A001542 -id:A001542 - cp's OEIS frontend

A115598 Consider all Pythagorean triples (X,X+1,Z) ordered by increasing Z; sequence gives Z-(X+1) values.

1, 8, 49, 288, 1681, 9800, 57121, 332928, 1940449, 11309768, 65918161, 384199200, 2239277041, 13051463048, 76069501249, 443365544448, 2584123765441, 15061377048200, 87784138523761, 511643454094368, 2982076586042449, 17380816062160328, 101302819786919521

Offset: 1

N. J. A. Sloane, Mar 14 2006

Old name was A001653(n) - A046090(n).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
L. J. Gerstein, Pythagorean triples and inner products, Math. Mag., 78 (2005), 205-213.
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Essentially a duplicate of A001108.

LinearRecurrence[{7,-7,1},{1,8,49},30] (* Harvey P. Dale, Oct 27 2013 *)
CoefficientList[Series[-(x + 1)/((x - 1) (x^2 - 6 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 28 2014 *)

a(n) = (-2+(3-2*sqrt(2))^n+(3+2*sqrt(2))^n)/4.
a(n) = 7*a(n-1)-7*a(n-2)+a(n-3).
G.f.: -x*(x+1) / ((x-1)*(x^2-6*x+1)).
a(n)^2 + (a(n)+1)^2 = A001542(n)^2 + 1^2. - Hermann Stamm-Wilbrandt, Jul 27 2014

Corrected and edited by Colin Barker, Jul 31 2013

A182188 A sequence of row differences for table A182119.

Original entry on oeis.org

1, -1, -11, -69, -407, -2377, -13859, -80781, -470831, -2744209, -15994427, -93222357, -543339719, -3166815961, -18457556051, -107578520349, -627013566047, -3654502875937, -21300003689579

Offset: 0

Kenneth J Ramsey, Apr 17 2012

This is a list of row differences corresponding to a difference of 1 in table A182119, column 0. If A181119(k+1,0) - A182119(k,0) = 1, then a(n) = A182119(k+1,n) - A182119(k,n).

If p is a prime of the form 8*n +- 3, then a(p) == 3 (mod p). If p is a prime of the form 8*n +- 1, then a(p) == -1 (mod p).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A000129, A001109, A001333, A001542, A005409, A182189, A182190, A182119.

m = 13;n = 3; c = 0;
list3 = Reap[While[c < 22, t = 6 n - m - 4; Sow[t];m = n; n = t;c++]][[2,1]]
Table[1 -Fibonacci[2*n, 2], {n,0,40}] (* G. C. Greubel, May 24 2021 *)

[1 - lucas_number1(2*n,2,-1) for n in (0..40)] # G. C. Greubel, May 24 2021

a(n) = 6*a(n-1) - a(n-2) - 4. [corrected by Klaus Purath, Mar 19 2021]
a(n) = -(A182189(n-1) + 2*A182190(n-1)).
a(n) = 2 - A182189(n).
From Klaus Purath, Mar 19 2021: (Start)
a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3).
a(n) = (-1)*Sum_{i=1..2*n-1} A001333(i) for n > 0.
a(n) = 1 - A001542(n) for n > 0.
a(n) = 1 - 2*A001109(n) for n > 0.
a(n) = (-1)*A005409(2*n) for n > 0. (End)
G.f.: (1 - 8*x + 3*x^2)/((1-x)*(1-6*x+x^2)). - Chai Wah Wu, Apr 08 2021
a(n) = 1 - Pell(2*n), where Pell(n) = A000129(n). - G. C. Greubel, May 24 2021

A204519 Square root of floor(A055851(n)/6).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 11, 20, 40, 109, 198, 396, 1079, 1960, 3920, 10681, 19402, 38804, 105731, 192060, 384120, 1046629, 1901198, 3802396, 10360559, 18819920, 37639840, 102558961, 186298002

Offset: 1

M. F. Hasler, Jan 15 2012

Base-6 analog of A031150 [base 10], A204512 [base 8], A204517 (base 7), A204521 [base 5], A001353 [base 3], A001542 [base 2]. For bases 4 and 9, the corresponding sequence contains all integers.

Cf. A055792, A055793, A055812, A204517, A204512, A204503 and A023110.

Sqrt[Floor[Select[Range[100000],IntegerQ[Sqrt[Quotient[#^2,6]]]&]^2/6]] (* Vaclav Kotesovec, Nov 26 2012 *)

b=6;for(n=1,2e9,issquare(n^2\b) & print1(sqrtint(n^2\b),","))

Conjecture (for n>=8): a(n) = 10*a(n-3) - a(n-6). - Vaclav Kotesovec, Nov 26 2012

Empirical g.f.: x^4*(x^3+4*x^2+2*x+1) / (x^6-10*x^3+1). - Colin Barker, Sep 15 2014

More terms from Vaclav Kotesovec, Nov 26 2012

A204521 Square root of floor(A055812(n) / 5).

Original entry on oeis.org

0, 0, 0, 1, 3, 4, 8, 21, 55, 72, 144, 377, 987, 1292, 2584, 6765, 17711, 23184, 46368, 121393, 317811, 416020, 832040, 2178309, 5702887, 7465176

Offset: 1

M. F. Hasler, Jan 15 2012

nonn

Or: Numbers whose square yields another square when written in base 5.
(For the first 3 terms, the above "base 5" interpretation is questionable, since they have only 1 digit in base 5. It is understood that dropping this digit yields 0.)
Base-5 analog of A031150 [base 10], A001353 [base 3], A001542 [base 2].
The square roots of A055812 are listed in A204520.

Cf. A055792, A055793, A204519, A204517, A204512, A204503 and A023110.

b=5;for(n=1,2e9,issquare(n^2\b) && print1(sqrtint(n^2\b),","))

Empirical g.f.: x^4*(x^5+3*x^4+8*x^3+4*x^2+3*x+1) / ((x^4-4*x^2-1)*(x^4+4*x^2-1)). - Colin Barker, Sep 15 2014

A207832 Numbers x such that 20*x^2 + 1 is a perfect square.

Original entry on oeis.org

0, 2, 36, 646, 11592, 208010, 3732588, 66978574, 1201881744, 21566892818, 387002188980, 6944472508822, 124613502969816, 2236098580947866, 40125160954091772, 720016798592704030

Offset: 0

Gary Detlefs, Feb 20 2012

Denote as {a,b,c,d} the second-order linear recurrence a(n) = c*a(n-1) + d*a(n-2) with initial terms a, b. The following sequences and recurrence formulas are related to integer solutions of k*x^2 + 1 = y^2.
.
   k             x                        y
   -  -----------------------  -----------------------
A001542 {0,2,6,-1}       A001541 {1,3,6,-1}
A001353 {0,1,4,-1}       A001075 {1,2,4,-1}
A060645 {0,4,18,-1}      A023039 {1,9,18,-1}
A001078 {0,2,10,-1}      A001079 {1,5,10,-1}
A001080 {0,3,16,-1}      A001081 {1,8,16,-1}
A001109 {0,1,6,-1}       A001541 {1,3,6,-1}
A084070 {0,1,38,-1}      A078986 {1,19,38,-1}
A001084 {0,3,20,-1}      A001085 {1,10,20,-1}
A011944 {0,2,14,-1}      A011943 {1,7,14,-1}
A075871 {0,180,1298,-1}  A114047 {1,649,1298,-1}
A068204 {0,4,30,-1}      A069203 {1,15,30,-1}
A001090 {0,1,8,-1}       A001091 {1,4,8,-1}
A121740 {0,8,66,-1}      A099370 {1,33,66,-1}
A202299 {0,4,34,-1}      A056771 {1,17,34,-1}
A174765 {0,39,340,-1}    A114048 {1,179,340,-1}
a(n)    {0,2,18,-1}      A023039 {1,9,18,-1}
A174745 {0,12,110,-1}    A114049 {1,55,110,-1}
A174766 {0,42,394,-1}    A114050 {1,197,394,-1}
A174767 {0,5,48,-1}      A114051 {1,24,48,-1}
A004189 {0,1,10,-1}      A001079 {1,5,10,-1}
A174768 {0,10,102,-1}    A099397 {1,51,102,-1}
The sequence of the c parameter is listed in A180495.

Bruno Berselli, Table of n, a(n) for n = 0..500
Hacène Belbachir, Soumeya Merwa Tebtoub, and 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 (18,-1).

Cf. A023039, A049660, A079962.

m:=16; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x/(1-18*x+x^2))); // Bruno Berselli, Jun 19 2019

readlib(issqr):for x from 1 to 720016798592704030 do if issqr(20*x^2+1) then print(x) fi od;

LinearRecurrence[{18, -1}, {0, 2}, 16] (* Bruno Berselli, Feb 21 2012 *)
Table[2 ChebyshevU[-1 + n, 9], {n, 0, 16}]  (* Herbert Kociemba, Jun 05 2022 *)

makelist(expand(((2+sqrt(5))^(2*n)-(2-sqrt(5))^(2*n))/(4*sqrt(5))), n, 0, 15); /* Bruno Berselli, Jun 19 2019 */

a(n) = 18*a(n-1) - a(n-2).
From Bruno Berselli, Feb 21 2012: (Start)
G.f.: 2*x/(1-18*x+x^2).
a(n) = -a(-n) = 2*A049660(n) = ((2 + sqrt(5))^(2*n)-(2 - sqrt(5))^(2*n))/(4*sqrt(5)). (End)
a(n) = Fibonacci(6*n)/4. - Bruno Berselli, Jun 19 2019
For n>=1, a(n) = A079962(6n-3). - Christopher Hohl, Aug 22 2021

A074821 Numbers k such that tau(k) = tau(2k+1).

Original entry on oeis.org

2, 3, 4, 5, 10, 11, 23, 27, 29, 34, 38, 41, 46, 53, 55, 57, 62, 76, 77, 83, 89, 91, 93, 106, 113, 118, 123, 129, 131, 133, 136, 143, 145, 159, 161, 173, 177, 179, 185, 191, 201, 203, 205, 206, 212, 213, 218, 226, 232, 233, 235, 239, 251, 259, 267, 281, 291, 293

Offset: 1

Benoit Cloitre, Sep 08 2002

nonn

If a term is a perfect square, then its square root must be in A001542. The first few such terms are the squares of 2, 2378, 93222358, 165326326037771920630... - Ivan Neretin, May 25 2016

Ivan Neretin, Table of n, a(n) for n = 1..10014

Cf. A000005 (tau), A001542, A005384 (subsequence), A099774.

Select[Range[300], DivisorSigma[0, #] == DivisorSigma[0, 2 # + 1] &] (* Ivan Neretin, May 25 2016 *)

isok(k) = numdiv(k) == numdiv(2*k+1); \\ Amiram Eldar, May 08 2025

A114619 a(n) = 2*A079291(n) (twice squares of Pell numbers).

Original entry on oeis.org

0, 2, 8, 50, 288, 1682, 9800, 57122, 332928, 1940450, 11309768, 65918162, 384199200, 2239277042, 13051463048, 76069501250, 443365544448, 2584123765442, 15061377048200, 87784138523762, 511643454094368

Offset: 0

Creighton Dement, Feb 17 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Cf. A000129, A001109, A001333, A001542, A079291.

Cf. also A090390, A108475, A114619, A116484.

[n le 3 select 2*(n-1)^2 else 5*Self(n-1) +5*Self(n-2) -Self(n-3): n in [1..31]]; // G. C. Greubel, Aug 18 2022

2*Fibonacci[Range[0, 30], 2]^2 (* G. C. Greubel, Aug 18 2022 *)

[2*lucas_number1(n,2,-1)^2 for n in (0..30)] # G. C. Greubel, Aug 18 2022

a(n) = 2*A000129(n)^2.
G.f.: 2*x*(1-x)/((1+x)*(1-6*x+x^2)).
a(n) = A001333(n)^2 - (-1)^n. - Antonio Pane (apane1(AT)spc.edu), Dec 15 2007

Entry revised by N. J. A. Sloane, Mar 15 2024

A120744 Least k>0 such that a centered polygonal number nk(k+1)/2+1 is a perfect square; or -1 if no such number exists.

Original entry on oeis.org

2, -1, 1, 3, 2, 7, 15, 1, 16, 8, 14, 4, 5, 15, 1, 2, 5, -1, 6, 3, 2, 39, 6, 1, 21, 7, 110, 3, 15, 7, 15, -1, 2, 8, 1, 4, 989, 8, 14, 2, 45, 15, 9, 4, 5, 335, 9, 1, 29, -1, 30, 15, 10, 415, 6, 2, 10, 32, 54, 3, 77, 55, 1, 5, 2, 7, 47750, 11, 15, 23, 47, -1, 48, 24, 16, 12, 5, 8, 2639, 1, 6720, 704, 38, 4, 2, 39, 505, 3, 13, 56, 9, 20, 13, 1631, 41

Offset: 1

Alexander Adamchuk, Apr 26 2007

sign

			a(5) = 2 because A129556(2) = 2>1 and A129556(1) = 0<1.

Max Alekseyev, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Centered Polygonal Number

Cf. A001542, A006451, A001921, A129444, A001570, A001652, A129556, A053606, A105038, A105040, A053141, A061278.

a(n) = -1 for n in A166259.

a(n) = 1 for n = k^2-1.

Edited and b-file provided by Max Alekseyev, Jan 20 2010

A321782 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt((A321768(n, k) + A321770(n, k))/2).

Original entry on oeis.org

2, 3, 5, 4, 4, 8, 7, 8, 12, 9, 7, 9, 6, 5, 11, 10, 13, 19, 14, 12, 16, 11, 11, 21, 18, 19, 29, 22, 16, 20, 13, 10, 18, 15, 14, 22, 17, 11, 13, 8, 6, 14, 13, 18, 26, 19, 17, 23, 16, 18, 34, 29, 30, 46, 35, 25, 31, 20, 17, 31, 26, 25, 39, 30, 20, 24, 15, 14, 30

Offset: 1

Rémy Sigrist, Nov 18 2018

This sequence and A321783 are related to a parametrization of the primitive Pythagorean triples in the tree described in A321768.

This sequence is "i" from the construction in A321768. It takes ternary digits of k-1 from most to least significant. Here the result is the same going instead least to most, due to how the relevant matrix product is related to its reversal. As a flat sequence this means a(A351702(n)) = a(n) unchanged. - Kevin Ryde, Mar 10 2022

			The first rows are:
   2
   3, 5, 4
   4, 8, 7, 8, 12, 9, 7, 9, 6

Rémy Sigrist, Rows n = 1..9, flattened
Kevin Ryde, Trees of Primitive Pythagorean Triples, section UAD Tree, "row-wise p".
Robert Saunders and Trevor Randall, The Family Tree of the Pythagorean Triplets Revisited, Mathematical Gazette, item 78.12, volume 78, July 1994, pages 190-193, see page 192 tree terms "m" by columns.
Index entries related to Pythagorean Triples

Cf. A000129, A321768, A321770, A321783.
Cf. A001542 (row sums).
Cf. A351702 (product reversal permutation).

M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (sqrtint((t[1, 1] + t[3, 1])/2))

Empirically:
- T(n, 1) = n + 1,
- T(n, (3^(n-1) + 1)/2) = A000129(n + 1),
- T(n, 3^(n-1)) = 2 * n.

A350984 a(0)=0, a(1)=18, a(2)=612; a(n) = 35*(a(n-1)-a(n-2))+a(n-3).

Original entry on oeis.org

0, 18, 612, 20790, 706248, 23991642, 815009580, 27686334078, 940520349072, 31950005534370, 1085359667819508, 36870278700328902, 1252504116143363160, 42548269670174018538, 1445388664669773267132, 49100666329102117063950, 1667977266524802206907168, 56662126395514172917779762, 1924844320180957076997604740

Offset: 0

N. J. A. Sloane, Mar 08 2022

P.-F. Teilhet, Query 2376, L'Intermédiaire des Mathématiciens, 11 (1904), 138-139.

Index entries for linear recurrences with constant coefficients, signature (34,-1).

Cf. A001541, A001542, A350983.
Equals 18*A029547(n-1) for n >= 1.
Equals 18*A091761.

CoefficientList[Series[18*x/(x^2 - 34*x + 1), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 03 2022 *)

From Chai Wah Wu, Mar 08 2022: (Start)
a(n) = 34*a(n-1) - a(n-2) for n > 1.
G.f.: 18*x/(x^2 - 34*x + 1). (End)

cp's OEIS Frontend

A115598 Consider all Pythagorean triples (X,X+1,Z) ordered by increasing Z; sequence gives Z-(X+1) values.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A182188 A sequence of row differences for table A182119.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A204519 Square root of floor(A055851(n)/6).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A204521 Square root of floor(A055812(n) / 5).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A207832 Numbers x such that 20*x^2 + 1 is a perfect square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

Formula

A074821 Numbers k such that tau(k) = tau(2k+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A114619 a(n) = 2*A079291(n) (twice squares of Pell numbers).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A120744 Least k>0 such that a centered polygonal number nk(k+1)/2+1 is a perfect square; or -1 if no such number exists.

Views