A007654 -id:A007654 - cp's OEIS frontend

A175497 Numbers k with the property that k^2 is a product of two distinct triangular numbers.

0, 6, 30, 35, 84, 180, 204, 210, 297, 330, 546, 840, 1170, 1189, 1224, 1710, 2310, 2940, 2970, 3036, 3230, 3900, 4914, 6090, 6930, 7134, 7140, 7245, 7440, 8976, 10710, 12654, 14175, 14820, 16296, 16380, 17220, 19866, 22770, 25172, 25944, 29103

Offset: 1

Zak Seidov, May 30 2010

nonn

From Robert G. Wilson v, Jul 24 2010: (Start)
Terms in the i-th row are products contributed with a factor A000217(i):
(1) 0, 6, 35, 204, 1189, 6930, 40391, 235416, 1372105, 7997214, 46611179, ...
(2) 30, 297, 2940, 29103, 288090, 2851797, 28229880, ...
(3) 84, 1170, 16296, 226974, 3161340, ...
(4) 180, 3230, 57960, 1040050, 18662940, ...
(5) 330, 7245, 159060, 3492075, 76666590, ...
(6) 546, 14175, 368004, 9553929, ...
(7) 840, 25172, 754320, 22604428, ...
(8) 210, 1224, 7134, 41580, 242346, 1412496, 8232630, 47983284, ...
(9) 1710, 64935, 2465820, 93636225, ...
(10) 2310, 96965, 4070220, ...
(11) 3036, 139590, 6418104, ...
(12) 3900, 194922, 9742200, ...
(13) 4914, 265265, 14319396, ...
(14) 6090, 353115, 20474580, ...
(15) 7440, 461160, 28584480, ...
(End)
Numbers m with property that m^2 is a product of two distinct triangular numbers T(i) and T(j) such that i and j are in the same row of the square array A(n, k) defined in A322699. - Onur Ozkan, Mar 17 2023

R. J. Mathar, Table of n, a(n) for n = 1..1000 replacing a b-file of _Robert G. Wilson v_ of 2010.
R. J. Mathar, OEIS A175497, Mar 16 2023
Project Euler, Problem 833. Square Triangle Products.
M. Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT], 2008. Theorem 5.6.

From Robert G. Wilson v, Jul 24 2010: (Start)
A001109 (with the exception of 1), A011945, A075848 and A055112 are all proper subsets.
Many terms are in common with A147779.
Cf. A001108, A132596, A007654, A001108, A132593. (End)
Cf. A152005 (two distinct tetrahedral numbers).

isA175497 := proc(n)
    local i,Ti,Tj;
    if n = 0 then
        return true;
    end if;
    for i from 1 do
        Ti := i*(i+1)/2 ;
        if Ti > n^2 then
            return false;
        else
            Tj := n^2/Ti ;
            if Tj <> Ti and type(Tj,'integer') then
                if isA000217(Tj) then  # code in A000217
                    return true;
                end if;
            end if;
        end if;
    end do:
end proc:
for n from 0 do
    if isA175497(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, May 26 2016

triangularQ[n_] := IntegerQ[Sqrt[8n + 1]];
okQ[n_] := Module[{i, Ti, Tj}, If[n == 0, Return[True]]; For[i = 1, True, i++, Ti = i(i+1)/2; If[Ti > n^2, Return[False], Tj = n^2/Ti; If[Tj != Ti && IntegerQ[Tj], If[ triangularQ[Tj], Return[True]]]]]];
Reap[For[k = 0, k < 30000, k++, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jun 13 2023, after R. J. Mathar *)

from itertools import count, islice, takewhile
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A175497_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda k:not k or any(map(lambda d: is_square((d<<3)+1) and is_square((k**2//d<<3)+1), takewhile(lambda d:d**2A175497_list = list(islice(A175497_gen(),20)) # Chai Wah Wu, Mar 13 2023

def A175497_list(n):
    def A322699_A(k, n):
        p, q, r, m = 0, k, 4*k*(k+1), 0
        while m < n:
            p, q, r = q, r, (4*k+3)*(r-q) + p
            m += 1
        return p
    def a(k, n, j):
        if n == 0: return 0
        p = A322699_A(k, n)*(A322699_A(k, n)+1)*(2*k+1) - a(k, n-1, 1)
        q = (4*k+2)*p - A322699_A(k, n)*(A322699_A(k, n)+1)//2
        m = 1
        while m < j: p, q = q, (4*k+2)*q - p; m += 1
        return p
    A = set([a(k, 1, 1) for k in range(n+1)])
    k, l, m = 1, 1, 2
    while True:
        x = a(k, l, m)
        if x < max(A):
            A |= {x}
            A  = set(sorted(A)[:n+1])
            m += 1
        else:
            if m == 1 and l == 1:
                if k > n:
                    return sorted(A)
                k += 1
            elif m > 1:
                l += 1; m = 1
            elif l > 1:
                k += 1; l, m = 1, 1
# Onur Ozkan, Mar 15 2023

a(n)^2 = A169836(n). - R. J. Mathar, Mar 12 2023

A373016 a(n) is the least positive integer k such that 3n^2 + 2n + k is a square.

Original entry on oeis.org

4, 9, 3, 8, 15, 1, 8, 17, 28, 4, 15, 28, 43, 9, 24, 41, 60, 16, 35, 56, 4, 25, 48, 73, 11, 36, 63, 92, 20, 49, 80, 113, 31, 64, 99, 9, 44, 81, 120, 20, 59, 100, 143, 33, 76, 121, 3, 48, 95, 144, 16, 65, 116, 169, 31, 84, 139, 196, 48, 105, 164, 8, 67, 128, 191, 25, 88, 153, 220, 44, 111, 180

Offset: 1

Claude H. R. Dequatre, May 20 2024

The scatterplot shows an interesting crosshatch structure where all terms are at the intersection of ascending and descending hatches.
Terms on each hatch are quite well fitted by a polynomial of degree 2.
For terms on ascending hatches, the parity of the term indices does not change on a given hatch but alternates from one hatch to the next and on the same hatch, the parity of two consecutive terms alternates.
For terms on descending hatches, the parity of the indices of two consecutive terms alternates on the same hatch and that of terms does not change on the same hatch but alternates from one hatch to the next.
All squares exclusively are in ascending order on the same ascending hatch at n = 6, 10, 14, 18, 22, ... but some squares can be also found at the intersection of other hatches.
The first differences of the indices of the terms located on ascending and descending hatches are respectively equal to 4 and 3. For terms that are on the ascending and descending hatches, the differences of order 2 quickly become constant and equal to 2 and 4, respectively.
The fixed points begin 3, 48, 675, 9408, etc. They are all divisible by 3 and their parity seems to alternate. It appears that they are the positive terms of A007654.

			a(1) = 4 because 3*1^2 + 2*1 = 5 and 5 + 1, 5 + 2, 5 + 3 are not squares, but 5 + 4 is. So, 4 is a term.
a(2) = 9 because 3*2^2 + 2*2 = 16 and 16 + 1, 16 + 2, 16 + 3, 16 + 4, 16 + 5, 16 + 6, 16 + 7, 16 + 8 are not squares, but 16 + 9 is. So, 9 is a term.

Cf. A000005, A000290, A005408.
Cf. A045944, A080883.
Cf. A373026, A373028.
Sequences with similar scatterplot and pin plot graphs: A141130, A141131, A141134, A141135.

a(n)=my(t=3*n^2+2*n); (sqrtint(t)+1)^2-t \\ Charles R Greathouse IV, May 21 2024

a(n) is the smallest square greater than 3*n^2 + 2*n, minus 3*n^2 + 2*n. - Charles R Greathouse IV, May 21 2024
1 <= a(n) <= floor(sqrt(12)*n) + 3. I believe both bounds are tight infinitely often. - Charles R Greathouse IV, May 21 2024
a(n) = A080883(A045944(n)). - Michel Marcus, May 22 2024

A132607 X-values of solutions to the equation X(X + 1) - 11Y^2 = 0.

Original entry on oeis.org

0, 99, 39600, 15760899, 6272798400, 2496558002499, 993623812196400, 395459780696164899, 157391999093261433600, 62641620179337354408099, 24931207439377173792990000, 9922557919251935832255612099

Offset: 0

Mohamed Bouhamida, Nov 14 2007

Harvey P. Dale, Table of n, a(n) for n = 0..384
Index entries for linear recurrences with constant coefficients, signature (399,-399,1).

Cf. A007654.

LinearRecurrence[{399,-399,1},{0,99,39600},20] (* Harvey P. Dale, Aug 12 2022 *)

a(n) = 398*a(n-1) - a(n-2) + 198; a(0)=0, a(1)=99.

G.f.: -99*x*(x+1)/((x-1)*(x^2 - 398*x + 1)). - Colin Barker, Oct 25 2012

More terms from Paolo P. Lava, Oct 07 2008

A132644 X-values of solutions to the equation X(X + 1) - 13Y^2 = 0.

Original entry on oeis.org

0, 324, 421200, 546717924, 709639444800, 921111452633124, 1195601955878350800, 1551890417618646705924, 2014352566467047545939200, 2614628079383810095982376324, 3393785232687619037537578530000

Offset: 0

Mohamed Bouhamida, Nov 14 2007

Index entries for linear recurrences with constant coefficients, signature (1299,-1299,1).

Cf. A007654.

CoefficientList[Series[-((324 x (1+x))/(-1+1299 x-1299 x^2+x^3)),{x,0,20}],x] (* or *) LinearRecurrence[{1299,-1299,1},{0,324,421200},20] (* Harvey P. Dale, Jul 24 2021 *)

a(n) = 1298*a(n-1) - a(n-2) + 648; a(0)=0, a(1)=324.

G.f.: -324*x*(x+1)/((x-1)*(x^2 - 1298*x + 1)). - Colin Barker, Oct 25 2012

More terms from Paolo P. Lava, Oct 07 2008

A132594 Values X satisfying the equation: X(X + 1) - 7*Y^2 = 0.

Original entry on oeis.org

0, 63, 16128, 4096575, 1040514048, 264286471743, 67127723308800, 17050177433963583, 4330677940503441408, 1099975146710440154175, 279389356586511295719168, 70963796597827158672514623

Offset: 0

Mohamed Bouhamida, Nov 14 2007

nonn

The full set of integer solutions to this equation consists of the pairs [X(i),Y(i)] = [1+-A001081(i), Y(i)=A001080(i)]. The present generates every second one of them: a(n) = [A001081(2n)-1]/2. - R. J. Mathar, Nov 20 2007

Index entries for linear recurrences with constant coefficients, signature (255, -255, 1).

Cf. A007654.

Cf. A001080, A001081.

LinearRecurrence[{255,-255,1},{0,63,16128},20] (* Harvey P. Dale, Dec 15 2012 *)

a(0)=0, a(1)=63 and a(n)=254*a(n-1) - a(n-2) + 126.
G.f.: -63*x*(1+x)/(-1+x)/(1-254*x+x^2). a(n) = [A001081(2n)-1]/2. - R. J. Mathar, Nov 20 2007
a(0)=0, a(1)=63, a(2)=16128, a(n)=255*a(n-1)-255*a(n-2)+a(n-3). - Harvey P. Dale, Dec 15 2012

More terms from Max Alekseyev, Nov 13 2009

A174906 a(n) is the index of the first triangular number T_i exceeding T_n such that the product of T_i*T_n is a perfect square.

Original entry on oeis.org

24, 48, 80, 120, 168, 224, 49, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1368, 1520, 1680, 1848, 2024, 2208, 242, 2600, 2808, 3024, 3248, 3480, 3720, 3968, 4224, 4488, 4760, 5040, 5328, 5624, 5928, 6240, 6560, 6888, 7224, 7568, 7920, 8280, 8648

Offset: 2

Robert G. Wilson v, Apr 01 2010

nonn

"You can find an infinite number of [different] triangular numbers such that when multipled together form a square number. For example, for every triangular number, T_n, there are an infinite number of other triangular numbers, T_m, such that T_n*T_m is a square. For example, T_2 * T_24 = 30^2."

Clifford A. Pickover, The Loom of God, Tapestries of Mathematics and Mysticism, Sterling, NY, 2009, page 33.

Cf. A132596, A007654, A132584.

tri[n_] := n (n + 1)/2; f[n_] := Block[{k = n + 1, t = tri@n}, While[ !IntegerQ@ Sqrt[ t*tri@k], k++ ]; k]; Table[ f@n, {n, 2, 46}]

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A175497 Numbers k with the property that k^2 is a product of two distinct triangular numbers.

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A373016 a(n) is the least positive integer k such that 3*n^2 + 2*n + k is a square.

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A132607 X-values of solutions to the equation X*(X + 1) - 11*Y^2 = 0.

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A132644 X-values of solutions to the equation X*(X + 1) - 13*Y^2 = 0.

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A132594 Values X satisfying the equation: X(X + 1) - 7*Y^2 = 0.

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A174906 a(n) is the index of the first triangular number T_i exceeding T_n such that the product of T_i*T_n is a perfect square.

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Mathematica

A373016 a(n) is the least positive integer k such that 3n^2 + 2n + k is a square.

A132607 X-values of solutions to the equation X(X + 1) - 11Y^2 = 0.

A132644 X-values of solutions to the equation X(X + 1) - 13Y^2 = 0.