A130253 -id:A130253 - cp's OEIS frontend

A130238 Partial sums of A130237.

0, 2, 8, 20, 36, 61, 91, 126, 174, 228, 288, 354, 426, 517, 615, 720, 832, 951, 1077, 1210, 1350, 1518, 1694, 1878, 2070, 2270, 2478, 2694, 2918, 3150, 3390, 3638, 3894, 4158, 4464, 4779, 5103, 5436, 5778, 6129, 6489, 6858, 7236, 7623, 8019, 8424, 8838

Offset: 0

Hieronymus Fischer, May 17 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000045, A130233, A130234, A130235, A130236, A130237, A130239, A130240, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

[(&+[j*Floor(Log(3/2 +j*Sqrt(5))/Log((1+Sqrt(5))/2)): j in [0..n]]): n in [0..70]]; // G. C. Greubel, Mar 18 2023

a[n_]:= a[n]= Sum[j*Floor[Log[GoldenRatio, 3/2 +j*Sqrt[5]]], {j,0,n}];
Table[a[n], {n,0,70}] (* G. C. Greubel, Mar 18 2023 *)

def A130238(n): return sum(j*int(log(3/2 +j*sqrt(5), golden_ratio)) for j in range(n+1))
[A130238(n) for n in range(71)] # G. C. Greubel, Mar 18 2023

a(n) = Sum_{k=0..n} A130237(k).
a(n) = (n*(n+1)*A130233(n) - (Fib(A130233(n)) - 1)*(Fib(A130233(n) + 1) - 1))/2.
G.f.: (1/(1-x)^3)*Sum_{k>=1} (Fib(k)*(1-x) + x)*x^Fib(k).

A335741 Number of Pell numbers (A000129) <= n.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Ovidiu Bagdasar, Jun 20 2020

nonn

The sequence is constant on the interval A000129(k) < n <= A000129(k+1).

			The Pell numbers A000129 are 0,1,2,5,12,29,70,...
We have a(2)=a(3)=a(4)=3, since there are three Pell numbers less than or equal to 2,3 and 4, respectively.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36.

Cf. A000129 (Pell Numbers), A108852 (Fibonacci), A130245 (Lucas), A130253 (Jacobsthal).

Partial sums of A105349.

Block[{a = 2, b = -1, nn = 105, u, v = {}}, u = {0, 1}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, ?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v] (* _Michael De Vlieger, Jun 11 2021 *)
Module[{pn=LinearRecurrence[{2,1},{0,1},9],nn=100},Accumulate[Table[If[ MemberQ[ pn,n],1,0],{n,0,nn}]]] (* Harvey P. Dale, Apr 10 2022 *)

a(n) = 1+floor(log_alpha(2*sqrt(2)*n+1)), n>=0, where alpha=1+sqrt(2).

A345380 Number of Jacobsthal-Lucas numbers m <= n.

Original entry on oeis.org

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

Offset: 0

Ovidiu Bagdasar, Jun 16 2021

			a(0)=0 since the least term in A014551 is 1.
a(1)=1 since A014551(0) = 2 is followed in that sequence by 1.
a(k)=2 for 2 <= k <= 4 since the first terms of A014551 are {2, 1, 5}.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36. See Section 5.6, pp. 35, Table 7.

Cf. A014551, A108852 (Fibonacci), A130245 (Lucas), A130253.

Block[{a = 1, b = -2, nn = 105, u, v = {}}, u = {2, a}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, _?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v]  (* or *)
{0}~Join~Accumulate@ ReplacePart[ConstantArray[0, Last[#]], Map[# -> 1 &, #]] &@ LinearRecurrence[{1, 2}, {2, 1}, 8] (* Michael De Vlieger, Jun 16 2021 *)

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A130238 Partial sums of A130237.

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A335741 Number of Pell numbers (A000129) <= n.

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A345380 Number of Jacobsthal-Lucas numbers m <= n.

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