A129641 -id:A129641 - cp's OEIS frontend

A129642 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+457)^2 = y^2.

0, 348, 495, 1371, 3255, 4088, 9140, 20096, 24947, 54383, 118235, 146508, 318072, 690228, 855015, 1854963, 4024047, 4984496, 10812620, 23454968, 29052875, 63021671, 136706675, 169333668, 367318320, 796785996, 986950047, 2140889163, 4644010215, 5752367528

Offset: 1

Mohamed Bouhamida, May 31 2007

Also values x of Pythagorean triples (x, x+457, y).
Corresponding values y of solutions (x, y) are in A160580.
Limit_{n->oo} a(n)/a(n-3) = 3+2*sqrt(2).
Limit_{n->oo} a(n)/a(n-1) = (601+276*sqrt(2))/457 for n mod 3 = {1, 2}.
Limit_{n->oo} a(n)/a(n-1) = (213651+31850*sqrt(2))/457^2 for n mod 3 = 0.

Harvey P. Dale, 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. A160580, A001652, A129641, A156035 (decimal expansion of 3+2*sqrt(2)), A160581 (decimal expansion of (601+276*sqrt(2))/457), A160582 (decimal expansion of (213651+31850*sqrt(2))/457^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,348,495,1371,3255,4088,9140},30] (* Harvey P. Dale, May 13 2012 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+914*n+208849), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+914 for n > 6; a(1)=0, a(2)=348, a(3)=495, a(4)=1371, a(5)=3255, a(6)=4088.
G.f.: x^2*(348+147*x+876*x^2-204*x^3-49*x^4-204*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 457*A001652(k) for k >= 0.
a(1)=0, a(2)=348, a(3)=495, a(4)=1371, a(5)=3255, a(6)=4088, a(7)=9140, a(n) = a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, May 13 2012

Edited and two terms added by Klaus Brockhaus, Jun 08 2009

A160578 Decimal expansion of (473+168*sqrt(2))/409.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Jun 08 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129641.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A160577.

			(473+168*sqrt(2))/409 = 1.73737867598699258728...

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

Cf. A129641, A160577, A002193 (decimal expansion of sqrt(2)), A160579 (decimal expansion of (204819+83570*sqrt(2))/409^2).

(473 +168*Sqrt(2))/409; // G. C. Greubel, Apr 08 2018

RealDigits[(473+168Sqrt[2])/409,10,120][[1]] (* Harvey P. Dale, Mar 05 2015 *)

(473 +168*sqrt(2))/409 \\ G. C. Greubel, Apr 08 2018

Equals (42+8*sqrt(2))/(42-8*sqrt(2)).

A160579 Decimal expansion of (204819+83570*sqrt(2))/409^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Jun 08 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A129641.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160577.

			(204819+83570*sqrt(2))/409^2 = 1.93091162419832230335...

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

Cf. A129641, A160577, A002193 (decimal expansion of sqrt(2)), A160578 (decimal expansion of (473+168*sqrt(2))/409).

(204819 +83570*Sqrt(2))/409^2; // G. C. Greubel, Apr 08 2018

RealDigits[(204819+83570*Sqrt[2])/409^2,10,120][[1]] (* Harvey P. Dale, Jul 16 2013 *)

(204819 +83570*sqrt(2))/409^2 \\ G. C. Greubel, Apr 08 2018

Equals (610+137*sqrt(2))/(610-137*sqrt(2)).

Equals (3+2*sqrt(2))*(42-8*sqrt(2))^2/(42+8*sqrt(2))^2.

A160577 Positive numbers y such that y^2 is of the form x^2+(x+409)^2 with integer x.

Original entry on oeis.org

305, 409, 641, 1189, 2045, 3541, 6829, 11861, 20605, 39785, 69121, 120089, 231881, 402865, 699929, 1351501, 2348069, 4079485, 7877125, 13685549, 23776981, 45911249, 79765225, 138582401, 267590369, 464905801, 807717425, 1559630965

Offset: 1

Klaus Brockhaus, Jun 08 2009

nonn

(-136, a(1)) and (A129641(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+409)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (473+168*sqrt(2))/409 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (204819+83570*sqrt(2))/409^2 for n mod 3 = 1.

			(-136, a(1)) = (-136, 305) is a solution: (-136)^2+(-136+409)^2 = 18496+74529 = 93025 = 305^2.
(A129641(1), a(2)) = (0, 409) is a solution: 0^2+(0+409)^2 = 167281 = 409^2.
(A129641(3), a(4)) = (611, 1189) is a solution: 611^2+(611+409)^2 = 373321+1040400 = 1413721 = 1189^2.

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

Cf. A129641, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160578 (decimal expansion of (473+168*sqrt(2))/409), A160579 (decimal expansion of (204819+83570*sqrt(2))/409^2).

LinearRecurrence[{0,0,6,0,0,-1},{305,409,641,1189,2045,3541},50] (* or *) Select[Table[Sqrt[x^2+(x+409)^2],{x,-140,10^6}],IntegerQ] (* The second program generates the first 16 terms of the sequence. To generate more, increase the x constant but the program may take a long time to run. *) (* Harvey P. Dale, Mar 14 2022 *)

{forstep(n=-136, 10000000, [3, 1], if(issquare(2*n^2+818*n+167281, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=305, a(2)=409, a(3)=641, a(4)=1189, a(5)=2045, a(6)=3541.
G.f.: (1-x)*(305+714*x+1355*x^2+714*x^3+305*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 409*A001653(k) for k >= 1.

A201916 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2737)^2 = y^2.

Original entry on oeis.org

0, 75, 203, 323, 552, 708, 1020, 1127, 1311, 1428, 1608, 1820, 1955, 2336, 2675, 3128, 3311, 3627, 3927, 4140, 4508, 4743, 5535, 6003, 6800, 7280, 7848, 8211, 8588, 9240, 9860, 11063, 11895, 13583, 14168, 15180, 15827, 16827, 18011, 18768, 20915, 22836

Offset: 1

T. D. Noe, Feb 09 2012

Note that 2737 = 7 * 17 * 23, the product of the first three distinct primes in A058529 (and A001132) and hence the smallest such number. This sequence satisfies a linear difference equation of order 55 whose 55 initial terms can be found by running the Mathematica program.
There are many sequences like this one. What determines the order of the linear difference equation? All primes p have order 7. For those p, it appears that p^2 has order 11, p^3 order 15, and p^i order 3+4*i. It appears that for semiprimes p*q (with p > q), the order is 19. What is the next term of the sequence beginning 3, 7, 19, 55, 163? This could be sequence A052919, which is 1 + 2*3^f, where f is the number of primes.
The crossref list is thought to be complete up to Feb 14 2012.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A001652 (1), A076296 (7), A118120 (17), A118337 (23), A118674 (31).
Cf. A129288 (41), A118675 (47), A118554 (49), A118673 (71), A129289 (73).
Cf. A118676 (79), A129298 (89), A129836 (97), A157119 (103), A161478 (113).
Cf. A129837 (119), A129992 (127), A129544 (137), A161482 (151).
Cf. A206426 (161), A130608 (167), A161486 (191), A185394 (193).
Cf. A129993 (199), A198294 (217), A130609 (223), A129625 (233).
Cf. A204765 (239), A129991 (241), A207058 (263), A129626 (281).
Cf. A205644 (287), A207059 (289), A129640 (313), A205672 (329).
Cf. A129999 (337), A118611 (343), A130610 (359), A207060 (401).
Cf. A129641 (409), A207061 (433), A130645 (439), A130004 (449).
Cf. A129642 (457), A129725 (521), A101152 (569), A130005 (577).
Cf. A207075 (479), A207076 (487), A207077 (497), A207078 (511).
Cf. A111258 (601), A115135 (617), A130013 (647), A130646 (727).
Cf. A122694 (761), A123654 (809), A129010 (833), A130647 (839).
Cf. A129857 (857), A130014 (881), A129974 (937), A129975 (953).
Cf. A130017 (967), A118630 (2401), A118576 (16807).

d = 2737; terms = 100; t = Select[Range[0, 55000], IntegerQ[Sqrt[#^2 + (#+d)^2]] &]; Do[AppendTo[t, t[[-1]] + 6*t[[-27]] - 6*t[[-28]] - t[[-54]] + t[[-55]]], {terms-55}]; t

a(n) = a(n-1) + 6*a(n-27) - 6*a(n-28) - a(n-54) + a(n-55), where the 55 initial terms can be computed using the Mathematica program.

G.f.: x^2*(73*x^53 +116*x^52 +100*x^51 +171*x^50 +104*x^49 +184*x^48 +57*x^47 +92*x^46 +55*x^45 +80*x^44 +88*x^43 +53*x^42 +139*x^41 +113*x^40 +139*x^39 +53*x^38 +88*x^37 +80*x^36 +55*x^35 +92*x^34 +57*x^33 +184*x^32 +104*x^31 +171*x^30 +100*x^29 +116*x^28 +73*x^27 -363*x^26 -568*x^25 -480*x^24 -797*x^23 -468*x^22 -792*x^21 -235*x^20 -368*x^19 -213*x^18 -300*x^17 -316*x^16 -183*x^15 -453*x^14 -339*x^13 -381*x^12 -135*x^11 -212*x^10 -180*x^9 -117*x^8 -184*x^7 -107*x^6 -312*x^5 -156*x^4 -229*x^3 -120*x^2 -128*x -75) / ((x -1)*(x^54 -6*x^27 +1)). - Colin Barker, May 18 2015

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A129642 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+457)^2 = y^2.

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A160578 Decimal expansion of (473+168*sqrt(2))/409.

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A160579 Decimal expansion of (204819+83570*sqrt(2))/409^2.

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A160577 Positive numbers y such that y^2 is of the form x^2+(x+409)^2 with integer x.

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A201916 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2737)^2 = y^2.

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