A156035 -id:A156035 - cp's OEIS frontend

A161482 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+151)^2 = y^2.

0, 96, 189, 453, 969, 1496, 3020, 6020, 9089, 17969, 35453, 53340, 105096, 207000, 311253, 612909, 1206849, 1814480, 3572660, 7034396, 10575929, 20823353, 40999829, 61641396, 121367760, 238964880, 359272749, 707383509, 1392789753

Offset: 1

Klaus Brockhaus, Jun 13 2009

nonn

Corresponding values y of solutions (x, y) are in A161483.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (187+78*sqrt(2))/151 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (24723+6758*sqrt(2))/151^2 for n mod 3 = 0.

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

Cf. A161483, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A161484 (decimal expansion of (187+78*sqrt(2))/151), A161485 (decimal expansion of (24723+6758*sqrt(2))/151^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,96,189,453,969,1496,3020},30] (* Harvey P. Dale, Jul 10 2023 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+302*n+22801), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+302 for n > 6; a(1)=0, a(2)=96, a(3)=189, a(4)=453, a(5)=969, a(6)=1496.
G.f.: x*(96+93*x+264*x^2-60*x^3-31*x^4-60*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 151*A001652(k) for k >= 0.

A161486 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+191)^2 = y^2.

Original entry on oeis.org

0, 69, 336, 573, 936, 2449, 3820, 5929, 14740, 22729, 35020, 86373, 132936, 204573, 503880, 775269, 1192800, 2937289, 4519060, 6952609, 17120236, 26339473, 40523236, 99784509, 153518160, 236187189, 581587200, 894769869, 1376600280

Offset: 1

Klaus Brockhaus, Jun 13 2009

nonn

Corresponding values y of solutions (x, y) are in A161487.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (209+60*sqrt(2))/191 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (52323+26522*sqrt(2))/191^2 for n mod 3 = 0.

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

Cf. A161487, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A161488 (decimal expansion of (209+60*sqrt(2))/191), A161489 (decimal expansion of (52323+26522*sqrt(2))/191^2).

Transpose[NestList[Flatten[{Rest[#],6#[[4]]-First[#]+382}]&,{0,69, 336, 573, 936,2449},40]][[1]]  (* Harvey P. Dale, Apr 01 2011 *)
LinearRecurrence[{1,0,6,-6,0,-1,1},{0,69,336,573,936,2449,3820},40] (* Harvey P. Dale, Mar 29 2016 *)

{forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+382*n+36481), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+382 for n > 6; a(1)=0, a(2)=69, a(3)=336, a(4)=573, a(5)=936, a(6)=2449.
G.f.: x*(69+267*x+237*x^2-51*x^3-89*x^4-51*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 191*A001652(k) for k >= 0.

A219014 Numerators in a product expansion for sqrt(2).

Original entry on oeis.org

6, 6726, 13765255184676885126

Offset: 0

Peter Bala, Nov 09 2012

a(3) has 96 digits and a(4) has 479 digits.
Iterating the algebraic identity sqrt(1 + 4/x) = (1 + 2*(x + 2)/(x^2 + 3*x + 1)) * sqrt(1 + 4/(x*(x^2 + 5*x + 5)^2)) produces a rapidly converging product expansion sqrt(1 + 4/x) = Product_{n >= 0} (1 + 2*a(n)/b(n)), where a(n) and b(n) are integer sequences when x is a positive integer.
The present case is when x = 4. The denominators b(n) are in A219015. See also A219010 (x = 1) and A219012 (x = 2).

			The first two terms of the product give 18 correct decimal places of sqrt(2): (1 + 2*6/29)*(1 + 2*6726/45232349) = 1.41421 35623 73095 048(5...).

Amiram Eldar, Table of n, a(n) for n = 0..4

Cf. A156035, A219010, A219012, A219015.

RecurrenceTable[{a[n+1] == a[n]^5 - 5*a[n]^3 + 5*a[n], a[0] == 6}, a, {n, 0, 3}] (* Amiram Eldar, Jul 20 2025 *)

Let tau = 3 + 2*sqrt(2). Then a(n) = tau^(5^n) + 1/tau^(5^n).

Recurrence equation: a(n+1) = a(n)^5 - 5*a(n)^3 + 5*a(n) with initial condition a(0) = 6.

A344576 a(n) = f(n,n) where f(0,n) = f(n,0) = Fibonacci(n) and f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1).

Original entry on oeis.org

0, 2, 10, 52, 278, 1510, 8288, 45834, 254922, 1424252, 7986550, 44921582, 253320352, 1431678194, 8106897418, 45982821860, 261206625526, 1485765938390, 8461264982176, 48237937154554, 275275548126890, 1572297656021292, 8987888015996790, 51417128080562142

Offset: 0

José María Grau Ribas, May 24 2021

nonn

a(n+1)/a(n) tends to A156035.

Cf. A000045, A001850, A008288, A156035.

F[0, 0] = 0; F[m_, 0] := Fibonacci[m]; F[0, n_] := Fibonacci[n];
F[m_, n_] := F[m, n] =   F[m - 1 , n ] + F[m , n - 1] +  F[m - 1, n - 1];
Table[F[n, n], {n, 0, 100}]

\\ here D(n,k) is A008288(n,k).
D(n, k) = {sum(d = 0, min(n,k), binomial(k, d)*binomial(n+k-d, k))}
a(n) = {2*sum(k=1, n, fibonacci(k)*(D(n-1,n-k) + D(n-1,n-k-1)))} \\ Andrew Howroyd, May 29 2021

a(n) = 2*Sum_{k=1..n} Fibonacci(k)*(A008288(n-1,n-k) + A008288(n-1,n-k-1)). - Andrew Howroyd, May 29 2021
G.f.: x*(3*x^2-18*x+3-(x+1)*sqrt(x^2-6*x+1))/((x^2-7*x+1)*(x^2-6*x+1)). - Alois P. Heinz, May 29 2021
a(n) = ((79-97*n+26*n^2)*a(n-1) + (-9+9*n-2*n^2)*a(n-4) + (107-111*n+26*n^2)*a(n-3) + (-322+352*n-88*n^2)*a(n-2)) / (5-7*n+2*n^2) for n >= 4. - José María Grau Ribas, Jun 19 2021

A156568 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=23, a(2)=115.

Original entry on oeis.org

23, 115, 667, 3887, 22655, 132043, 769603, 4485575, 26143847, 152377507, 888121195, 5176349663, 30169976783, 175843511035, 1024891089427, 5973503025527, 34816127063735, 202923259356883, 1182723429077563

Offset: 1

Klaus Brockhaus, Feb 11 2009, Feb 16 2009

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

Second trisection of A156567. Equals 23*A001653.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156569, A156570.

{m=19; v=concat([23, 115], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

a(n) = 23*((2+sqrt(2))*(3-2*sqrt(2))^n +(2-sqrt(2))*(3+2*sqrt(2))^n)/4.
G.f.: 23*x*(1-x)/(1-6*x+x^2). [corrected by Klaus Brockhaus, Sep 22 2009]
Limit_{n -> oo} a(n)/a(n-1) = 3+2*sqrt(2).

A156569 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=37, a(2)=205.

Original entry on oeis.org

37, 205, 1193, 6953, 40525, 236197, 1376657, 8023745, 46765813, 272571133, 1588660985, 9259394777, 53967707677, 314546851285, 1833313400033, 10685333548913, 62278687893445, 362986793811757, 2115642074977097

Offset: 1

Klaus Brockhaus, Feb 11 2009, Feb 16 2009

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

Third trisection of A156567.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156568, A156570.

LinearRecurrence[{6,-1},{37,205},30] (* Harvey P. Dale, Aug 18 2014 *)

{m=19; v=concat([37, 205], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

a(n) = ((34+7*sqrt(2))*(3-2*sqrt(2))^n+(34-7*sqrt(2))*(3+2*sqrt(2))^n)/4.
G.f.: x*(37-17*x)/(1-6*x+x^2). [corrected by Klaus Brockhaus, Sep 22 2009]
Limit_{n -> oo} a(n)/a(n-1) = 3+2*sqrt(2).

A156570 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=17, a(2)=65.

Original entry on oeis.org

17, 65, 373, 2173, 12665, 73817, 430237, 2507605, 14615393, 85184753, 496493125, 2893773997, 16866150857, 98303131145, 572952636013, 3339412684933, 19463523473585, 113441728156577, 661186845465877, 3853679344638685

Offset: 1

Klaus Brockhaus, Feb 11 2009, Feb 16 2009

nonn

lim_{n -> infinity} a(n)/a(n-1) = 3+2*sqrt(2).

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

First trisection of A156567.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156568, A156569.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((74+47*r2)*(3-2*r2)^n+(74-47*r2)*(3+2*r2)^n)/4: n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ];

{m=20; v=concat([17, 65], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

a(n) = ((74+47*sqrt(2))*(3-2*sqrt(2))^n+(74-47*sqrt(2))*(3+2*sqrt(2))^n)/4.

G.f.: x*(17-37*x)/(1-6*x+x^2).

G.f. corrected by Klaus Brockhaus, Sep 22 2009

A157348 Positive numbers y such that y^2 is of the form x^2+(x+281)^2 with integer x.

Original entry on oeis.org

229, 281, 365, 1009, 1405, 1961, 5825, 8149, 11401, 33941, 47489, 66445, 197821, 276785, 387269, 1152985, 1613221, 2257169, 6720089, 9402541, 13155745, 39167549, 54802025, 76677301, 228285205, 319409609, 446908061, 1330543681

Offset: 1

Klaus Brockhaus, Apr 12 2009

(-60, a(1)) and (A129626(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.

			(-60, a(1)) = (-60, 229) is a solution: (-60)^2+(-60+281)^2 = 3600+48841 = 52441 = 229^2.
(A129626(1), a(2)) = (0, 281) is a solution: 0^2+(0+281)^2 = 78961 = 281^2.
(A129626(3), a(4)) = (559, 1009) is a solution: 559^2+(559+281)^2 = 312481+705600 = 1018081 = 1009^2.

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

Cf. A129626, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157349 (decimal expansion of (297+68*sqrt(2))/281), A157350 (decimal expansion of (130803+73738*sqrt(2))/281^2).

{forstep(n=-60, 200000000, [3, 1], if(issquare(2*n^2+562*n+78961, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=229, a(2)=281, a(3)=365, a(4)=1009, a(5)=1405, a(6)=1961.
G.f.: x*(1-x)*(229+510*x+875*x^2+510*x^3+229*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 281*A001653(k) for k >= 1.
Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (297+68*sqrt(2))/281 for n mod 3 = {0, 2}.
Limit_{n -> oo} a(n)/a(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 1.

A157472 Decimal expansion of (627 + 238*sqrt(2))/23^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Mar 12 2009

			(627 + 238*sqrt(2))/23^2 = 1.82151763297693123178...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A118337, A156567, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A156571 (decimal expansion of (27+10*sqrt(2))/23).

SetDefaultRealField(RealField(100)); (627+238*sqrt(2))/23^2; // G. C. Greubel, Aug 17 2018

RealDigits[(627+238Sqrt[2])/23^2,10,120][[1]] (* Harvey P. Dale, Feb 03 2015 *)

(627 + 238*sqrt(2))/23^2 \\ G. C. Greubel, Aug 17 2018

Equals (34 + 7*sqrt(2))/(34 - 7*sqrt(2)) = (3+2*sqrt(2))*(10- 2*sqrt(2) )^2/(10+2*sqrt(2))^2.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, where b is A118337.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, where b is A156567.

A157649 Decimal expansion of (387 + 182*sqrt(2))/17^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Mar 11 2009

Lim_{n -> infinity} b(n)/b(n-1) = (387 + 182*sqrt(2))/17^2 for n mod 3 = 1, b = A155923.

			(387 + 182*sqrt(2))/17^2 = 2.22971234723841971931...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A118120, A155923, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A156163 (decimal expansion of (19+6*sqrt(2))/17).

SetDefaultRealField(RealField(100)); (387 + 182*Sqrt(2))/17^2; // G. C. Greubel, Aug 17 2018

RealDigits[(387 + 182*Sqrt[2])/17^2, 10, 100][[1]] (* G. C. Greubel, Aug 17 2018 *)

(387 + 182*sqrt(2))/17^2 \\ G. C. Greubel, Aug 17 2018

Equals (26 + 7*sqrt(2))/(26 - 7*sqrt(2)) = (3 + 2*sqrt(2))/((19 + 6*sqrt(2))/17)^2 = (3 + 2*sqrt(2))*(6 - sqrt(2))^2/(6 + sqrt(2))^2.

cp's OEIS Frontend

A161482 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+151)^2 = y^2.

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A161486 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+191)^2 = y^2.

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A219014 Numerators in a product expansion for sqrt(2).

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A344576 a(n) = f(n,n) where f(0,n) = f(n,0) = Fibonacci(n) and f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1).

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A156568 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=23, a(2)=115.

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A156569 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=37, a(2)=205.

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A156570 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=17, a(2)=65.

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A157348 Positive numbers y such that y^2 is of the form x^2+(x+281)^2 with integer x.

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