A157646 -id:A157646 - cp's OEIS frontend

A118674 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31)^2 = y^2.

0, 9, 60, 93, 140, 429, 620, 893, 2576, 3689, 5280, 15089, 21576, 30849, 88020, 125829, 179876, 513093, 733460, 1048469, 2990600, 4274993, 6111000, 17430569, 24916560, 35617593, 101592876, 145224429, 207594620, 592126749, 846430076

Offset: 1

Mohamed Bouhamida, May 19 2006

Also values x of Pythagorean triples (x, x+31, y).
Corresponding values y of solutions (x, y) are in A157646.
For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (33 + 8*sqrt(2))/31 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1539 + 850*sqrt(2))/31^2 for n mod 3 = 0.

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

cf. A157646, A066436 (primes of the form 2*n^2-1), A118673, A129836, A001652, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3 + 2*sqrt(2)), A157647 (decimal expansion of (33 + 8*sqrt(2))/31), A157648 (decimal expansion of (1539 + 850*sqrt(2))/31^2).

I:=[0,9,60,93,140,429,620]; [n le 7 select I[n] else Self(n-1) - 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..50]]; // G. C. Greubel, Mar 31 2018

ClearAll[a]; Evaluate[Array[a, 6]] = {0, 9, 60, 93, 140, 429}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 62; Table[a[n], {n, 1, 31}] (* Jean-François Alcover, Dec 27 2011, after given formula *)
LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,9,60,93,140,429,620}, 50] (* G. C. Greubel, Mar 31 2018 *)

{forstep(n=0, 850000000, [1, 3], if(issquare(2*n^2+62*n+961), print1(n, ",")))};

a(n) = 6*a(n-3) - a(n-6) + 62 for n > 6; a(1)=0, a(2)=9, a(3)=60, a(4)=93, a(5)=140, a(6)=429.
G.f.: x*(9 + 51*x + 33*x^2 - 7*x^3 - 17*x^4 - 7*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
a(3*k + 1) = 31*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Mar 11 2009

A157647 Decimal expansion of (33+8*sqrt(2))/31.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Mar 11 2009

lim_{n -> infinity} b(n)/b(n-1) = (33+8*sqrt(2))/31 for n mod 3 = {1, 2}, b = A118674.

lim_{n -> infinity} b(n)/b(n-1) = (33+8*sqrt(2))/31 for n mod 3 = {0, 2}, b = A157646.

			(33+8*sqrt(2))/31 = 1.42947446770918581904...

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

A118674, A157646, A002193 (decimal expansion of sqrt(2)), A157648 (decimal expansion of (1539+850*sqrt(2))/31^2).

(33+8*Sqrt(2))/31; // G. C. Greubel, Mar 30 2018

with(MmaTranslator[Mma]): Digits:=100:
RealDigits(evalf((33+8*sqrt(2))/31))[1]; # Muniru A Asiru, Mar 31 2018

RealDigits[(33+8*Sqrt[2])/31,10,120][[1]] (* Harvey P. Dale, Mar 02 2014 *)

(33+8*sqrt(2))/31 \\ G. C. Greubel, Mar 30 2018

(33+8*sqrt(2))/31 = (8+sqrt(2))/(8-sqrt(2)).

A157648 Decimal expansion of (1539+850*sqrt(2))/31^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Mar 11 2009

Limit_{n -> oo} b(n)/b(n-1) = (1539+850*sqrt(2))/31^2 for n mod 3 = 0, b = A118674.

Limit_{n -> oo} b(n)/b(n-1) = (1539+850*sqrt(2))/31^2 for n mod 3 = 1, b = A157646.

			2.85232208950794046980...

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

Cf. A118674, A157646, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157647 (decimal expansion of (33+8*sqrt(2))/31).

(1539+850*Sqrt(2))/31^2; // G. C. Greubel, Mar 30 2018

with(MmaTranslator[Mma]): Digits:=100:
RealDigits(evalf((1539+850*sqrt(2))/31^2))[1]; # Muniru A Asiru, Mar 31 2018

RealDigits[(1539+850*Sqrt[2])/31^2, 10, 100][[1]] (* G. C. Greubel, Mar 30 2018 *)

(1539+850*sqrt(2))/31^2 \\ G. C. Greubel, Mar 30 2018

Equals (50+17*sqrt(2))/(50-17*sqrt(2)).

Equals (3+2*sqrt(2))*(8-sqrt(2))^2/(8+sqrt(2))^2.

cp's OEIS Frontend

A118674 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31)^2 = y^2.

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A157647 Decimal expansion of (33+8*sqrt(2))/31.

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A157648 Decimal expansion of (1539+850*sqrt(2))/31^2.

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