Eli Jaffe - cp's OEIS frontend

A370202 a(n) = a(n-3) + a(n-2) + gcd(a(n-2), a(n-1)) with a(1) = a(2) = a(3) = 1.

1, 1, 1, 3, 3, 7, 7, 17, 15, 25, 37, 41, 63, 79, 105, 143, 185, 249, 329, 435, 579, 767, 1015, 1347, 1783, 2363, 3131, 4147, 5495, 7279, 9643, 12775, 16923, 22419, 29701, 39343, 52121, 69045, 91465, 121171, 160511, 212637, 281683, 373149, 494321, 654833, 867471

Offset: 1

Eli Jaffe, Feb 11 2024

The ratio between consecutive terms (a(n)/a(n-1)) appears to approach the plastic constant A060006.

Eli Jaffe, Table of n, a(n) for n = 1..1000

Cf. A060006, A083658.

from math import gcd
def terms(n):
  nums = [1,1,1]
  for i in range(n-3):
    new_num = nums[i] + nums[i+1] + gcd(nums[i+1], nums[i+2])
    nums.append(new_num)
  return nums

A271310 Decimal expansion of the leftmost root of Im(W(z)/log(z)) = Re(W(z)/log(z)) (negated), where W(z) denotes the Lambert W function.

Original entry on oeis.org

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

Offset: 0

Eli Jaffe, Mar 27 2016

			-0.42131506848449048984606891964560158397494449...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Lambert W-Function
Wikipedia, Lambert W function

f:= z-> Re(LambertW(-z)/ln(-z))-Im(LambertW(-z)/ln(-z)):
Digits:= 200:
fsolve(f(x), x=0.4..1.0);  # Alois P. Heinz, May 04 2016

FindRoot[Im[ProductLog[z]/Log[z]] - Re[ProductLog[z]/Log[z]] == 0, {z, -0.42241, -0.416207}, WorkingPrecision ->100 ]

A263727 Largest square number less than or equal to the n-th Fibonacci number.

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 4, 9, 16, 25, 49, 81, 144, 225, 361, 576, 961, 1521, 2500, 4096, 6724, 10816, 17689, 28561, 46225, 74529, 121104, 196249, 316969, 514089, 831744, 1345600, 2175625, 3523129, 5702544, 9223369, 14922769, 24157225, 39087504, 63234304, 102333456

Offset: 0

Eli Jaffe, Oct 24 2015

			For a(8), Fibonacci(8) = 21, the largest square under 21 is 16, so a(8) = 16.

Cf. A000045, A048760, A061287.

Floor[Sqrt[Fibonacci[Range[40]]]]^2 (* Alonso del Arte, Oct 24 2015 *)

a(n) = sqrtint(fibonacci(n))^2; \\ Michel Marcus, Oct 25 2015

a(n) = floor(sqrt(Fibonacci(n)))^2.
a(n) = A061287(n)^2. - Michel Marcus, Oct 25 2015
a(n) = A048760(A000045(n)). - Michel Marcus, Nov 11 2015

A263651 Numbers n such that the difference between n and the largest square less than n is a nonzero square.

Original entry on oeis.org

2, 5, 8, 10, 13, 17, 20, 26, 29, 34, 37, 40, 45, 50, 53, 58, 65, 68, 73, 80, 82, 85, 90, 97, 101, 104, 109, 116, 122, 125, 130, 137, 145, 148, 153, 160, 170, 173, 178, 185, 194, 197, 200, 205, 212, 221, 226, 229, 234, 241, 250, 257, 260, 265, 272, 281, 290, 293, 298, 305

Offset: 1

Eli Jaffe, Oct 22 2015

Numbers n such that A053186(n) is a positive square. - Michel Marcus, Oct 23 2015
Numbers of the form a^2 + b^2 where a >= 1 and 1 <= b^2 <= 2a. - Robert Israel, Oct 23 2015
Numbers n such that A053610(n) = 2. - Thomas Ordowski, May 22 2016

			For n=5, the largest square less than 5 is 4, and the difference between 4 and 5 is 1, which is also square.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A053186.

N:= 1000: # to get all terms <= N
sort([seq(seq(a^2 + b^2, b=1..min(floor(sqrt(2*a)),floor(sqrt(N-a^2)))),a=1..floor(sqrt(N-1)))]); # Robert Israel, Oct 23 2015

Select[Range@ 305, And[IntegerQ@ Sqrt[# - Floor[Sqrt@ #]^2], ! IntegerQ@ Sqrt@ #] &] (* Michael De Vlieger, Oct 23 2015 *)

isok(n) = (d = (n - sqrtint(n)^2)) && issquare(d); \\ Michel Marcus, Oct 23 2015

More terms from Michel Marcus, Oct 23 2015

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User: Eli Jaffe

A370202 a(n) = a(n-3) + a(n-2) + gcd(a(n-2), a(n-1)) with a(1) = a(2) = a(3) = 1.

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A271310 Decimal expansion of the leftmost root of Im(W(z)/log(z)) = Re(W(z)/log(z)) (negated), where W(z) denotes the Lambert W function.

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A263727 Largest square number less than or equal to the n-th Fibonacci number.

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A263651 Numbers n such that the difference between n and the largest square less than n is a nonzero square.

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