%I #16 May 18 2015 08:10:12

%S 0,75,203,323,552,708,1020,1127,1311,1428,1608,1820,1955,2336,2675,

%T 3128,3311,3627,3927,4140,4508,4743,5535,6003,6800,7280,7848,8211,

%U 8588,9240,9860,11063,11895,13583,14168,15180,15827,16827,18011,18768,20915,22836

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

%C 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.

%C 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.

%C The crossref list is thought to be complete up to Feb 14 2012.

%H Colin Barker, <a href="/A201916/b201916.txt">Table of n, a(n) for n = 1..1000</a>

%F 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.

%F 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

%t 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

%Y Cf. A001652 (1), A076296 (7), A118120 (17), A118337 (23), A118674 (31).

%Y Cf. A129288 (41), A118675 (47), A118554 (49), A118673 (71), A129289 (73).

%Y Cf. A118676 (79), A129298 (89), A129836 (97), A157119 (103), A161478 (113).

%Y Cf. A129837 (119), A129992 (127), A129544 (137), A161482 (151).

%Y Cf. A206426 (161), A130608 (167), A161486 (191), A185394 (193).

%Y Cf. A129993 (199), A198294 (217), A130609 (223), A129625 (233).

%Y Cf. A204765 (239), A129991 (241), A207058 (263), A129626 (281).

%Y Cf. A205644 (287), A207059 (289), A129640 (313), A205672 (329).

%Y Cf. A129999 (337), A118611 (343), A130610 (359), A207060 (401).

%Y Cf. A129641 (409), A207061 (433), A130645 (439), A130004 (449).

%Y Cf. A129642 (457), A129725 (521), A101152 (569), A130005 (577).

%Y Cf. A207075 (479), A207076 (487), A207077 (497), A207078 (511).

%Y Cf. A111258 (601), A115135 (617), A130013 (647), A130646 (727).

%Y Cf. A122694 (761), A123654 (809), A129010 (833), A130647 (839).

%Y Cf. A129857 (857), A130014 (881), A129974 (937), A129975 (953).

%Y Cf. A130017 (967), A118630 (2401), A118576 (16807).

%K nonn,easy

%O 1,2

%A _T. D. Noe_, Feb 09 2012