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a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=37, a(2)=205.
+0
3
37, 205, 1193, 6953, 40525, 236197, 1376657, 8023745, 46765813, 272571133, 1588660985, 9259394777, 53967707677, 314546851285, 1833313400033, 10685333548913, 62278687893445, 362986793811757, 2115642074977097
FORMULA
a(n) = ((34+7*sqrt(2))*(3-2*sqrt(2))^n+(34-7*sqrt(2))*(3+2*sqrt(2))^n)/4.
Limit_{n -> oo} a(n)/a(n-1) = 3+2*sqrt(2).
MATHEMATICA
LinearRecurrence[{6, -1}, {37, 205}, 30] (* Harvey P. Dale, Aug 18 2014 *)
PROG
(PARI) {m=19; v=concat([37, 205], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}
Positive numbers y such that y^2 is of the form x^2+(x+647)^2 with integer x.
+0
3
613, 647, 685, 2993, 3235, 3497, 17345, 18763, 20297, 101077, 109343, 118285, 589117, 637295, 689413, 3433625, 3714427, 4018193, 20012633, 21649267, 23419745, 116642173, 126181175, 136500277, 679840405, 735437783, 795581917
COMMENTS
(-35,a(1)) and ( A130013(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.
FORMULA
a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=613, a(2)=647, a(3)=685, a(4)=2993, a(5)=3235, a(6)=3497.
G.f.: (1-x)*(613+1260*x+1945*x^2+1260*x^3+613*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 647* 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) = (649+36*sqrt(2))/647 for n mod 3 = {0, 2}.
Limit_{n -> oo} a(n)/a(n-1) = (1084467+707402*sqrt(2))/647^2 for n mod 3 = 1.
EXAMPLE
(-35, a(1)) = (-35, 613) is a solution: (-35)^2+(-35+647)^2 = 1225+374544 = 375769 = 613^2.
( A130013(1), a(2)) = (0, 647) is a solution: 0^2+(0+647)^2 = 418609 = 647^2. ( A130013(3), a(4)) = (1768, 2993) is a solution: 1768^2+(1768+647)^2 = 3125824+5832225 = 8958049 = 2993^2.
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {613, 647, 685, 2993, 3235, 3497}, 30] (* Harvey P. Dale, Jun 22 2022 *)
PROG
(PARI) {forstep(n=-36, 10000000, [1, 3], if(issquare(2*n^2+1294*n+418609, &k), print1(k, ", ")))}
CROSSREFS
Cf. A130013, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159642 (decimal expansion of (649+36*sqrt(2))/647), A159643 (decimal expansion of (1084467+707402*sqrt(2))/647^2).
Positive numbers y such that y^2 is of the form x^2+(x+577)^2 with integer x.
+0
3
545, 577, 613, 2657, 2885, 3133, 15397, 16733, 18185, 89725, 97513, 105977, 522953, 568345, 617677, 3047993, 3312557, 3600085, 17765005, 19306997, 20982833, 103542037, 112529425, 122296913, 603487217, 655869553, 712798645, 3517381265
COMMENTS
(-33,a(1)) and ( A130005(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+577)^2 = y^2.
FORMULA
a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=545, a(2)=577, a(3)=613, a(4)=2657, a(5)=2885, a(6)=3133.
G.f.: (1-x)*(545+1122*x+1735*x^2+1122*x^3+545*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 577* 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) = (579+34*sqrt(2))/577 for n mod 3 = {0, 2}.
Limit_{n -> oo} a(n)/a(n-1) = (855171+556990*sqrt(2))/577^2 for n mod 3 = 1.
EXAMPLE
(-33, a(1)) = (-33, 545) is a solution: (-33)^2+(-33+577)^2 = 1089+295936 = 297025 = 545^2.
( A130005(1), a(2)) = (0, 577) is a solution: 0^2+(0+577)^2 = 332929 = 577^2. ( A130005(3), a(4)) = (1568, 2657) is a solution: 1568^2+(1568+577)^2 = 2458624+4601025 = 7059649 = 2657^2.
PROG
(PARI) {forstep(n=-36, 50000000, [3, 1], if(issquare(2*n^2+1154*n+332929, &k), print1(k, ", ")))}
CROSSREFS
Cf. A130005, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159627 (decimal expansion of (579+34*sqrt(2))/577), A159628 (decimal expansion of (855171+556990*sqrt(2))/577^2).
Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+343)^2 = y^2.
+0
5
0, 77, 132, 245, 392, 585, 728, 1029, 1428, 1725, 2352, 3185, 4292, 5117, 6860, 9177, 10904, 14553, 19404, 25853, 30660, 40817, 54320, 64385, 85652, 113925, 151512, 179529, 238728, 317429, 376092, 500045, 664832, 883905, 1047200, 1392237, 1850940, 2192853
COMMENTS
Also values x of Pythagorean triples (x, x+343, y); 343=7^3.
Corresponding values y of solutions (x, y) are in A157246. Limit_{n -> oo} a(n)/a(n-7) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 7 = {1, 2, 4, 5, 6}.
Limit_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 7 = {0, 3}.
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,6,-6,0,0,0,0,0,-1,1).
FORMULA
a(n) = 6*a(n-7)-a(n-14)+686 for n > 14; a(1)=0, a(2)=77, a(3)=132, a(4)=245, a(5)=392, a(6)=585, a(7)=728, a(8)=1029, a(9)=1428, a(10)=1725, a(11)=2352, a(12)=3185, a(13)=4292, a(14)=5117.
G.f.: x*(77+55*x+113*x^2+147*x^3+193*x^4+143*x^5+301*x^6-63*x^7 -33*x^8-51*x^9-49*x^10-51*x^11-33*x^12-63*x^13)/((1-x)*(1-6*x^7+x^14)).
a(7*k+1) = 343* A001652(k) for k >= 0.
EXAMPLE
132^2+(132+343)^2 = 17424+225625 = 243049 = 493^2.
MATHEMATICA
LinearRecurrence[{1, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 0, -1, 1}, {0, 77, 132, 245, 392, 585, 728, 1029, 1428, 1725, 2352, 3185, 4292, 5117, 6860}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
PROG
(PARI) {forstep(n=0, 1400000, [1, 3], if(issquare(n^2+(n+343)^2), print1(n, ", ")))}
Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2401)^2 = y^2.
+0
4
0, 539, 924, 1220, 1715, 2744, 3503, 4095, 5096, 7203, 9996, 12075, 13703, 16464, 22295, 26640, 30044, 35819, 48020, 64239, 76328, 85800, 101871, 135828, 161139, 180971, 214620, 285719, 380240, 450695, 505899, 599564, 797475, 944996, 1060584
COMMENTS
Also values x of Pythagorean triples (x, x+2401, y); 2401=7^4.
Corresponding values y of solutions (x, y) are in A157247. Limit_{n -> oo} a(n)/a(n-9) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 9 = {1, 2, 6}.
Limit_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 9 = {0, 3, 5, 7}.
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))^3 / ((9+4*sqrt(2))/7)^7 for n mod 9 = {4, 8}.
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1).
FORMULA
a(n) = 6*a(n-9)-a(n-18)+4802 for n > 18; a(1)=0, a(2)=539, a(3)=924, a(4)=1220, a(5)=1715, a(6)=2744, a(7)=3503, a(8)=4095, a(9)=5096, a(10)=7203, a(11)=9996, a(12)=12075, a(13)=13703, a(14)=16464,a (15)=22295, a(16)=26640, a(17)=30044, a(18)=35819.
G.f.: x*(539+385*x+296*x^2+495*x^3+1029*x^4+759*x^5+592*x^6 +1001*x^7+2107*x^8-441*x^9-231*x^10-148*x^11-209*x^12-343*x^13 -209*x^14-148*x^15-231*x^16-441*x^17) / ((1-x)*(1-6*x^9+x^18)).
a(9*k+1) = 2401* A001652(k) for k >= 0.
EXAMPLE
924^2+(924+2401)^2 = 853776+11055625 = 11909401 = 3451^2.
PROG
(PARI) {forstep(n=0, 1100000, [3 , 1], if(issquare(n^2+(n+2401)^2), print1(n, ", ")))}
Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+16807)^2 = y^2.
+0
4
0, 2145, 3773, 6468, 8540, 12005, 19208, 24521, 28665, 35672, 41148, 50421, 61388, 69972, 84525, 95921, 115248, 156065, 186480, 210308, 250733, 282405, 336140, 399797, 449673, 534296, 600600, 713097, 950796, 1127973, 1266797, 1502340
COMMENTS
Also values x of Pythagorean triples (x, x+16807, y); 16807 = 7^5.
Corresponding values y of solutions (x, y) are in A156713. Limit_{n -> oo} a(n)/a(n-11) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 11 = {1, 2, 4, 6, 8, 10}.
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))^3 / ((9+4*sqrt(2))/7)^7 for n mod 11 = {0, 3, 5, 9}.
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 11 = 7.
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,0,0,-1,1).
FORMULA
a(n) = 6*a(n-11)-a(n-22)+33614 for n > 22; a(1) = 0, a(2) = 2145, a(3) = 3773, a(4) = 6468, a(5) = 8540, a(6) = 12005, a(7) = 19208, a(8) = 24521, a(9) = 28665, a(10) = 35672, a(11) = 41148, a(12) = 50421, a(13) = 61388, a(14) = 69972, a(15) = 84525, a(16) = 95921, a(17) = 115248, a(18) = 156065, a(19) = 186480, a(20) = 210308, a(21) = 250733, a(22) = 282405.
G.f.: x*(2145+1628*x+2695*x^2+2072*x^3+3465*x^4+7203*x^5+5313*x^6+4144*x^7+7007*x^8+5476*x^9+9273*x^10-1903*x^11-1184*x^12-1617*x^13-1036*x^14-1463*x^15-2401*x^16-1463*x^17 -1036*x^18-1617*x^19-1184*x^20-1903*x^21 )/((1-x)*(1-6*x^11+x^22)).
PROG
(PARI) {forstep(n=0, 1600000, [1, 3], if(issquare(2*n^2 + 33614*n + 282475249), print1(n, ", ")))}
Positive numbers y such that y^2 is of the form x^2+(x+857)^2 with integer x.
+0
4
697, 857, 1117, 3065, 4285, 6005, 17693, 24853, 34913, 103093, 144833, 203473, 600865, 844145, 1185925, 3502097, 4920037, 6912077, 20411717, 28676077, 40286537, 118968205, 167136425, 234807145, 693397513, 974142473, 1368556333
COMMENTS
(-185, a(1)) and ( A129857(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2. lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (907+210*sqrt(2))/857 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1208787+678878*sqrt(2))/857^2 for n mod 3 = 1.
FORMULA
a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=697, a(2)=857, a(3)=1117, a(4)=3065, a(5)=4285, a(6)=6005.
G.f.: (1-x)*(697+1554*x+2671*x^2+1554*x^3+697*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 857* A001653(k) for k >= 1.
EXAMPLE
(-185, a(1)) = (-185, 697) is a solution: (-185)^2+(-185+857)^2 = 34225+451584 = 485809 = 697^2.
( A129857(1), a(2)) = (0, 857) is a solution: 0^2+(0+857)^2 = 734449 = 857^2. ( A129857(3), a(4)) = (1696, 3065) is a solution: 1696^2+(1696+857)^2 = 2876416+6517809 = 9394225 = 3065^2.
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {697, 857, 1117, 3065, 4285, 6005}, 50] (* G. C. Greubel, May 14 2018 *)
PROG
(PARI) {forstep(n=-188, 10000000, [3, 1], if(issquare(2*n^2 +1714*n +734449, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(697+1554*x+2671*x^2+1554*x^3 +697*x^4 )/(1-6*x^3+x^6)) \\ G. C. Greubel, May 14 2018 (Magma) I:=[697, 857, 1117, 3065, 4285, 6005]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..40]]; // G. C. Greubel, May 14 2018
CROSSREFS
Cf. A129857, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160207 (decimal expansion of (907+210*sqrt(2))/857), A160208 (decimal expansion of (1208787+678878*sqrt(2))/857^2).
Positive numbers y such that y^2 is of the form x^2+(x+601)^2 with integer x.
+0
4
425, 601, 1261, 1289, 3005, 7141, 7309, 17429, 41585, 42565, 101569, 242369, 248081, 591985, 1412629, 1445921, 3450341, 8233405, 8427445, 20110061, 47987801, 49118749, 117210025, 279693401, 286285049, 683150089, 1630172605
COMMENTS
(-297, a(1)) and ( A111258(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+601)^2 = y^2. lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (843+418*sqrt(2))/601 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (361299+5950*sqrt(2))/601^2 for n mod 3 = 1.
FORMULA
a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=425, a(2)=601, a(3)=1261, a(4)=1289, a(5)=3005, a(6)=7141.
G.f.: (1-x)*(425 +1026*x +2287*x^2 +1026*x^3 +425*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 601* A001653(k) for k >= 1.
EXAMPLE
(-297, a(1)) = (-297, 425) is a solution: (-297)^2+(-297+601)^2 = 88209+92416 = 180625 = 425^2.
( A111258(1), a(2)) = (0, 601) is a solution: 0^2+(0+601)^2 = 361201 = 601^2. ( A111258(3), a(4)) = (560, 1289) is a solution: 560^2+(560+601)^2 = 313600+1347921 = 1661521 = 1289^2.
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {425, 601, 1261, 1289, 3005, 7141}, 50] (* G. C. Greubel, Apr 22 2018 *)
PROG
(PARI) {forstep(n=-300, 10000000, [3, 1], if(issquare(2*n^2+1202*n+361201, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(425 +1026*x +2287*x^2 +1026*x^3 +425*x^4 )/(1-6*x^3+x^6)) \\ G. C. Greubel, Apr 22 2018 (Magma) I:=[425, 601, 1261, 1289, 3005, 7141]; [n le 6 select I[n] else 5*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 22 2018
CROSSREFS
Cf. A111258, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160099 (decimal expansion of (843+418*sqrt(2))/601), A160100 (decimal expansion of (361299+5950*sqrt(2))/601^2).
Positive numbers y such that y^2 is of the form x^2+(x+953)^2 with integer x.
+0
3
845, 953, 1093, 3977, 4765, 5713, 23017, 27637, 33185, 134125, 161057, 193397, 781733, 938705, 1127197, 4556273, 5471173, 6569785, 26555905, 31888333, 38291513, 154779157, 185858825, 223179293, 902119037, 1083264617, 1300784245
COMMENTS
(-116, a(1)) and ( A129975(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+953)^2 = y^2. lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (969+124*sqrt(2))/953 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1947891+1218490*sqrt(2))/953^2 for n mod 3 = 1.
FORMULA
a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=845, a(2)=953, a(3)=1093, a(4)=3977, a(5)=4765, a(6)=5713.
G.f.: (1-x)*(845+1798*x+2891*x^2+1798*x^3+845*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 953* A001653(k) for k >= 1.
EXAMPLE
(-116, a(1)) = (-116, 845) is a solution: (-116)^2+(-116+953)^2 = 13456+700569 = 714025 = 845^2.
( A129975(1), a(2)) = (0, 953) is a solution: 0^2+(0+953)^2 = 908209 = 953^2. ( A129975(3), a(4)) = (2295, 3977) is a solution: 2295^2+(2295+953)^2 = 5267025+10549504 = 15816529 = 3977^2.
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {845, 953, 1093, 3977, 4765, 5713}, 30] (* Harvey P. Dale, Feb 18 2024 *)
PROG
(PARI) {forstep(n=-116, 10000000, [3, 1], if(issquare(2*n^2+1906*n+908209, &k), print1(k, ", ")))}
CROSSREFS
Cf. A129975, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160213 (decimal expansion of (969+124*sqrt(2))/953), A160214 (decimal expansion of (1947891+1218490*sqrt(2))/953^2).
Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 41)^2 = y^2.
+0
12
0, 36, 39, 123, 319, 336, 820, 1960, 2059, 4879, 11523, 12100, 28536, 67260, 70623, 166419, 392119, 411720, 970060, 2285536, 2399779, 5654023, 13321179, 13987036, 32954160, 77641620, 81522519, 192071019, 452528623, 475148160, 1119472036
COMMENTS
Also values x of Pythagorean triples (x, x+41, y).
Corresponding values y of solutions (x, y) are in A157257. lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (7 + 2*sqrt(2))/(7 - 2*sqrt(2)) for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3 + 2*sqrt(2))*(7 - 2*sqrt(2))^2/(7 + 2*sqrt(2))^2 for n mod 3 = 0.
FORMULA
a(n) = 6*a(n-3) - a(n-6) + 82 for n > 6; a(1)=0, a(2)=36, a(3)=39, a(4)=123, a(5)=319, a(6)=336.
G.f.: x*(36 + 3*x + 84*x^2 - 20*x^3 - x^4 - 20*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
a(3*k + 1) = 41* A001652(k) for k >= 0.
MATHEMATICA
LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 36, 39, 123, 319, 336, 820}, 40] (* Harvey P. Dale, Jan 18 2015 *)
PROG
(PARI) forstep(n=0, 1200000000, [3 , 1], if(issquare(2*n^2+82*n+1681), print1(n, ", ")))
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(36+3*x+84*x^2-20*x^3-x^4-20*x^5)/((1-x)*(1-6*x^3+ x^6)))); // G. C. Greubel, May 07 2018
CROSSREFS
Cf. A157257, A001652, A156035 (decimal expansion of 3 + 2*sqrt(2)), A157258 (decimal expansion of 7 + 2*sqrt(2)), A157259 (decimal expansion of 7 - 2*sqrt(2)), A157260 (decimal expansion of (7 + 2*sqrt(2))/(7 - 2*sqrt(2))).
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