457310, 518665, 585276, 657359, 735130, 818805, 908600, 1004731, 1107414, 1216865, 1333300, 1456935, 1587986

Offset: 0

Ralf Steiner, Nov 25 2021

Euclid of Alexandria, Elements, VII Def. 17, p. 194, 300 BCE.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A000447, A002492, A016921.

nterms=50;Table[36n^3+36n^2+11n+1,{n,0,nterms-1}] (* Paolo Xausa, Nov 25 2021 *)

a(n) = subst(m*(m+1)*(m+2)/6, 'm, 6*n+1); \\ Michel Marcus, Dec 16 2021

def A349682(n): return n*(n*(36*n + 36) + 11) + 1 # Chai Wah Wu, Dec 27 2021

a(n) = 1 + 11*n + 36*n^2 + 36*n^3 = (1 + 2*n)*(1 + 3*n)*(1 + 6*n).
G.f.: (1 + 80*x + 125*x^2 + 10*x^3)/(1 - x)^4. - Stefano Spezia, Nov 29 2021
From Elmo R. Oliveira, Aug 22 2025: (Start)
E.g.f.: exp(x)*(1 + 83*x + 144*x^2 + 36*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
From Amiram Eldar, Aug 31 2025: (Start)
Sum_{n>=0} 1/a(n) = Pi/(4*sqrt(3)) + 2*log(2) - 3*log(3)/4.
Sum_{n>=0} (-1)^n/a(n) = (3/4 - 1/sqrt(3))*Pi + sqrt(3)*log(2 + sqrt(3))/2 - log(2). (End)

A334909 Area/6 of primitive Pythagorean triangles given in A334638 as triples.

Original entry on oeis.org

1, 35, 770, 14260, 244776, 4053840, 65979040, 1064678720, 17107266176, 274296689920, 4393395202560, 70331527418880, 1125602147608576, 18012016334950400, 288211318352814080, 4611533554425610240, 73785756576381566976, 1180581862943988449280

Offset: 0

Ralf Steiner, May 16 2020

See A334638 for these triangles, and also for the Firstov reference.
For primitive Pythagorean triangle (x, y, z) = (u^2 - v^2, 2*u*v, u^2 + v^2) the area is A = x*y/2 = u*v*(u^2 - v^2) = z*h/2 with altitude h, and h is an irreducible fraction.
From A334638 follows A(n)/6 = (x(n)/3)*(y(n)/4) = A171477(n)*A010036(n), for n >= 0. See the formula section.
Limit_{n->infinity} A(n)/(3*2^(4*n+3)) = 1. See the formula section. - Wolfdieter Lang, Jun 14 2020

			a(0) = 3*4/12 = 1 for (3, 4, 5).

Index entries for linear recurrences with constant coefficients, signature (30,-280,960,-1024).

Cf. A010036, A171477, A334638.

Table[ 2^(-1 + n) (-1 + 3 2^n) (-1 + 2^(1 + n)) (-1 + 2^(2 + n))/3, {n, 0, 17}]

Vec((1 + 5*x) / ((1 - 2*x)*(1 - 4*x)*(1 - 8*x)*(1 - 16*x)) + O(x^20)) \\ Colin Barker, May 17 2020

a(n) = 2^(n-1)*(3*2^n - 1)*(2^(n+1) - 1)*(2^(n+2) - 1)/3.
a(n) = 2^(4*n+2)*(1 - 13/(3*2^(n+2)) + 3/2^(2*n+3) - 1/(3*2^(3*(n+1)))), for n >= 0.
From Colin Barker: (Start)
G.f.: (1 + 5*x) / ((1 - 2*x)*(1 - 4*x)*(1 - 8*x)*(1 - 16*x)).
a(n) = 30*a(n-1) - 280*a(n-2) + 960*a(n-3) - 1024*a(n-4) for n > 3. (End)

Edited by Wolfdieter Lang, Jun 14 2020

A334638 Three-column array pPT read by rows: subsequence of primitive Pythagorean triples (x, y, z) with x = A153893^2 - A000079^2, y = 2A153893A000079, z = A153893^2 + A000079^2, ordered by increasing z.

Original entry on oeis.org

3, 4, 5, 21, 20, 29, 105, 88, 137, 465, 368, 593, 1953, 1504, 2465, 8001, 6080, 10049, 32385, 24448, 40577, 130305, 98048, 163073, 522753, 392704, 653825, 2094081, 1571840, 2618369, 8382465, 6289408, 10479617, 33542145, 25161728, 41930753, 134193153, 100655104, 167747585, 536821761, 402636800, 671039489, 2147385345, 1610579968, 2684256257

Offset: 0

Ralf Steiner, May 07 2020

Let [h21] = {{1, 3}, {0, 2}} be the matrix [h_2]*[h_1] in Firstov's notation, from eqs. (24) and (39). Then primitive Pythagorean triples (pPT) (x(n), y(n), z(n)) = (u(n)^2 - v(n)^2, 2*u(n)*v(n), u(n)^2 + v(n)^2), with u(n) and v(n) of different parity, gcd(u(n), v(n)) = 1, and u(n) > v(n) > 0, are generated by (u(n), v(n))^T = [h21]^n*(2,1)^T (T for transpose).
For n > 0: (x(n), y(n), z(n)) = (1, 0, 1) (mod 4). Thus some z are Pythagorean primes (A002144).
The triples converge to the proportion (4:3:5) with:
lim_{n->infinity} x(n)/y(n) = 4/3, lim_{n->infinity} y(n)/z(n) = 3/5.
Altitude h(n) = x(n)*y(n)/z(n) is an irreducible fraction because of primitivity.
From Wolfdieter Lang, Jun 13 2020: (Start)
[h21]^n = sqrt(2)^n*(S(n, 3/sqrt(2))*[1_3] + S(n-1, 3/sqrt(2))*(1/sqrt(2))*([h21] - 3*[1_3])) with the Chebyshev S polynomials (A049310).
u(n) = sqrt(2)^n*(2*S(n, 3/sqrt(2)) - (1/sqrt(2))*S(n-1, 3/sqrt(2)))
     = A153893(n),
v(n) = sqrt(2)^n*(S(n, 3/sqrt(2)) - (1/sqrt(2))*S(n-1, 3/sqrt(2)))
     = A000079(n). Proof from the recurrence, using the Cayley-Hamilton theorem.
With the monic Chebyshev T polynomials, called R in A127672:
x(n)/3 = 2^(n+1)*(R(2*(n+1), 3/sqrt(2)) - (sqrt(2)/3)*R(2*n+1,3/sqrt(2)) - 1) = A171477(n),
y(n)/4 = 3*2^(n-1)*(sqrt(2)*R(2*n+1,3/sqrt(2)) - R(2*n,3/sqrt(2)) - 1/3)
= A010036(n),
z(n) = 3*2^(n+1)*((3/sqrt(2))*R(2*n+1, 3/sqrt(2)) - (4/3)*R(2*n,3/sqrt(2)) - 1).
Using 2^n*Rnx(2*n, 3/sqrt(2)) = A052539(n) = 2^(2*n) + 1, and
  2^(n)*(sqrt(2)/3)*Rnx(2*n+1, 3/sqrt(2)) = A007583(n) = (2^(2*n + 1) + 1)/3,
produces the explicit formulas given by the author in the formula section.
G.f.s for {x(n)} G0(x) = 3/((1 - 4*x)*(1 - 2*x)*(1 - x)), for {y(n)} G1(x) =  4*(1-x)/((1 - 4*x)*(1 - 2*x)), and for {z(n)} = (5 - 6*x + 4*x^2)/((1 - 4*x)*(1 - 2*x)*(1 - x)). This produces the g.f. for the array, read as sequence {a(n)}: G(x) = G0(x^3) + x*G1(x^3) + x^2*G2(x^3) given in the formula section by Colin Barker.
(End)

			The three-column array pPT(n,k) begins:
n\k        0        1         2
-------------------------------
0:         3        4         5
1:        21       20        29
2:       105       88       137
3:       465      368       593
4:      1953     1504      2465
5:      8001     6080     10049
6:     32385    24448     40577
7:    130305    98048    163073
8:    522753   392704    653825
9:   2094081  1571840   2618369
10:  8382465  6289408  10479617
... - _Wolfdieter Lang_, Jun 13 2020

V. E. Firstov, A Special Matrix Transformation Semigroup of Primitive Pairs and the Genealogy of Pythagorean Triples; Mathematical Notes, volume 84, number 2, August 2008, pages 263-279; Link of the page (for the Russian article).
Ralf Steiner, Weitere spezielle Folge primitiver pythagoraischer Dreiecke, ResearchGate.
Wikipedia, Tree of primitive Pythagorean triples.
Index entries for linear recurrences with constant coefficients, signature (0,0,7,0,0,-14,0,0,8).

Cf. A000079, A010035, A049310, A083329, A093357, A010036, A134057, A153893, A171477.

h21={{1, 3}, {0, 2}}; l = {}; Do[v = MatrixPower[h21, n, {2, 1}]; p = v[[1]]; q = v[[2]];
a = p^2 - q^2; b = 2 p q; c = p^2 + q^2; l = AppendTo[l, {a, b, c}], {n, 0, 14}]; l // Flatten

Vec((3 + 4*x + 5*x^2 - 8*x^4 - 6*x^5 + 4*x^7 + 4*x^8) / ((1 - x)*(1 + x + x^2)*(1 - 2*x^3)*(1 - 4*x^3)) + O(x^35)) \\ Colin Barker, Jun 12 2020

The three-column array PT(n, k) is for k = 0, 1, 2:  x(n), y(n), z(n), for n >= 0, with
x(n) = a(3*n + 0) = A153893(n)^2 - A000079(n)^2 = 1 - 3*2^(n+1) + 2^(2*n+3) = binomial(2^(n+2) - 1, 2) = 3*A171477(n),
y(n) = a(3*n + 1) = 2*A153893(n)*A000079(n) = 2^(n+1)*(-1 + 3*2^n) = 4*A010036(n),
z(n) = a(3*n + 2) = A153893(n)^2 + A000079(n)^2 = 1 - 6*2^n + 10*2^(2*n).
From Colin Barker, May 08 2020: (Start)
G.f. (read as sequence {a(n)}): (3 + 4*x + 5*x^2 - 8*x^4 - 6*x^5 + 4*x^7 + 4*x^8) / ((1 - x)*(1 + x + x^2)*(1 - 2*x^3)*(1 - 4*x^3)).
a(n) = 7*a(n-3) - 14*a(n-6) + 8*a(n-9), for n > 8.
(End)

Edited, and corrected proportion by Wolfdieter Lang, Jun 13 2020

Minor grammatical edits. - N. J. A. Sloane, Sep 12 2020

A334908 Area/6 of primitive Pythagorean triangles generated by {{2, 0}, {1, -1}}^n * {{2}, {1}}, for n >= 0.

Original entry on oeis.org

1, 10, 220, 3080, 52976, 818720, 13333440, 211474560, 3398520576, 54257082880, 869067996160, 13897453373440, 222420341682176, 3558236809994240, 56935698394234880, 910939899548958720, 14575288593717067776, 233202615903456460800

Offset: 0

Ralf Steiner, May 16 2020

Matrix {{2, 0}, {1, -1}} is [g_{-2}] given by Firstov in eq. (24).
These primitive Pythagorean triples are also given by Lee Price as (M_2)^n (3,4,5)^T (T for transposed), with M_2 = {{2, 1, 1}, {2, -2, 2}, {2, -1, 3}}.
For a primitive Pythagorean triangle (x, y, z) = (u^2-v^2, 2*u*v, u^2+v^2) the area is A = x*y/2 = u*v*(u^2 - v^2) = z*h/2 with altitude h, and h is an irreducible fraction. Here:
x(n) = A084175(n+2).
y(n) = 4*(A084175(n+1) - A084175(n)) = A054881(n+2).
     = 2*A192382(n+1) = 4*A003683(n+1).
z(n) = A084175(n+2) + 2*A084175(n+1) - 4*A084175(n).
     = A108924(n+2)/2 = A084175(n+2) + 2*A139818(n+1).
     = A000302(n+1) + A139818(n+1).
u(n) = A000079(n+1) = 2^(n+1).
v(n) = A001045(n+1) = (2^(n+1) + (-1)^n)/3.
For the area A(n): Limit_{n -> oo} (3^3/(2^(4*n+7)))*A(n) = 1. See the formula section. - Wolfdieter Lang, Jun 14 2020

			a(0) = 3*4/12 = 1 for the triangle (3, 4, 5).

G. C. Greubel, Table of n, a(n) for n = 0..825
V. E. Firstov, A Special Matrix Transformation Semigroup of Primitive Pairs and the Genealogy of Pythagorean Triples; Mathematical Notes, volume 84, number 2, August 2008, pages 263-279; Link of the page (for the Russian article).
H. Lee Price, The Pythagorean Tree: A New Species, arXiv:0809.4324 [math.HO], 2008-2011
R. Steiner, Spezielle Folge primitiver pythagoräischer Dreiecke, researchgate.net, 2020
Index entries for linear recurrences with constant coefficients, signature (10,120,-320,-1024).

Cf. A000079, A000302, A001045, A003683, A054881, A084175, A108924, A139818, A192382.

[(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81: n in [0..40]]; // G. C. Greubel, Feb 18 2023

Table[(2^(2*n+1)*(2^(2*n+5) -3) + (-2)^n*(3*2^(2*n+3) -1))/3^4, {n,0,40}]

[(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81 for n in range(41)] # G. C. Greubel, Feb 18 2023

a(n) = ( 2^(4*n+6) - 3*2^(2*n+1) - 3*(-2)^(3*n+3) - (-2)^n )/3^4.
G.f.: 1 / ((1 + 2*x)*(1 - 4*x)*(1 + 8*x)*(1 - 16*x)). - Colin Barker, Jun 11 2020
E.g.f.: (1/81)*(24*exp(-8*x) - exp(-2*x) - 6*exp(4*x) + 64*exp(16*x)). - G. C. Greubel, Feb 18 2023

A326917 Nonnegative numbers of the form 8*T(x) - T(y) with 0 <= x, 0 <= y, where T() denotes a triangular number.

Original entry on oeis.org

0, 2, 3, 5, 7, 8, 9, 12, 14, 15, 18, 20, 21, 23, 24, 25, 27, 29, 32, 33, 34, 35, 38, 42, 44, 45, 47, 48, 52, 53, 54, 57, 59, 60, 62, 63, 65, 70, 71, 74, 75, 77, 78, 79, 80, 84, 88, 89, 90, 92, 93, 96, 98, 99, 102, 104, 105, 107, 110, 113, 114, 115, 117, 119

Offset: 1

Ralf Steiner, Oct 21 2019

nonn

When incremented by 1 this is also the difference between an odd square (1 + 8*T) and a triangular number T.

			8*A000217(1) - A000217(2) = 8*1 - 3 = 5 = a(4).

Cf. A000217 (T), A175035, A016754 (odd squares).

T[n_] := n (n + 1)/2;Select[Union[Flatten[Table[8 T[x] - T[y], {x, 0, 15}, {y, 0, 100}]]],115 >= # >= 0 &]

a(n) = A175035(n) - 1.

A309388 Numbers y such that x(x+1) + y(y+1) = z*(z+1) does not have a solution in positive integers x, z with x <= y.

Original entry on oeis.org

1, 3, 4, 7, 8, 11, 12, 15, 16, 19, 23, 28, 31, 32, 36, 40, 43, 47, 52, 59, 60, 63, 64, 67, 71, 72, 79, 83, 87, 88, 96, 100, 103, 107, 108, 112, 127, 128, 131, 136, 139, 148, 151, 156, 163, 167, 172, 176, 179, 180, 183, 187, 191, 192, 196, 199, 211, 223, 227

Offset: 1

Ralf Steiner, Aug 02 2019

nonn

The similar sequence A027861 (complement of A012132) is related to primes.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Robert Israel)

Complement of A308395.

Cf. A012132, A027861.

filter:= proc(y) local S;
  S:= map(t -> subs(t, x), [isolve(x*(x+1)+y*(y+1)=z*(z+1))]);
  select(t -> t>0 and t<=y, S) = []
end proc:
select(filter, [$1..300]); # Robert Israel, Aug 06 2019

max = 500; lst = {}; For[x = 1, x < max, x++,
For[y = x, y < max, y++,
  For[z = y, z < max, z++,
   If[x (x + 1) + y (y + 1) == z (z + 1),
    lst = AppendTo[lst, y]]]]]; lst =
Select[Union[lst], # < max/2 &]; Complement[Range[Length[lst]], lst]

from sympy import integer_nthroot
A309388_list, y, w = [], 1, 0
while len(A309388_list) < 10000:
    w += y
    z = 0
    for x in range(1,y+1):
        z += x
        if integer_nthroot(8*(w+z)+1,2)[1]:
            break
    else:
        A309388_list.append(y)
    y += 1 # Chai Wah Wu, Aug 07 2019

A308395 Numbers y such that x(x+1) + y(y+1) = z*(z+1) is solvable in positive integers x, z with x <= y.

Original entry on oeis.org

2, 5, 6, 9, 10, 13, 14, 17, 18, 20, 21, 22, 24, 25, 26, 27, 29, 30, 33, 34, 35, 37, 38, 39, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 61, 62, 65, 66, 68, 69, 70, 73, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 89, 90, 91, 92, 93, 94, 95, 97

Offset: 1

Ralf Steiner, Jul 31 2019

nonn

			14 is a term because 14*15 + 14*15 = 20*21.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A012132.

max = 220; lst = {}; For[x = 1, x < max, x++,
For[y = x, y < max, y++,
  For[z = y, z < max, z++,
   If[x (x + 1) + y (y + 1) == z (z + 1),
    lst = AppendTo[lst, y]]]]]; Select[Union[lst], # < max/2 &]

from sympy import integer_nthroot
A308395_list, y, w = [], 1, 0
while len(A308395_list) < 10000:
    w += y
    z = 0
    for x in range(1,y+1):
        z += x
        if integer_nthroot(8*(w+z)+1,2)[1]:
            A308395_list.append(y)
            break
    y += 1 # Chai Wah Wu, Aug 02 2019

A319673 Primes that are neither a twin prime nor a Sophie Germain or safe prime.

Original entry on oeis.org

37, 67, 79, 97, 127, 157, 163, 211, 223, 257, 277, 307, 317, 331, 337, 353, 367, 373, 379, 389, 397, 401, 409, 439, 449, 457, 487, 499, 541, 547, 557, 577, 607, 613, 631, 647, 673, 677, 691, 701, 709, 727, 733, 739, 751, 757, 769, 773, 787, 797, 853, 877, 907, 919, 929, 937, 941, 947, 967, 971, 977, 991, 997

Offset: 1

Ralf Steiner, Sep 25 2018

nonn

			37 is prime, but it is not a twin prime (neither 35 nor 39 are prime), it is not a Sophie Germain prime (2*37 + 1 = 75 is not prime), and it is not a safe prime ((37 - 1)/2 = 18 is not prime).  So 37 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A001097, A005384, A005385.

Filtered([1..1000],p->IsPrime(p) and not IsPrime(p-2) and not IsPrime(p+2) and not IsPrime(2*p+1) and not IsPrime((p-1)/2)); # Muniru A Asiru, Sep 27 2018

[p: p in PrimesUpTo(1000) | not IsPrime(p-2) and not IsPrime(p+2)and not IsPrime(2*p+1)and not IsPrime((p-1) div 2)]; // Vincenzo Librandi, Oct 25 2018

select(p->isprime(p) and not isprime(p-2) and not isprime(p+2) and not isprime(2*p+1) and not isprime((p-1)/2),[$1..1000]); # Muniru A Asiru, Sep 27 2018

Select[Prime@ Range@ PrimePi[10^3], NoneTrue[{# - 2, # + 2, 2 # + 1, (# - 1)/2}, PrimeQ] &] (* Michael De Vlieger, Sep 26 2018 *)

isok(p) = isprime(p) && !isprime(p-2) && !isprime(p+2) && !isprime(2*p+1) && !isprime((p-1)/2); \\ Michel Marcus, Sep 26 2018

A000040\(A001097 U A005384 U A005385).

A319506 Number of numbers of the form 2p or 3p between consecutive triangular numbers T(n - 1) < {2,3}*p <= T(n) with p prime.

Original entry on oeis.org

0, 0, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 4, 3, 3, 3, 4, 5, 2, 4, 3, 5, 5, 2, 6, 3, 5, 5, 5, 5, 6, 4, 3, 7, 5, 6, 6, 5, 6, 7, 4, 5, 6, 6, 7, 6, 7, 9, 6, 6, 7, 8, 5, 6, 7, 9, 7, 7, 8, 7, 11, 7, 8, 8, 7, 6, 11, 5, 12, 7, 7, 7, 11, 11, 7, 12, 10, 9, 10, 7, 9, 9, 8, 10, 12, 10, 7, 10, 9, 12, 9, 11, 10, 13, 14, 10, 7

Offset: 1

Ralf Steiner, Sep 21 2018

nonn

1) It is conjectured that for k >= 1 each left-sided half-open interval (T(2*k - 1), T(2*k + 1)] and (T(2*k), T(2*(k + 1))] contains at least one composite c_2 = 2*p_i and c_3 = 3*p_j each, p_i, p_j prime, i != j.
2) It is conjectured that for k >= 3 each left-sided half-open interval (T(k - 1), T(k)] contains at least one composite c_2 = 2*p_i or c_3 = 3*p_j, p_i, p_j prime, i != j.
3) It is conjectured that for k >= 2 each left-sided half-open interval (T(2*k - 1), T(2*k)] contains at least one composite c_3 = 3*p_j, p_j prime.
4) It is conjectured that for k >= 1 each left-sided half-open interval (T(2*k), T(2*k + 1)] contains at least one composite c_2 = 2*p_i, p_i prime.

			a(3) = 2 since (T(3 - 1),T(3)] = {4 = 2*2,5,6 = 2*3 = 3*2}, 2,3 prime.

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

Cf. A000040 (primes), A000217 (triangular numbers).

Table[Count[
  Select[Range[(n - 1) n/2 + 1, n (n + 1)/2],
   PrimeQ[#/2] || PrimeQ[#/3] &], _Integer], {n, 1, 100}]
p23[{a_,b_}]:=Module[{r=Range[a+1,b]},Count[Union[Join[r/2,r/3]], ?PrimeQ]]; p23/@Partition[Accumulate[Range[0,100]],2,1] (* _Harvey P. Dale, May 02 2020 *)

isok1(n, k) = ((n%k) == 0) && isprime(n/k);
isok2(n) = isok1(n,2) || isok1(n,3);
t(n) = n*(n+1)/2;
a(n) = sum(i=t(n-1)+1, t(n), isok2(i)); \\ Michel Marcus, Oct 12 2018

A316676 Ordered set of products (P_s(k) + 1)*(P_s'(k') + 1), s,s' >= 3, k,k' in {3,4} with nontrivial polygonal numbers P_s(k).

Original entry on oeis.org

49, 70, 77, 91, 100, 110, 112, 119, 121, 130, 133, 143, 154, 160, 161, 169, 170, 175, 176, 187, 190, 196, 203, 208, 209, 217, 220, 221, 230, 238, 242, 245, 247, 250, 253, 256, 259, 272, 275, 280, 286, 287, 289, 290, 299, 301, 304, 308, 310, 319, 322, 323, 325, 329, 340, 341, 343, 350, 352, 361, 364

Offset: 1

Ralf Steiner, Jul 10 2018

nonn

Conjecture: All odd numbers d >= 17 excluding d in {P_s(k), s >= 3, k >= 5; P_s(k) - 1, s >= 3, k >= 4; a(n)} are accurate the primes p = d >= 17.

Cf. A090466 (nontrivial polygonal numbers).

pn[s_, k_] := s (k - 1) k/2 - (k - 1)^2 + 1;
lst = {}; Do[n = (pn[s, k] + 1) (pn[ss, ks] + 1);
lst = Union[lst, {n}], {s, 3, 20}, {ss, 3, 20}, {k, 3, 4}, {ks, 3,
  4}]; Take[lst, 70]

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User: Ralf Steiner

A349682 a(n) = A000292(6*n + 1) where A000292 are the tetrahedral numbers.

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A334909 Area/6 of primitive Pythagorean triangles given in A334638 as triples.

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A334638 Three-column array pPT read by rows: subsequence of primitive Pythagorean triples (x, y, z) with x = A153893^2 - A000079^2, y = 2*A153893*A000079, z = A153893^2 + A000079^2, ordered by increasing z.

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A334908 Area/6 of primitive Pythagorean triangles generated by {{2, 0}, {1, -1}}^n * {{2}, {1}}, for n >= 0.

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A326917 Nonnegative numbers of the form 8*T(x) - T(y) with 0 <= x, 0 <= y, where T() denotes a triangular number.

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A309388 Numbers y such that x*(x+1) + y*(y+1) = z*(z+1) does not have a solution in positive integers x, z with x <= y.

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Maple

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A308395 Numbers y such that x*(x+1) + y*(y+1) = z*(z+1) is solvable in positive integers x, z with x <= y.

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A334638 Three-column array pPT read by rows: subsequence of primitive Pythagorean triples (x, y, z) with x = A153893^2 - A000079^2, y = 2A153893A000079, z = A153893^2 + A000079^2, ordered by increasing z.

A309388 Numbers y such that x(x+1) + y(y+1) = z*(z+1) does not have a solution in positive integers x, z with x <= y.

A308395 Numbers y such that x(x+1) + y(y+1) = z*(z+1) is solvable in positive integers x, z with x <= y.

A319506 Number of numbers of the form 2p or 3p between consecutive triangular numbers T(n - 1) < {2,3}*p <= T(n) with p prime.