A106328 -id:A106328 - cp's OEIS frontend

A108931 a(2n) = -A106328(n), a(2n+1) = A054488(n).

0, 1, -3, 8, -18, 47, -105, 274, -612, 1597, -3567, 9308, -20790, 54251, -121173, 316198, -706248, 1842937, -4116315, 10741424, -23991642, 62605607, -139833537, 364892218, -815009580, 2126747701, -4750223943, 12395593988, -27686334078, 72246816227

Offset: 0

Creighton Dement, Jul 20 2005

Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A106328, A054488.

seriestolist(series(x*(2*x-1)*(x-1)/((x^2+2*x-1)*(x^2-2*x-1)), x=0,30)); -or- Floretion Algebra Multiplication Program, FAMP Code: 1jescycrokseq[ + 'jj' + 'kk' + 'ij'] Roktype is set to: Y[sqa.Findk()] = Y[sqa.Findk()] + 1/2;

G.f. x*(2*x-1)*(x-1)/((x^2+2*x-1)*(x^2-2*x-1))

A156035 Decimal expansion of 3 + 2*sqrt(2).

Original entry on oeis.org

5, 8, 2, 8, 4, 2, 7, 1, 2, 4, 7, 4, 6, 1, 9, 0, 0, 9, 7, 6, 0, 3, 3, 7, 7, 4, 4, 8, 4, 1, 9, 3, 9, 6, 1, 5, 7, 1, 3, 9, 3, 4, 3, 7, 5, 0, 7, 5, 3, 8, 9, 6, 1, 4, 6, 3, 5, 3, 3, 5, 9, 4, 7, 5, 9, 8, 1, 4, 6, 4, 9, 5, 6, 9, 2, 4, 2, 1, 4, 0, 7, 7, 7, 0, 0, 7, 7, 5, 0, 6, 8, 6, 5, 5, 2, 8, 3, 1, 4, 5, 4, 7, 0, 0, 2

Offset: 1

Klaus Brockhaus, Feb 02 2009

Limit_{n -> oo} b(n+1)/b(n) = 3+2*sqrt(2) for b = A155464, A155465, A155466.
Limit_{n -> oo} b(n)/b(n-1) = 3+2*sqrt(2) for b = A001652, A001653, A002315, A156156, A156157, A156158. - Klaus Brockhaus, Sep 23 2009
From Richard R. Forberg, Aug 14 2013: (Start)
Ratios b(n+1)/b(n) for all sequences of the form b(n) = 6*b(n-1) - b(n-2), for any initial values of b(0) and b(1), converge to this ratio.
Ratios b(n+1)/b(n) for all sequences of the form b(n) = 5*b(n-1) + 5*b(n-2) + b(n-3), for all b(0), b(1) and b(2) also converge to 3 + 2*sqrt(2). For example see A084158 (Pell Triangles).
Ratios of alternating values, b(n+2)/b(n), for all sequences of the form b(n) = 2*b(n-1) + b(n-2), also converge to 3 + 2*sqrt(2). These include A000129 (Pell Numbers). Also see A014176. (End)
Let ABCD be a square inscribed in a circle. When P is the midpoint of the arc AB, then the ratio (PC*PD)/(PA*PB) is equal to 3+2*sqrt(2). See the Mathematical Reflections link. - Michel Marcus, Jan 10 2017
Limit of ratios of successive terms of A001652 when n-> infinity. - Harvey P. Dale, Jun 16 2017; improved by Bernard Schott, Feb 28 2022
A quadratic integer with minimal polynomial x^2 - 6x + 1. - Charles R Greathouse IV, Jul 11 2020
Ratio between radii of the large circumscribed circle R and the small internal circle r drawn on the Sangaku tablet at Isaniwa Jinjya shrine in Ehime Prefecture (pictures in links). - Bernard Schott, Feb 25 2022

			3 + 2*sqrt(2) = 5.828427124746190097603377448...

Diogo Queiros-Condé and Michel Feidt, Fractal and Trans-scale Nature of Entropy, Iste Press and Elsevier, 2018, page 45.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Mathematical Reflections, Solution to Problem J286, Issue 1, 2014, p. 5.
Bernard Schott, Sangaku at Isaniwa Jinya, The six circles.
Terakoya Suzu, Sangaku (mathematics tablet) II, Sangaku at Isaniwa Jinya shrine.
Wikipedia, Sangaku.
Bernard Ycart, Les Sangakus, Sangaku du Temple Isaniwa Jinya (in French).
Index entries for algebraic numbers, degree

Cf. A002193 (sqrt(2)), A090488, A010466, A014176.
Cf. A155464, A155465, A155466.
Cf. A104178 (decimal expansion of log_10(3+2*sqrt(2))).
Cf. A000129, A001109, A001541, A001542, A001652, A001653, A002315, A005319, A075870, A038723, A038725, A038761, A054488, A054489, A075848, A077413, A084158, A106328, A156156, A156157, A156158.
Cf. A242412 (sangaku).

SetDefaultRealField(RealField(100));  3 + 2*Sqrt(2); // G. C. Greubel, Aug 21 2018

RealDigits[N[3+2*Sqrt[2],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

3+sqrt(8) \\ Charles R Greathouse IV, Jun 10 2011

Equals 1 + A090488 = 3 + A010466. - R. J. Mathar, Feb 19 2009
Equals exp(arccosh(3)), since arccosh(x) = log(x+sqrt(x^2-1)). - Stanislav Sykora, Nov 01 2013
Equals (1+sqrt(2))^2, that is, A014176^2. - Michel Marcus, May 08 2016
The periodic continued fraction is [5; [1, 4]]. - Stefano Spezia, Mar 17 2024

A276600 Values of m such that m^2 + 6 is a triangular number (A000217).

Original entry on oeis.org

0, 2, 3, 7, 15, 20, 42, 88, 117, 245, 513, 682, 1428, 2990, 3975, 8323, 17427, 23168, 48510, 101572, 135033, 282737, 592005, 787030, 1647912, 3450458, 4587147, 9604735, 20110743, 26735852, 55980498, 117214000, 155827965, 326278253, 683173257, 908231938

Offset: 1

Colin Barker, Sep 07 2016

2*a(n+2) gives the y members of all positive solutions (x(n), y(n)), proper and improper, of the Pell equation x^2 - 2*y^2 = 7^2, n >= 0. The corresponding x members are x(n) = A106525(n). - Wolfdieter Lang, Sep 29 2016

			7 is in the sequence because 7^2 + 6 = 55, which is a triangular number.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A000217, A230044.
Cf. A001109 (k=0), A106328 (k=1), A077241 (k=2), A276598 (k=3), A276599 (k=5), A276601 (k=9), A276602 (k=10), where k is the value added to n^2.
Cf. A275794, A275796, A106525.

I:=[0,2,3,7,15,20]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..41]]; // G. C. Greubel, Sep 15 2021

LinearRecurrence[{0,0,6,0,0,-1}, {0,2,3,7,15,20}, 41] (* G. C. Greubel, Sep 15 2021 *)

concat(0, Vec(x^2*(2+3*x+7*x^2+3*x^3+2*x^4)/(1-6*x^3+x^6) + O(x^40)))

def A276600_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2*(2+3*x+7*x^2+3*x^3+2*x^4)/(1-6*x^3+x^6) ).list()
a=A276600_list(41); a[1:] # G. C. Greubel, Sep 15 2021

a(n) = 6*a(n-3) - a(n-6) for n>6.
G.f.: x^2*(2 + 3*x + 7*x^2 + 3*x^3 + 2*x^4)/(1 - 6*x^3 + x^6).
From Wolfdieter Lang, Sep 29 2016: (Start)
Trisection:
a(2+3*n) = 15*S(n-1,6) - 2*S(n-2,6) = A275794(n),
a(3+3*n) = 20*S(n-1,6) - 3*S(n-2,6) = A275796(n),
a(4+3*n) = 7*(6*S(n-1,6) - S(n-2,6)) = 7*A001109(n+1) for n >= 0, with the Chebyshev polynomials S(n, 6) = A001109(n+1), n >= -1, with S(-2, 6) = -1.
(End)

A164055 Triangular numbers that are one plus a perfect square.

Original entry on oeis.org

1, 10, 325, 11026, 374545, 12723490, 432224101, 14682895930, 498786237505, 16944049179226, 575598885856165, 19553418069930370, 664240615491776401, 22564627508650467250, 766533094678624110085, 26039560591564569275626

Offset: 1

Tanya Khovanova & Alexey Radul, Aug 08 2009

a(n+1) is the second element of the n-th set of three consecutive triangular numbers whose product is a perfect square. - Arkadiusz Wesolowski, Apr 27 2012

The triangular numbers of this sequence satisfy the Diophantine equation T(k) = k*(k+1)/2 = m^2 + 1, which is equivalent to (2k+1)^2 - 2*(2m)^2 = 9. Now, with x=2k+1 and y=2m, the Pell-Fermat equation x^2 - 2*y^2 = 9 appears. The solutions x and y of this equation are respectively in A106329 and A075848. The indices k=(x-1)/2 of the triangular numbers of this sequence are in A072221, while the indices m=y/2 of the corresponding square numbers are in A106328. - Bernard Schott, Mar 09 2019

			10 is in this sequence because it is a triangular number A000217(4) and is equal to a square plus 1: 10 = 3^2 + 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..500
K. B. Subramaniam, Almost Square Triangular Numbers, The Fibonacci Quarterly, Vol. 37, No. 3 (1999), pp. 194-197.
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A001110, A106329, A075848 (solutions of x^2 - 2 * y^2 = 9).

Cf. A072221 (indices of the triangular terms of this sequence), A106328 (indices of the corresponding square numbers).

a164055 n = a164055_list !! (n-1)
a164055_list = 1 : 10 : 325 : zipWith (+) a164055_list
   (map (* 35) $ tail $ zipWith (-) (tail a164055_list) a164055_list)
-- Reinhard Zumkeller, Apr 29 2012

LinearRecurrence[{35,-35,1}, {1,10,325}, 50] (* G. C. Greubel, Sep 09 2017 *)

my(x='x+O('x^50)); Vec(x*(1-25*x+10*x^2)/((1-x)*(x^2-34*x+1))) \\ G. C. Greubel, Sep 09 2017

A000217 INTERSECT A002522.
a(n) = A000217(A072221(n-1)).
From R. J. Mathar, Sep 22 2009: (Start)
a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3).
G.f.: x*(1-25*x+10*x^2)/((1-x)*(x^2-34*x+1)). (End)
A010054(a(n)) * A010052(a(n) - 1) = 1. - Reinhard Zumkeller, Apr 29 2012
a(n) = (A106329(n) - 1)*(A106329(n) + 1)/8 = (A106328(n))^2 + 1. - Bernard Schott, Mar 10 2019
a(n) = (14 + 9*(17+12*sqrt(2))^(1-n) - 9*(-17+12*sqrt(2))*(17+12*sqrt(2))^n) / 32. - Colin Barker, Mar 23 2019
a(n) = 9 * A001110(n) + 1 (Subramaniam, 1999). - Amiram Eldar, Jan 13 2022

Comment molded into formula by R. J. Mathar, Sep 22 2009

A276598 Values of m such that m^2 + 3 is a triangular number (A000217).

Original entry on oeis.org

0, 5, 30, 175, 1020, 5945, 34650, 201955, 1177080, 6860525, 39986070, 233055895, 1358349300, 7917039905, 46143890130, 268946300875, 1567533915120, 9136257189845, 53250009223950, 310363798153855, 1808932779699180, 10543232880041225, 61450464500548170

Offset: 1

Colin Barker, Sep 07 2016

			5 is in the sequence because 5^2 + 3 = 28, which is a triangular number.

Colin Barker, Table of n, a(n) for n = 1..1000
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, Integer sequences and ellipse chains inside a hyperbola, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A000129, A000217, A230044.
Cf. A001109 (k=0), A106328 (k=1), A077241 (k=2), A276599 (k=5), A276600 (k=6), A276601 (k=9), A276602 (k=10), where k is the value added to n^2.
Cf. A328791 (the resulting triangular numbers).

[n le 2 select 5*(n-1) else 6*Self(n-1) - Self(n-2): n in [1..31]]; // G. C. Greubel, Sep 15 2021

CoefficientList[Series[5*x/(1 - 6*x + x^2), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 07 2016 *)
LinearRecurrence[{6,-1},{0,5},30] (* Harvey P. Dale, Apr 26 2019 *)
(5/2)*Fibonacci[2*Range[30] -2, 2] (* G. C. Greubel, Sep 15 2021 *)

concat(0, Vec(5*x^2/(1-6*x+x^2) + O(x^30)))

a(n)=([0,1;-1,6]^n*[-5;0])[1,1] \\ Charles R Greathouse IV, Sep 07 2016

[(5/2)*lucas_number1(2*n-2, 2, -1) for n in (1..30)] # G. C. Greubel, Sep 15 2021

a(n) = 5*A001109(n-1).
a(n) = 5*( (3 - 2*sqrt(2))*(3 + 2*sqrt(2))^n - (3 + 2*sqrt(2))*(3 - 2*sqrt(2))^n )/(4*sqrt(2)).
a(n) = 6*a(n-1) - a(n-2) for n>2.
G.f.: 5*x^2 / (1-6*x+x^2).
a(n) = (5/2)*A000129(2*n-2). - G. C. Greubel, Sep 15 2021

A226074 Numbers k such that k^2+1 is a triangular number and k^2+2 is a prime number.

Original entry on oeis.org

0, 3, 105, 4116315, 9721683943673162520781155, 35285058531373239343849920668944097883186271103498415

Offset: 1

Alex Ratushnyak, May 25 2013

nonn

a(7) has 191 decimal digits: 222247847...170444715.
Corresponding primes and indices of triangular numbers: A226072 and A226073.
Subsequence of A106328. - Max Alekseyev, Jan 30 2014

Cf. A226069-A226073.

A276599 Values of n such that n^2 + 5 is a triangular number (A000217).

Original entry on oeis.org

1, 4, 10, 25, 59, 146, 344, 851, 2005, 4960, 11686, 28909, 68111, 168494, 396980, 982055, 2313769, 5723836, 13485634, 33360961, 78600035, 194441930, 458114576, 1133290619, 2670087421, 6605301784, 15562409950, 38498520085, 90704372279, 224385818726

Offset: 1

Colin Barker, Sep 07 2016

			4 is in the sequence because 4^2 + 5 = 21, which is a triangular number.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A000129, A000217, A230044.

Cf. A001109 (k=0), A106328 (k=1), A077241 (k=2), A276598 (k=3), A276600 (k=6), A276601 (k=9), A276602 (k=10), where k is the value added to n^2.

I:=[1,4,10,25]; [n le 4 select I[n] else 6*Self(n-2) - Self(n-4): n in [1..31]]; // G. C. Greubel, Sep 15 2021

CoefficientList[Series[(1+x)*(1+3*x+x^2)/((1+2*x-x^2)*(1-2*x-x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 07 2016 *)
LinearRecurrence[{0,6,0,-1},{1,4,10,25},30] (* Harvey P. Dale, Feb 13 2017 *)

Vec(x*(1+x)*(1+3*x+x^2)/((1+2*x-x^2)*(1-2*x-x^2)) + O(x^40))

a(n)=([0,1;-1,6]^(n\2)*if(n%2,[1;10],[-1;4]))[1,1] \\ Charles R Greathouse IV, Sep 07 2016

def P(n): return lucas_number1(n, 2, -1)
[(1/4)*(P(n+2) + P(n+1) + (-1)^n*(3*P(n) - 7*P(n-1))) for n in (1..30)] # G. C. Greubel, Sep 15 2021

a(n) = 6*a(n-2) - a(n-4) for n>4.
G.f.: x*(1+x)*(1+3*x+x^2) / ((1+2*x-x^2)*(1-2*x-x^2)).
a(n) = (1/4)*(P(n+2) + P(n+1) + (-1)^n*(3*P(n) - 7*P(n-1))), where P(n) = A000129(n). - G. C. Greubel, Sep 15 2021

A276601 Values of k such that k^2 + 9 is a triangular number (A000217).

Original entry on oeis.org

1, 6, 12, 37, 71, 216, 414, 1259, 2413, 7338, 14064, 42769, 81971, 249276, 477762, 1452887, 2784601, 8468046, 16229844, 49355389, 94594463, 287664288, 551336934, 1676630339, 3213427141, 9772117746, 18729225912, 56956076137, 109161928331, 331964339076

Offset: 1

Colin Barker, Sep 07 2016

			6 is in the sequence because 6^2+9 = 45, which is a triangular number.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A000129, A000217, A230044.

Cf. A001109 (k=0), A106328 (k=1), A077241 (k=2), A276598 (k=3), A276599 (k=5), A276600 (k=6), A276602 (k=10), where k is the value added to n^2.

I:=[1,6,12,37]; [n le 2 select I[n] else 6*Self(n-2) - Self(n-4): n in [1..31]]; // G. C. Greubel, Sep 15 2021

CoefficientList[Series[(1+x)*(1+5*x+x^2)/((1+2*x-x^2)*(1-2*x-x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 07 2016 *)
LinearRecurrence[{0,6,0,-1}, {1,6,12,37}, 31] (* G. C. Greubel, Sep 15 2021 *)

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

def P(n): return lucas_number1(n, 2, -1)
[(1/4)*(5*P(n+1) - P(n) + (-1)^n*(P(n-1) + 5*P(n-2))) for n in (1..30)] # G. C. Greubel, Sep 15 2021

a(n) = 6*a(n-2) - a(n-4) for n>4.
G.f.: x*(1+x)*(1+5*x+x^2) / ((1+2*x-x^2)*(1-2*x-x^2)).
a(n) = (1/4)*(5*P(n+1) - P(n) + (-1)^n*(P(n-1) + 5*P(n-2))), where P(n) = A000129(n). - G. C. Greubel, Sep 15 2021

A276602 Values of k such that k^2 + 10 is a triangular number (A000217).

Original entry on oeis.org

0, 9, 54, 315, 1836, 10701, 62370, 363519, 2118744, 12348945, 71974926, 419500611, 2445028740, 14250671829, 83059002234, 484103341575, 2821561047216, 16445262941721, 95850016603110, 558654836676939, 3256079003458524, 18977819184074205, 110610836100986706

Offset: 1

Colin Barker, Sep 07 2016

			9 is in the sequence because 9^2+10 = 91, which is a triangular number.

Colin Barker, Table of n, a(n) for n = 1..1000
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Soumeya M. Tebtoub, Hacène Belbachir and László Németh, Integer sequences and ellipse chains inside a hyperbola, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A000129, A000217, A230044.

Cf. A001109 (k=0), A106328 (k=1), A077241 (k=2), A276598 (k=3), A276599 (k=5), A276600 (k=6), A276601 (k=9), where k is the value added to n^2.

[n le 2 select 9*(n-1) else 6*Self(n-1) - Self(n-2): n in [1..31]]; // G. C. Greubel, Sep 15 2021

CoefficientList[Series[9*x/(1 - 6*x + x^2), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 07 2016 *)
(9/2)*Fibonacci[2*(Range[30] -1), 2] (* G. C. Greubel, Sep 15 2021 *)

concat(0, Vec(9*x^2/(1-6*x+x^2) + O(x^30)))

[(9/2)*lucas_number1(2*n-2, 2, -1) for n in (1..30)] # G. C. Greubel, Sep 15 2021

a(n) = (9/(4*sqrt(2))*( (3 - 2*sqrt(2))*(3 + 2*sqrt(2))^n - (3 + 2*sqrt(2))*(3 - 2*sqrt(2))^n) ).
a(n) = 9*A001109(n-1).
a(n) = 6*a(n-1) - a(n-2) for n>2.
G.f.: 9*x^2 / (1-6*x+x^2).
a(n) = (9/2)*A000129(2*n-2). - G. C. Greubel, Sep 15 2021

A125652 Numbers m such that m^2=A125650(k) for some k (belonging A125651).

Original entry on oeis.org

1, 3, 9, 105, 306, 3567, 10395, 121173, 353124, 4116315, 11995821, 139833537, 407504790, 4750223943, 13843167039, 161367780525, 470260174536, 5481754313907, 15975002767185, 186218278892313, 542679833909754

Offset: 1

Alexander Adamchuk, Nov 29 2006, corrected Dec 14 2006

nonn

Indices k such that a(n)^2=A125650(k) are listed in A125651.

3 divides a(n) for n>1. For n>1 a(n) = 3*A041053(2n-3), where A041053(n) = {1, 1, 2, 3, 32, 35, 67, 102, 1087, 1189, 2276, 3465, ...} Denominators of continued fraction convergents to sqrt(32). - Alexander Adamchuk, Jan 19 2007

			a(2)=3 because A125650(3)=9=3^2; a(3)=9 because A125650(24)=81=9^2.

Cf. A125650, A125651, A106328.

Cf. A041053.

a(2k)=A106328(2k); for k>0, a(2k+1)=A106328(2k+1)/2.
a(n) = sqrt(A125650(A125651(n))).
a(n) = 3*A041053(2n-3) for n>1. - Alexander Adamchuk, Jan 19 2007

Edited by Max Alekseyev, Jan 11 2007

cp's OEIS Frontend

A108931 a(2n) = -A106328(n), a(2n+1) = A054488(n).

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A156035 Decimal expansion of 3 + 2*sqrt(2).

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A276600 Values of m such that m^2 + 6 is a triangular number (A000217).

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A164055 Triangular numbers that are one plus a perfect square.

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A276598 Values of m such that m^2 + 3 is a triangular number (A000217).

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A226074 Numbers k such that k^2+1 is a triangular number and k^2+2 is a prime number.

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A276599 Values of n such that n^2 + 5 is a triangular number (A000217).

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