A046176 -id:A046176 - cp's OEIS frontend

A278310 Numbers m such that T(m) + 3*T(m+1) is a square, where T = A000217.

3, 143, 4899, 166463, 5654883, 192099599, 6525731523, 221682772223, 7530688524099, 255821727047183, 8690408031080163, 295218051329678399, 10028723337177985443, 340681375412721826703, 11573138040695364122499, 393146012008229658338303, 13355391270239113019379843

Offset: 1

Bruno Berselli, Nov 17 2016

Equivalently, both m+1 and 2*m+3 are squares for nonnegative m.
Corresponding triangular numbers T(m): 6, 10296, 12002550, 13855048416, 15988853699286, 18451128064030200, 21292585958400815526, ...
Square roots of T(m) + 3*T(m+1) are listed by A082405 (after 0).
Negative values of m for which T(m) + 3*T(m+1) is a square: -1, -2, -26, -842, -28562, -970226, -32959082, ...

			3 is in the sequence because T(3) + 3*T(4) = 6 + 3*10 = 6^2.
For n=5 is a(5) = 5654883, therefore floor(sqrt(5654883)) = 2377 = A182189(5) - 2 = 2379 - 2.

Colin Barker, Table of n, a(n) for n = 1..650
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Subsequence of A000466.
Cf. A000217, A001109, A029546, A046176, A082405, A156164, A182189.
Cf. A278438: numbers m such that T(m) + 2*T(m+1) is a square.
Cf. A078522: numbers m such that 3*T(m) + T(m+1) is a square.
Cf. similar sequences with closed form ((1 + sqrt(2))^(4*r) + (1 - sqrt(2))^(4*r))/8 + k/4: A084703 (k=-1), A076218 (k=3), this sequence (k=-5).

Iv:=[3,143]; [n le 2 select Iv[n] else 34*Self(n-1)-Self(n-2)+40: n in [1..20]];

P:=proc(q) local n; for n from 3 to q do if type(sqrt(2*n^2+5*n+3),integer) then print(n); fi; od; end: P(10^9); # Paolo P. Lava, Nov 18 2016

Table[((1 + Sqrt[2])^(4 n) + (1 - Sqrt[2])^(4 n))/8 - 5/4, {n, 1, 20}]
RecurrenceTable[{a[1] == 3, a[2] == 143, a[n] == 34 a[n - 1] - a[n - 2] + 40}, a, {n, 1, 20}]
LinearRecurrence[{35, -35, 1}, {3, 143, 4899}, 50] (* G. C. Greubel, Nov 20 2016 *)

Vec(x*(3 + 38*x - x^2)/((1 - x)*(1 - 34*x + x^2)) + O(x^50)) \\ G. C. Greubel, Nov 20 2016

def A278310():
    a, b = 3, 143
    yield a
    while True:
        yield b
        a, b = b, 34*b - a + 40
a = A278310(); print([next(a) for  in range(18)]) # _Peter Luschny, Nov 18 2016

O.g.f.: x*(3 + 38*x - x^2)/((1 - x)*(1 - 34*x + x^2)).
E.g.f.: (exp((1-sqrt(2))^4*x) + exp((1+sqrt(2))^4*x) - 10*exp(x))/8 + 1.
a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) for n>3.
a(n) = 34*a(n-1) - a(n-2) + 40 for n>2.
a(n) = a(-n) = ((1 + sqrt(2))^(4*n) + (1 - sqrt(2))^(4*n))/8 - 5/4.
a(n) = 4*A001109(n)^2 - 1.
a(n) = -A029546(n) + 38*A029546(n-1) + 3*A029546(n-2) for n>1.
Lim_{n -> infinity} a(n)/a(n-1) = A156164.
Floor(sqrt(a(n))) = A182189(n) - 2.
a(n) - a(n-1) = 4*A046176(n) for n>1.

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)

A082405 a(n) = 34*a(n-1) - a(n-2); a(0)=0, a(1)=6.

Original entry on oeis.org

0, 6, 204, 6930, 235416, 7997214, 271669860, 9228778026, 313506783024, 10650001844790, 361786555939836, 12290092900109634, 417501372047787720, 14182756556724672846, 481796221556591089044, 16366888776367372354650

Offset: 0

Lekraj Beedassy, Apr 23 2003

Sequence refers to inradius of primitive Pythagorean triangle with consecutive legs, even followed by odd. It has semiperimeter A046176(n+1) and area a(n)*A046176(n+1).

Indranil Ghosh, Table of n, a(n) for n = 0..652
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Circle chains inside the arbelos and integer sequences, Int'l J. Geom. (2023) Vol. 12, No. 1, 71-82.
Serge Perrine, About the diophantine equation z^2 = 32y^2 - 16, SCIREA Journal of Mathematics (2019) Vol. 4, Issue 5, 126-139.
Index entries for linear recurrences with constant coefficients, signature (34, -1).

Cf. A046176.

a[0] = 1; a[1] = 6; a[n_] := 34 a[n-1] - a[n-2]; Table[a[n], {n,0,15}] (* or *) LinearRecurrence[{34,-1}, {1,6}, 16] (* Indranil Ghosh, Feb 18 2017 *)

For n > 1, a(n)/2 = A001652(2*n-1) - Sum_{k=0..n-1} A001333(4*k); e.g., 6930/2 = 4059 - (17+577). - Charlie Marion, Jul 31 2003
a(n) = A001109(2n).
G.f.: 6*x/(1 - 34*x + x^2). - Philippe Deléham, Nov 18 2008
a(n) = 6*A029547(n-1). - R. J. Mathar, Jun 07 2016

A284876 Positive integers that are square roots of products a(a+d)(a+2*d) with coprime a > 0, d >= 0.

Original entry on oeis.org

1, 35, 120, 1189, 1547, 1560, 2737, 4080, 8400, 13175, 24360, 29520, 31080, 39997, 40391, 52633, 62279, 65773, 80520, 93023, 131040, 133055, 133560, 185640, 212219, 240240, 241345, 379680, 385440, 393805, 399960, 434231, 449497, 471240, 510229, 555360, 585395

Offset: 1

Jonathan Sondow, Apr 05 2017

nonn

The main entry for this sequence is A284666, formed by the triples a, a+d, a+2*d. The pairs a, d form A284874.

sqrt((1+d)*(1+2*d)) is a member if and only if d is in A078522. The values of sqrt((1+d)*(1+2*d)) form the subsequence A046176.

			gcd(1,24)=1 and 1*(1+24)*(1+2*24) = 25*49 = (5*7)^2, so 5*7 = 35 is a member.
gcd(18,7)=1 and 18*(18+7)*(18+2*7) = 18*25*32 = 9*25*64 = (3*5*8)^2, so 3*5*8 = 120 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..416 (terms < 10^9)

Cf. A046176, A078522, A284666, A284874.

nn = 50000; t = {};
p[a_, b_, c_] := a b c; Do[
If[p[a, a + d, a + 2 d] <= 2 nn^2 && GCD[a, d] == 1 &&
   IntegerQ[Sqrt[p[a, a + d, a + 2 d]]],
  AppendTo[t, Sqrt[p[a, a + d, a + 2 d]]]], {a, 1, nn}, {d, 0, nn}]; Sort[t]

is(n,s)={!fordiv(n*=n,a,a^3>n && return;issquare(n\a*8+a^2,&s) && (s-=3*a)%4==0 && gcd(s\4,a)==1 && break)} \\ M. F. Hasler, Apr 06 2017

a(k+1)^2 = A284666(3*k+1)*A284666(3*k+2)*A284666(3*k+3) = A284874(2*k+1)*(A284874(2*k+1) + A284874(2*k+2))*(A284874(2*k+1) + 2*A284874(2*k+2)) for k >= 0.

a(19)-a(37) from Giovanni Resta, Apr 06 2017

A111649 a(n) = A001541(n)A001653(n+1) + A001541(n)A002315(n) + A001653(n+1)*A002315(n).

Original entry on oeis.org

3, 71, 2379, 80783, 2744211, 93222359, 3166815963, 107578520351, 3654502875939, 124145519261543, 4217293152016491, 143263821649299119, 4866752642924153523, 165326326037771920631, 5616228332641321147899

Offset: 0

Charlie Marion, Aug 24 2005

nonn

			a(1) = 71 = 3*5 + 3*7 + 5*7.

Ray Chandler, Table of n, a(n) for n = 0..652
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A001541, A001653, A002315, A111647, A111648.

LinearRecurrence[{35, -35, 1}, {3, 71, 2379}, 20] (* Paolo Xausa, Feb 06 2024 *)

a(n) = A001653(2n+2) - 2*A002315(n)^2, e.g., 2379 = 5741 - 2*41^2;
a(n) = A001652(2n) + A002315(n)^2 + 2, e.g., 2379 = 696 + 41^2 + 2;
a(n) = 2*A046176(n+1)+1, e.g., 2379 = 2*1189 + 1.
G.f.: (x^2+34*x-3) / ((x-1)*(x^2-34*x+1)). - Colin Barker, Dec 14 2014 [adjusted for corrected term and empirical g.f. confirmed for more terms and recurrence of source sequences. - Ray Chandler, Feb 05 2024]

a(3) = 80783 corrected by Ray Chandler, Feb 05 2024

A195539 Numerators b(n) of Pythagorean approximations b(n)/a(n) to sqrt(8).

Original entry on oeis.org

12, 35, 408, 1189, 13860, 40391, 470832, 1372105, 15994428, 46611179, 543339720, 1583407981, 18457556052, 53789260175, 627013566048, 1827251437969, 21300003689580, 62072759630771, 723573111879672, 2108646576008245, 24580185800219268

Offset: 1

Clark Kimberling, Sep 20 2011

nonn

See A195500 for discussion and list of related sequences; see A195538 for Mathematica program.

Conjecture: a(n) = +34*a(n-2) -a(n-4) with a(2n+1) = A000129(4n+4), a(2n)=A046176(n+1). - R. J. Mathar, Sep 21 2011

Cf. A195500, A195538, A195540.

A284666 List of 3-term arithmetic progressions of coprime positive integers whose product is a square.

Original entry on oeis.org

1, 1, 1, 1, 25, 49, 18, 25, 32, 1, 841, 1681, 49, 169, 289, 50, 169, 288, 49, 289, 529, 128, 289, 450, 98, 625, 1152, 289, 625, 961, 800, 841, 882, 162, 1681, 3200, 288, 1369, 2450, 529, 1369, 2209, 1, 28561, 57121, 49, 5329, 10609, 961, 1681, 2401, 289, 2809, 5329

Offset: 1

Jonathan Sondow, Mar 31 2017

This is a 3-column table read by rows a, a+d, a+2*d. Each row has product a square. The rows are ordered by the products. The square roots of the products form A284876, which contains A046176. The pairs a,d form A284874.
Goldbach proved that a product of 3 consecutive positive integers is never a square.
Euler proved that a product of 4 consecutive positive integers is never a square.
Erdos and Selfridge (1975) proved that a product of 2 or more consecutive positive integers is never a square or a higher power.
Saradha (1998) proved that 18, 25, 32 is the only arithmetic progression a, a+d, ..., a+(k-1)*d whose product is a square if a>=1, 1=3 with gcd(a,d)=1. In 1997 she showed that the product is not a square or a higher power if a>=1, 1=3 with gcd(a,d)=1.
(1, 1+d, 1+2*d) is in the table if and only if d is in A078522. - Robert Israel, Apr 05 2017 - Jonathan Sondow, Apr 06 2017

			18*(18+7)*(18+2*7) = 18*25*32 = 9*25*64 = (3*5*8)^2 and gcd(18,25,32) = 1, so 18,25,32 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..1248 (triples with product < 10^18)
P. Erdős and J.L. Selfridge, The product of consecutive integers is never a power, Illinois J. Math., 19 (1975), 292-301.
N. Saradha, On perfect powers in products with terms from arithmetic progressions, Acta Arith., 82 (1997), 147-172.
N. Saradha, Squares in products with terms in an arithmetic progression, Acta Arith., 86 (1998), 27-43.

Cf. A046176, A078522, A284874, A284876.

N:= 10^11: # to get all triples where the product <= N
Res:= [1,0]:
for a from 1 to floor(N^(1/3)) do
  for d from 1 while a*(a+d)*(a+2*d) <= N do
     if igcd(a,d) = 1 and issqr(a*(a+d)*(a+2*d)) then
       Res:= Res, [a,d]
     fi
  od
od:
Res:= sort([Res], (s,t) -> s[1]*(s[1]+s[2])*(s[1]+2*s[2]) <= t[1]*(t[1]+t[2])*(t[1]+2*t[2])):
map(t -> (t[1],t[1]+t[2],t[1]+2*t[2]), Res); # Robert Israel, Apr 05 2017

nn = 50000; t = {};
p[a_, b_, c_] := a *b*c; Do[
If[p[a, a + d, a + 2 d] <= 2 nn^2 && GCD[a, d] == 1 &&
   IntegerQ[Sqrt[p[a, a + d, a + 2 d]]],
  AppendTo[t, {a, a + d, a + 2 d}]], {a, 1, nn}, {d, 0, nn}];
Sort[t, p[#1[[1]], #1[[2]], #1[[3]]] <
    p[#2[[1]], #2[[2]], #2[[3]]] &] // Flatten

a(3*k+1) = A284874(2*k+1) and a(3*k+2) = A284874(2*k+1)+A284874(2*k+2) and a(3*k+3) = A284874(2*k+1)+2*A284874(2*k+2) and a(3*k+1)*a(3*k+2)*a(3*k+3) = A284876(k+1)^2 for k>=0.

A157880 Expansion of 136x^2 / (-x^3+1155x^2-1155*x+1).

Original entry on oeis.org

0, 136, 157080, 181270320, 209185792336, 241400223085560, 278575648254944040, 321476056685982336736, 370983090839975361649440, 428114165353274881361117160, 494043375834588373115367553336, 570125627598949629300252795432720, 657924480205812037624118610561805680

Offset: 1

Paul Weisenhorn, Mar 08 2009

This sequence is part of a solution of a more general problem involving two equations, three sequences a(n), b(n), c(n) and a constant A:
    A    * c(n)+1 = a(n)^2,
   (A+1) * c(n)+1 = b(n)^2, for details see comment in A157014.
A157880 is the c(n) sequence for A=8.

Colin Barker, Table of n, a(n) for n = 1..325
Index entries for linear recurrences with constant coefficients, signature (1155,-1155,1).

8*A157880(n)+1 = A077420(n-1)^2.

9*A157880(n)+1 = A046176(n)^2.

LinearRecurrence[{1155,-1155,1},{0,136,157080},20] (* Harvey P. Dale, Dec 04 2019 *)

concat(0, Vec(136*x^2/(-x^3+1155*x^2-1155*x+1) + O(x^20))) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = round(-((577+408*sqrt(2))^(-n)*(-1+(577+408*sqrt(2))^n)*(17+12*sqrt(2)+(-17+12*sqrt(2))*(577+408*sqrt(2))^n))/288) \\ Colin Barker, Jul 25 2016

G.f.: 136*x^2/(-x^3+1155*x^2-1155*x+1).
c(1) = 0, c(2) = 136, c(3) = 1155*c(2), c(n) = 1155 * (c(n-1)-c(n-2)) + c(n-3) for n>3.
a(n) = -((577+408*sqrt(2))^(-n)*(-1+(577+408*sqrt(2))^n)*(17+12*sqrt(2)+(-17+12*sqrt(2))*(577+408*sqrt(2))^n))/288. - Colin Barker, Jul 25 2016

Edited by Alois P. Heinz, Sep 09 2011

A004294 Expansion of (1+2x+x^2)/(1-34x+x^2).

Original entry on oeis.org

1, 36, 1224, 41580, 1412496, 47983284, 1630019160, 55372668156, 1881040698144, 63900011068740, 2170719335639016, 73740557400657804, 2505008232286726320, 85096539340348037076, 2890777329339546534264, 98201332658204234127900, 3335954533049604413814336

Offset: 0

N. J. A. Sloane

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, middle display.

Vincenzo Librandi, Table of n, a(n) for n = 0..600
J. M. Alonso, Growth functions of amalgams, in Alperin, ed., Arboreal Group Theory, Springer, pp. 1-34, esp. p. 32.
Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Index entries for linear recurrences with constant coefficients, signature (34,-1).

Pairwise sums of A046176.

CoefficientList[Series[(1+2*x+x^2)/(1-34*x+x^2),{x,0,20}],x] (* Vincenzo Librandi, Jun 14 2012 *)
LinearRecurrence[{34,-1},{1,36,1224},20] (* Harvey P. Dale, Mar 29 2019 *)

Vec((1+2*x+x^2)/(1-34*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

From Colin Barker, Apr 16 2016: (Start)
a(n) = 3*((17+12*sqrt(2))^(1-n)*(-1+(17+12*sqrt(2))^(2*n)))/(48+34*sqrt(2)) for n>0.
a(n) = 34*a(n-1) - a(n-2) for n>2.
(End)
a(n) = (-(-1)^(2^n) + 3*sqrt(2)*sinh(n*log(17+12*sqrt(2))) + 1)/2. - Ilya Gutkovskiy, Apr 16 2016

A280181 Indices of centered 9-gonal numbers (A060544) that are also squares (A000290).

Original entry on oeis.org

1, 17, 561, 19041, 646817, 21972721, 746425681, 25356500417, 861374588481, 29261379507921, 994025528680817, 33767606595639841, 1147104598723073761, 38967788749988868017, 1323757712900898438801, 44968794449880558051201, 1527615253583038075302017

Offset: 1

Colin Barker, Dec 28 2016

Also positive integers y in the solutions to 2*x^2 - 9*y^2 + 9*y - 2 = 0, the corresponding values of x being A046176.

Consider all ordered triples of consecutive integers (k, k+1, k+2) such that k is a square and k+1 is twice a square; then the values of k are the squares of the NSW numbers (A002315), the values of k+1 are twice the squares of the odd Pell numbers (A001653), and the values of k+2 are thrice the terms of this sequence. (See the Example section.) - Jon E. Schoenfield, Sep 06 2019

			17 is in the sequence because the 17th centered 9-gonal number is 1225, which is also the 35th square.
From _Jon E. Schoenfield_, Sep 06 2019: (Start)
The following table illustrates the relationship between the NSW numbers (A002315), the odd Pell numbers (A001653), and the terms of this sequence:
.
  |  A002315(n-1)^2  |   2*A001653(n)^2  |
n |   = 3*a(n) - 2   |    = 3*a(n) - 1   |       3*a(n)
--+------------------+-------------------+-------------------
1 |    1^2 =       1 |   1^2*2 =       2 |      1*3 =       3
2 |    7^2 =      49 |   5^2*2 =      50 |     17*3 =      51
3 |   41^2 =    1681 |  29^2*2 =    1682 |    561*3 =    1683
4 |  239^2 =   57121 | 169^2*2 =   57122 |  19041*3 =   57123
5 | 1393^2 = 1940449 | 985^2*2 = 1940450 | 646817*3 = 1940451
(End)

Colin Barker, Table of n, a(n) for n = 1..650
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A000290, A001653, A002315, A046176, A046177, A056771, A060544, A077420.

Cf. A156164.

LinearRecurrence[{35, -35, 1}, {1, 17, 561}, 50] (* G. C. Greubel, Dec 28 2016 *)

Vec(x*(1 - 18*x + x^2) / ((1 - x)*(1 - 34*x + x^2)) + O(x^20))

a(n) = (6 + (3-2*sqrt(2))*(17+12*sqrt(2))^(-n) + (3+2*sqrt(2))*(17+12*sqrt(2))^n) / 12.
a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 - 18*x + x^2) / ((1 - x)*(1 - 34*x + x^2)).
a(n) = (A002315(n-1)^2 + 2)/3 = (2*A001653(n)^2 + 1)/3. - Jon E. Schoenfield, Sep 06 2019
a(n) = A077420(floor((n-1)/2)) * A056771(floor(n/2)). - Jon E. Schoenfield, Sep 08 2019
E.g.f.: -1+(1/12)*(6*exp(x)+(3-2*sqrt(2))*exp((17-12*sqrt(2))*x)+(3+2*sqrt(2))*exp((17+12*sqrt(2))*x)). - Stefano Spezia, Sep 08 2019
Limit_{n->oo} a(n+1)/a(n) = 17 + 12*sqrt(2) = A156164. - Andrea Pinos, Oct 07 2022

cp's OEIS Frontend

A278310 Numbers m such that T(m) + 3*T(m+1) is a square, where T = A000217.

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

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PARI

Formula

A082405 a(n) = 34*a(n-1) - a(n-2); a(0)=0, a(1)=6.

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Mathematica

Formula

A284876 Positive integers that are square roots of products a*(a+d)*(a+2*d) with coprime a > 0, d >= 0.

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Mathematica

PARI

Formula

Extensions

A111649 a(n) = A001541(n)*A001653(n+1) + A001541(n)*A002315(n) + A001653(n+1)*A002315(n).

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Mathematica

Formula

Extensions

A195539 Numerators b(n) of Pythagorean approximations b(n)/a(n) to sqrt(8).

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A284666 List of 3-term arithmetic progressions of coprime positive integers whose product is a square.

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Maple

Mathematica

A284876 Positive integers that are square roots of products a(a+d)(a+2*d) with coprime a > 0, d >= 0.

A111649 a(n) = A001541(n)A001653(n+1) + A001541(n)A002315(n) + A001653(n+1)*A002315(n).

A157880 Expansion of 136x^2 / (-x^3+1155x^2-1155*x+1).

A004294 Expansion of (1+2x+x^2)/(1-34x+x^2).