A274680
Values of n such that 2*n+1 and 4*n+1 are both triangular numbers.
Original entry on oeis.org
0, 16065, 545751, 21394547226, 726784809030, 28491418065071115, 967869505172593485, 37942420317086720855700, 1288925370210688376036076, 50528452330120333959563160501, 1716479960463788790499334882595, 67289447366315927998308608003134830
Offset: 1
16065 is in the sequence because 2*16065+1 = 32131, 4*16065+1 = 64261, and 32131 and 64261 are both triangular numbers.
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Rest@ CoefficientList[Series[459 x^2 (35 + 1154 x + 35 x^2)/((1 - x) (1 - 1154 x + x^2) (1 + 1154 x + x^2)), {x, 0, 12}], x] (* Michael De Vlieger, Jul 02 2016 *)
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isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(4*n+1, 3)
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concat(0, Vec(459*x^2*(35+1154*x+35*x^2)/((1-x)*(1-1154*x+x^2)*(1+1154*x+x^2)) + O(x^20)))
A274756
Values of n such that 2*n+1 and 6*n+1 are both triangular numbers.
Original entry on oeis.org
0, 945, 13167, 35578242, 495540990, 1338951572595, 18649189618605, 50390103447476100, 701843601611053692, 1896381151803363988917, 26413182084381205040235, 71368408216577696911440390, 994033693861758668873164410, 2685878672926303893761783662455
Offset: 1
945 is in the sequence because 2*945+1 = 1891, 6*945+1 = 5671, and 1891 and 5671 are both triangular numbers.
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isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(6*n+1, 3)
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concat(0, Vec(63*x^2*(15+194*x+15*x^2)/((1-x)*(1-194*x+x^2)*(1+194*x+x^2)) + O(x^20)))
A274832
Values of n such that 2*n+1 and 7*n+1 are both triangular numbers (A000217).
Original entry on oeis.org
0, 27, 297, 24570, 267030, 22064157, 239792967, 19813588740, 215333817660, 17792580624687, 193369528466037, 15977717587380510, 173645621228683890, 14347972600887073617, 155933574493829667507, 12884463417879004727880, 140028176249837812737720
Offset: 1
27 is in the sequence because 2*27+1 = 55, 7*27+1 = 190, and 55 and 190 are both triangular numbers.
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LinearRecurrence[{1, 898, -898, -1, 1}, {0, 27, 297, 24570, 267030}, 20] (* Paolo Xausa, Oct 21 2024 *)
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isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(7*n+1, 3)
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concat(0, Vec(27*x^2*(1+10*x+x^2)/((1-x)*(1-30*x+x^2)*(1+30*x+x^2)) + O(x^20)))
A279042
Numbers k such that 2*k+1 and 10*k+1 are both triangular numbers (A000217).
Original entry on oeis.org
4455, 30537, 461938302, 3166172226, 47894687058501, 328275068740587, 4965816943137597372, 34036215673995404100, 514865832250497683700195, 3528942913182916419190605, 53382319214430283898266055610, 365887859090594924500524938502
Offset: 1
4455 is in the sequence because 2*4455+1 = 8911 and 10*4455+1 = 44551 are both triangular numbers.
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LinearRecurrence[{1, 103682, -103682, -1, 1}, {4455, 30537, 461938302, 3166172226, 47894687058501}, 20] (* Vincenzo Librandi, Dec 05 2016 *)
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Vec(81*x*(55 + 322*x + 55*x^2) / ((1 - x)*(1 - 322*x + x^2)*(1 + 322*x + x^2)) + O(x^15))
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isok(k) = ispolygonal(2*k+1, 3) & ispolygonal(10*k+1, 3)
Showing 1-4 of 4 results.
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