A122931 -id:A122931 - cp's OEIS frontend

A191797 a(n) = binomial(F(n), 2) where F(n) = A000045(n).

0, 0, 0, 1, 3, 10, 28, 78, 210, 561, 1485, 3916, 10296, 27028, 70876, 185745, 486591, 1274406, 3337236, 8738290, 22879230, 59901985, 156830905, 410597496, 1074972528, 2814337800, 7368069528, 19289917153, 50501756955, 132215475106, 346144864780, 906219437046

Offset: 0

Emeric Deutsch, Jun 21 2011

			a(7) = binomial(13,2) = 78.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Cf. A000045, A000071, A000217, A001622, A056014, A094825 (binomial transform), A122931.

with(combinat): seq(binomial(fibonacci(n), 2), n = 0 .. 30);

Table[Binomial[Fibonacci[n], 2], {n, 0, 39}] (* Alonso del Arte, Apr 04 2013 *)

a(n) = binomial(fibonacci(n), 2); \\ Michel Marcus, Sep 07 2015

concat(vector(3), Vec(x^3 / ((1+x)*(1-x-x^2)*(1-3*x+x^2)) + O(x^40))) \\ Colin Barker, Mar 26 2017

from sympy import binomial, fibonacci
def a(n): return binomial(fibonacci(n), 2) # Indranil Ghosh, Mar 26 2017

a(n) = 3*a(n-1) + 1*a(n-2) - 5*a(n-3) - 1*a(n-4) + 1*a(n-5).
G.f.: x^3/(1-3*x-x^2+5*x^3+x^4-x^5) = x^3/((1+x)*(1-x-x^2)*(1-3*x+x^2)).
a(n) + a(n+1) = A056014(n+1). - R. J. Mathar, Jun 24 2011
a(n) = (2*F(n)^2 - F(n+4) + 3*F(n+1))/4, F(n) = A000045(n). - Gary Detlefs, Jan 05 2013
a(n) = Sum_{k=1..n-2} A122931(k). - J. M. Bergot, Apr 05 2013
a(n) = A000217(A000071(n)). - Peter M. Chema, Mar 26 2017
a(n) = (2^(-1-n)*(-(-1)^n*2^(1+n) + sqrt(5)*(1-sqrt(5))^n + (3-sqrt(5))^n - sqrt(5)*(1+sqrt(5))^n + (3+sqrt(5))^n)) / 5. - Colin Barker, Mar 26 2017
a(n) ~ phi^(2*n) / 10, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 26 2025

A122930 Triangular array read by rows, based on the Zeckendorf expansion of n and containing the golden rectangle sequence A001629.

Original entry on oeis.org

1, 0, 2, 1, 0, 6, 1, 2, 0, 15, 2, 2, 6, 0, 40, 3, 4, 6, 15, 0, 104, 5, 6, 12, 15, 40, 0, 273, 8, 10, 18, 30, 40, 104, 0, 714, 13, 16, 30, 45, 80, 104, 273, 0, 1870, 21, 26, 48, 75, 120, 208, 273, 714, 0, 4895, 34, 42, 78, 120, 200, 312, 546, 714, 1870, 0, 12816

Offset: 1

Alford Arnold, Sep 20 2006

			The array begins:
  1
  0 2
  1 0 6
  1 2 0 15
  2 2 6 0 40
  3 4 6 15 0 104
  5 6 12 15 40 0 273
Row five is:
  2 2 6 0 40
because the values
  1 2 3 5 8 in Zeckendorf's expansion occur
  2 1 2 0 5 times for natural numbers 8 through 12.

Amiram Eldar, Table of n, a(n) for n = 1..210 (rows n=1..20, flattened)

Cf. A001629, A014417, A122931.

seq[rownum_] := Plus @@@ SplitBy[(#*Fibonacci[Range[2, Length[#] + 1]]) & /@ Reverse /@ IntegerDigits[FromDigits /@ Select[Tuples[{0, 1}, rownum], SequenceCount[#, {1, 1}] == 0 &]], Length] // Flatten; seq[11] (* Amiram Eldar, Jun 28 2025 *)

More terms from Amiram Eldar, Jun 28 2025

A301809 Group the natural numbers such that the first group is (1) then (2),(3),(4,5),(6,7,8),... with the n-th group containing F(n) sequential terms where F(n) is the n-th Fibonacci number (A000045(n)). Sequence gives the sum of terms in the n-th group.

Original entry on oeis.org

1, 2, 3, 9, 21, 55, 140, 364, 945, 2465, 6435, 16821, 43992, 115102, 301223, 788425, 2063817, 5402651, 14143524, 37026936, 96935685, 253777537, 664392743, 1739393929, 4553778096, 11921922650, 31211961195, 81713914569, 213929707485, 560075086495, 1466295355580, 3838810662436, 10050136117497

Offset: 1

Frank M Jackson, Mar 27 2018

a(n) is the sum of all nodes at height n-1 within a binary tree structure recursively built from the Hofstadter G-sequence (see comments for A005206).

			a(7) = 14 + 15 + 16 + ... + 21 = (F(9)+1)*F(6)/2 = 140.

Colin Barker, Table of n, a(n) for n = 1..1000
Wikipedia, Hofstadter sequence.
Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1, 1).

Cf. A000045, A005206, A122931.

Cf. A000217, A033192.

[1] cat [(Fibonacci(n+2)+1)*Fibonacci(n-1) div 2 : n in [2..35] ]; // Vincenzo Librandi, Apr 18 2018

a[n_] := If[n==1, 1, (Fibonacci[n+2]+1)Fibonacci[n-1]/2]; Array[a, 50]
Join[{1}, LinearRecurrence[{3, 1, -5, -1, 1}, {2, 3, 9, 21, 55}, 40]] (* Vincenzo Librandi, Apr 18 2018 *)

a(n) = if (n==1, 1, (fibonacci(n+2)+1)*fibonacci(n-1)/2); \\ Michel Marcus, Apr 21 2018

Vec(x*(1 - x)*(1 - 4*x^2 - x^3 + x^4) / ((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)) + O(x^60)) \\ Colin Barker, May 11 2018

a(1) = 1 and for n > 1, a(n) = (F(n+2)+1)*F(n-1)/2, where F(n) is the n-th Fibonacci number (A000045).
From Colin Barker, Mar 27 2018: (Start)
G.f.: x*(1 - x)*(1 - 4*x^2 - x^3 + x^4) / ((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)).
a(n) = 3*a(n-1) + a(n-2) - 5*a(n-3) - a(n-4) + a(n-5) for n>6. (End)
a(n) = A033192(n+1) - A033192(n) for n > 1. - J.S. Seneschal, Jul 07 2025

cp's OEIS Frontend

A191797 a(n) = binomial(F(n), 2) where F(n) = A000045(n).

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A122930 Triangular array read by rows, based on the Zeckendorf expansion of n and containing the golden rectangle sequence A001629.

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A301809 Group the natural numbers such that the first group is (1) then (2),(3),(4,5),(6,7,8),... with the n-th group containing F(n) sequential terms where F(n) is the n-th Fibonacci number (A000045(n)). Sequence gives the sum of terms in the n-th group.

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Magma

Mathematica

PARI

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