A372050 -id:A372050 - cp's OEIS frontend

A372051 a(n) is the index of the Lucas number that is a ratio of the sum of the first A000217(n) Fibonacci numbers divided by the largest possible Fibonacci number.

1, 0, 3, 5, 9, 11, 16, 20, 23, 29, 33, 39, 47, 53, 62, 70, 77, 87, 95, 105, 117, 127, 140, 152, 163, 177, 189, 203, 219, 233, 250, 266, 281, 299, 315, 333, 353, 371, 392, 412, 431, 453, 473, 495, 519, 541, 566, 590, 613, 639, 663, 689, 717, 743, 772, 800, 827, 857, 885, 915, 947, 977, 1010, 1042

Offset: 1

Tanya Khovanova and MIT PRIMES STEP junior group, Apr 17 2024

nonn

The sum of the first n Fibonacci numbers is sequence A000071.
When we divide the sum by the largest possible Fibonacci number, we always get a Lucas number.
A000217() are the triangular numbers.

			The sum of the first ten Fibonacci numbers is 143. The largest Fibonacci that divides this sum is 13, the seventh Fibonacci number. After the division we get 143/13 = 11, the fifth Lucas number. Thus, as 10 is the fourth triangular number, a(4) = 5.

Cf. A000032, A000045, A000071, A000217, A372048, A372049, A372050.

A372718 Triangular numbers that are 2 mod 4, halved.

Original entry on oeis.org

3, 5, 33, 39, 95, 105, 189, 203, 315, 333, 473, 495, 663, 689, 885, 915, 1139, 1173, 1425, 1463, 1743, 1785, 2093, 2139, 2475, 2525, 2889, 2943, 3335, 3393, 3813, 3875, 4323, 4389, 4865, 4935, 5439, 5513, 6045, 6123, 6683, 6765, 7353, 7439, 8055, 8145, 8789, 8883, 9555, 9653

Offset: 1

Tanya Khovanova and the PRIMES STEP senior group, May 11 2024

The sum of the first 2*a(n) numbers of any Fibonacci-like sequence equals its (a(n)+2)-nd term times the a(n)-th Lucas number.

			10 is a triangular number that has a remainder of 2 when divided by 4. Therefore, its half, 5, is in this sequence. Moreover, the sum of the first 5*2 Fibonacci numbers is 143 (not counting zero). This sum is a product of 13, which is the (5+2 = 7)-th term of the Fibonacci sequence times 11, which is the fifth Lucas number.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000032, A000045, A000217. A372048, A372049, A372050, A372051, A372070.

Select[Table[n (n + 1)/2, {n, 200}], Mod[#, 4] == 2 &]/2

a(n) = A000217(A047457(n))/2 = A372070(n)/2.

cp's OEIS Frontend

A372051 a(n) is the index of the Lucas number that is a ratio of the sum of the first A000217(n) Fibonacci numbers divided by the largest possible Fibonacci number.

Views

Author

Keywords

Comments

Examples

Crossrefs

A372718 Triangular numbers that are 2 mod 4, halved.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula