Hans J. H. Tuenter - cp's OEIS frontend

User: Hans J. H. Tuenter

Hans J. H. Tuenter has authored 12 sequences. Here are the ten most recent ones:

A386760 Numbers k such that the number of decimal digits of the Lucas number L(k) is greater than the number of decimal digits of the Fibonacci number F(k).

5, 6, 10, 11, 15, 16, 20, 24, 25, 29, 30, 34, 35, 39, 44, 48, 49, 53, 54, 58, 59, 63, 67, 68, 72, 73, 77, 78, 82, 83, 87, 91, 92, 96, 97, 101, 102, 106, 111, 115, 116, 120, 121, 125, 126, 130, 134, 135, 139, 140, 144, 145, 149, 150, 154, 158, 159, 163, 164, 168

Offset: 1

Hans J. H. Tuenter, Aug 13 2025

The difference in the number of decimal digits, A055642(L(k))-A055642(F(k)) = A060384(k)-A386758(k) is either zero or one. In fact, this difference is ceiling(beta-{k*alpha}), with alpha and beta as defined in the Formula section. This implies that, asymptotically, a fraction of beta=0.349485... of the Lucas numbers has one more decimal digit than the corresponding Fibonacci number. This gives the asymptotic behavior of the sequence as a(n)~n/beta. Conjecture: abs(a(n)-n/beta)

			5 is a term since F(5)=5 has length 1 decimal digit, but L(5)=11 has length 2 decimal digits which is greater.

Hans J. H. Tuenter, Table of n, a(n) for n = 1..1000

Cf. A000032, A000045, A001622, A055642, A060384, A097348, A386758.

Select[Range[168],IntegerLength[LucasL[#]]>IntegerLength[Fibonacci[#]]&] (* James C. McMahon, Aug 28 2025 *)

The sequence consists of the integers k>=2, for which {k*alpha}A097348, and phi = (1+sqrt(5))/2 = A001622.

A386758 Number of decimal digits in the n-th Lucas number.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17

Offset: 0

Hans J. H. Tuenter, Aug 06 2025

As F(n)<=L(n), the number of decimal digits of the Lucas number L(n) is at least as large as the number of decimal digits of the Fibonacci number F(n). Furthermore, the difference is at most one. The indices for which the difference is one is A386760.

			L(0)=2 has one digit, so that a(0)=1; L(5)=11 has two digits, so that a(5)=2.

Hans J. H. Tuenter, Table of n, a(n) for n = 0..10000

Cf. A000032, A001622, A055642, A060384, A097348.

Number of digits of L(p^n): A094057 (p=2), A114469 (p=10).

a:= n-> 1+floor(n*log[10]((1+sqrt(5))/2)):
seq(a(n), n=0..81);

a[n_] := IntegerLength[LucasL[n]]; Array[a, 100, 0] (* Amiram Eldar, Aug 16 2025 *)

a(n) = A055642(A000032(n)).

a(n) = 1 + floor(n*log_10(phi)), where log_10(phi) = A097348, and phi = (1+sqrt(5))/2 = A001622.

A363753 a(n) = Sum_{k=0..n} (-1)^kF(k-1)F(k)*F(k+1)/2, where F(n) is the Fibonacci number A000045(n).

Original entry on oeis.org

0, 0, 1, -2, 13, -47, 213, -879, 3762, -15873, 67342, -285098, 1207966, -5116586, 21674919, -91815276, 388937619, -1647563169, 6979194475, -29564334305, 125236542640, -530510487155, 2247278519916, -9519624520452, 40325776676748, -170822731106052, 723616701297373

Offset: 0

Hans J. H. Tuenter, Jun 19 2023

Alternating sum of the product of three consecutive Fibonacci numbers, divided by two.
Can also be seen as the alternating sum of the Fibonomial coefficients (n+1,3), A001655.
This sequence is part of a suite of sums over triple products of Fibonacci numbers. Subba Rao (1953) gives closed-form expressions for several Fibonacci sums of this type.

K. Subba Rao, Some properties of Fibonacci numbers, The American Mathematical Monthly, 60(10):680-684, December 1953.
Index entries for linear recurrences with constant coefficients, signature (-2,9,-3,-4,1).

Cf. A000045, A001655.
Other sequences with the product of three Fibonacci numbers as a summand (the sequence may have a shifted [and scaled] version of the summand given here).
A005968: F(k)^3,  A119284: (-1)^k*F(k)^3,  A215037: F(k-1)*F(k)*F(k+1),
A363753: (-1)^k*F(k-1)*F(k)*F(k+1),  A163198: F(2k)^3,  A163200: F(2k+1)^3,
A256178: F(2k)*F(2k+1)*F(2k+2),  this sequence: (-1)^k*F(k-1)*F(k)*F(k+1),
A363754: F(2k-1)*F(2k)*F(2k+1).

LinearRecurrence[{-2, 9, -3, -4, 1}, {0, 0, 1, -2, 13}, 27]

a(n) = ((-1)^n*(F(n+1)^3 - F(n)^3) + F(n+2) - 2)/8.
a(n) = ((-1)^n*F(3*n+1) + 4*F(n+2) - 5)/20.
a(n) = -2*a(n-1) + 9*a(n-2) - 3*a(n-3) - 4*a(n-4) + a(n-5).
a(-n) = A215037(n-3).
G.f.: x^2/((1 - x)*(1 + 4*x - x^2)*(1 - x - x^2)).
20*a(n) = (-1)^n*A033887(n) + 4*A000045(n+2) - 5. - R. J. Mathar, Jun 27 2023

A363754 a(n) = Sum_{k=0..n} F(2k-1)F(2k)F(2k+1)/2, where F(n) is the Fibonacci number A000045(n).

Original entry on oeis.org

0, 1, 16, 276, 4917, 88132, 1581196, 28372701, 509125596, 9135883240, 163936760185, 2941725767256, 52787126964456, 947226559367881, 16997290941068152, 305004010378316172, 5473074895864584141, 98210344115173624636, 1762313119177232976916, 31623425801074947486405

Offset: 0

Hans J. H. Tuenter, Jun 19 2023

This is one of the triple Fibonacci sums that were considered by Subba Rao (1953).

Taking any of the given closed-form expressions for a(n) with Fibonacci numbers, one can extend a(n) to negative indices by using the property F(-n)=(-1)^(n+1). This gives a(-n)=a(n-1).

K. Subba Rao, Some properties of Fibonacci numbers, The American Mathematical Monthly, 60(10):680-684, December 1953.
Index entries for linear recurrences with constant coefficients, signature (22,-77,77,-22,1).

Cf. A000045, A256178, A363753.

LinearRecurrence[{22, -77, 77, -22, 1}, {0, 1, 16, 276, 4917}]

a(n) = (F(2n+1)^3 + F(2n+1) - 2)/8.
a(n) = (F(6*n+3)+8*F(2*n+1)-10)/40.
a(n) = 22*a(n-1) - 77*a(n-2) + 77*a(n-3) - 22*a(n-4) + a(n-5).
G.f.: x*(1 - 6*x + x^2)/((1 - x)*(1 - 3*x + x^2)*(1 - 18*x + x^2)).

A363499 a(n) = Sum_{k=0..n} floor(sqrt(k))^5.

Original entry on oeis.org

0, 1, 2, 3, 35, 67, 99, 131, 163, 406, 649, 892, 1135, 1378, 1621, 1864, 2888, 3912, 4936, 5960, 6984, 8008, 9032, 10056, 11080, 14205, 17330, 20455, 23580, 26705, 29830, 32955, 36080, 39205, 42330, 45455, 53231, 61007, 68783, 76559, 84335, 92111, 99887

Offset: 0

Hans J. H. Tuenter, Jun 05 2023

Partial sums of the fifth powers of the terms of A000196.

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Sums of powers of A000196: A022554 (1st), A174060 (2nd), A363497 (3rd), A363498 (4th), this sequence (5th).

Table[(n + 1) #^5 - (1/84) # (# + 1)*(2 # + 1)*(3 # - 1)*(10 #^3 - 7 # + 4) &[Floor@ Sqrt[n]], {n, 0, 42}] (* Michael De Vlieger, Jun 10 2023 *)

from math import isqrt
def A363499(n): return (m:=isqrt(n))**5 *(n+1) - (m*(m+1)*(2*m+1)*(3*m-1)*(10*m**3-7*m+4))//84 # Karl-Heinz Hofmann, Jul 17 2023

a(n) = (n+1)*m^5 - (1/84)*m*(m+1)*(2*m+1)*(3*m-1)*(10*m^3-7*m+4), where m = floor(sqrt(n)).

A363497 a(n) = Sum_{k=0..n} floor(sqrt(k))^3.

Original entry on oeis.org

0, 1, 2, 3, 11, 19, 27, 35, 43, 70, 97, 124, 151, 178, 205, 232, 296, 360, 424, 488, 552, 616, 680, 744, 808, 933, 1058, 1183, 1308, 1433, 1558, 1683, 1808, 1933, 2058, 2183, 2399, 2615, 2831, 3047, 3263, 3479, 3695, 3911, 4127, 4343, 4559, 4775, 4991, 5334

Offset: 0

Hans J. H. Tuenter, Jun 05 2023

Partial sums of the third powers of the terms of A000196.

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Sums of powers of A000196: A022554 (1st), A174060 (2nd), this sequence (3rd), A363498 (4th), A363499 (5th).

Table[(n + 1) #^3 - (1/60) # (# + 1) (3 # - 1) (12 #^2 + 7 # - 4) &[Floor@ Sqrt[n]], {n, 0, 50}] (* Michael De Vlieger, Jun 10 2023 *)

a(n) = sum(k=0, n, sqrtint(k)^3); \\ Michel Marcus, Jun 06 2023

from math import isqrt
A363497 = [0]
for n in range(1,50): A363497.append(A363497[-1] + isqrt(n)**3)
print(A363497) # Karl-Heinz Hofmann, Jun 14 2023

from math import isqrt
def A363497(n):return (m:=isqrt(n))**3*(n+1)-(m*(m+1)*(3*m-1)*(12*m**2+7*m-4))//60
# Karl-Heinz Hofmann, Jun 14 2023

a(n) = (n+1)*m^3 - (1/60)*m*(m+1)*(3*m-1)*(12*m^2+7*m-4), where m = floor(sqrt(n)).

A363498 a(n) = Sum_{k=0..n} floor(sqrt(k))^4.

Original entry on oeis.org

0, 1, 2, 3, 19, 35, 51, 67, 83, 164, 245, 326, 407, 488, 569, 650, 906, 1162, 1418, 1674, 1930, 2186, 2442, 2698, 2954, 3579, 4204, 4829, 5454, 6079, 6704, 7329, 7954, 8579, 9204, 9829, 11125, 12421, 13717, 15013, 16309, 17605, 18901, 20197, 21493, 22789

Offset: 0

Hans J. H. Tuenter, Jun 05 2023

Partial sums of the fourth powers of the terms of A000196.

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Sums of powers of A000196: A022554 (1st), A174060 (2nd), A363497 (3rd), this sequence (4th), A363499 (5th).

Table[(n + 1) #^4 - (1/30) # (# + 1)*(20 #^4 + 4 #^3 - 14 #^2 + 4 # + 1) &[Floor@ Sqrt[n]], {n, 0, 45}] (* Michael De Vlieger, Jun 10 2023 *)

from math import isqrt
def A363498(n):
    return (m:=isqrt(n))**4 *(n+1) - (m*(m+1)*(20*m**4+4*m**3-14*m**2+4*m+1))//30
print([A363498(n) for n in range(0,46)]) # Karl-Heinz Hofmann, Jul 15 2023

a(n) = (n+1)*m^4 - (1/30)*m*(m+1)*(20*m^4+4*m^3-14*m^2+4*m+1), where m = floor(sqrt(n)).

A134470 Continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 8, 1, 5, 1, 1, 1, 12, 5, 1, 1, 5, 1, 12, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 2, 2, 2, 1, 11, 1, 6, 1, 3, 2, 1, 1, 1, 1, 1, 2, 6, 7, 1, 4, 2, 1, 1, 1, 13, 1, 1, 1, 2, 4, 2, 11, 1, 2, 5, 1, 8, 1, 78, 10, 1, 64, 1, 29, 1, 3, 1, 1, 1, 2, 1, 12, 1, 2, 1, 4, 1, 2, 1, 2, 32, 1, 92, 1, 14, 1, 10, 12, 2, 3, 16, 2, 1, 1, 1, 1, 8, 3, 15, 1, 2, 2, 1, 4, 4, 2, 8, 1, 1557, 3, 1, 69, 1, 5, 3, 11, 1, 1

Offset: 0

Hans J. H. Tuenter, Oct 27 2007

G. C. Greubel, Table of n, a(n) for n = 0..10000
Hans J. H. Tuenter, Overshoot in the Case of Normal Variables: Chernoff's Integral, Latta's Observation and Wijsman's Sum, Sequential Analysis, 26(4) (2007) 481-488.

Cf. A134469 (Decimal expansion), A134471 (Numerators of continued fraction convergents), A134472 (Denominators of continued fraction convergents).

Digits:=100; cfrac(-Zeta(1/2)/sqrt(2*Pi),30,'quotients');

ContinuedFraction[ -Zeta[1/2]/Sqrt[2 \[Pi]], 100] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)

default(realprecision,1000);
c=-zeta(1/2)/sqrt(2*Pi); /* == 0.582597157... (A134469) */
contfrac(c) /* gives 967 terms */

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

A134471 Numerators of the convergents of the continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

Original entry on oeis.org

0, 1, 1, 3, 4, 7, 60, 67, 395, 462, 857, 1319, 16685, 84744, 101429, 186173, 1032294, 1218467, 15653898, 16872365, 32526263, 49398628, 81924891, 213248410, 295173301, 508421711, 803595012, 1312016723, 3427628458, 11594902097, 26617432652, 64829767401

Offset: 1

Hans J. H. Tuenter, Oct 27 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Hans J. H. Tuenter, Overshoot in the Case of Normal Variables: Chernoff's Integral, Latta's Observation and Wijsman's Sum, Sequential Analysis, 26(4) (2007) 481-488.

Cf. A134469 (Decimal expansion), A134470 (Continued fraction expansion), A134472 (Denominators of continued fraction convergents).

Numerator[Convergents[-Zeta[1/2]/Sqrt[2Pi],30]] (* Harvey P. Dale, Sep 07 2015 *)

More terms from Harvey P. Dale, Sep 07 2015

A134472 Denominators of the convergents of the continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

Original entry on oeis.org

1, 1, 2, 5, 7, 12, 103, 115, 678, 793, 1471, 2264, 28639, 145459, 174098, 319557, 1771883, 2091440, 26869163, 28960603, 55829766, 84790369, 140620135, 366030639, 506650774, 872681413, 1379332187, 2252013600, 5883359387, 19902091761, 45687542909, 111277177579, 268241898067

Offset: 13

Hans J. H. Tuenter, Oct 27 2007

G. C. Greubel, Table of n, a(n) for n = 13..1012
Hans J. H. Tuenter, Overshoot in the Case of Normal Variables: Chernoff's Integral, Latta's Observation and Wijsman's Sum, Sequential Analysis, 26(4) (2007) 481-488.

Cf. A134469 (Decimal expansion), A134470 (Continued fraction expansion), A134471 (Numerators of continued fraction convergents).

Denominator[Convergents[-Zeta[1/2]/Sqrt[2 Pi], 50]] (* G. C. Greubel, Mar 28 2018 *)

Terms a(33) onward added by G. C. Greubel, Mar 28 2018

cp's OEIS Frontend

User: Hans J. H. Tuenter

A386760 Numbers k such that the number of decimal digits of the Lucas number L(k) is greater than the number of decimal digits of the Fibonacci number F(k).

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A386758 Number of decimal digits in the n-th Lucas number.

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A363753 a(n) = Sum_{k=0..n} (-1)^k*F(k-1)*F(k)*F(k+1)/2, where F(n) is the Fibonacci number A000045(n).

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A363754 a(n) = Sum_{k=0..n} F(2k-1)*F(2k)*F(2k+1)/2, where F(n) is the Fibonacci number A000045(n).

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A363499 a(n) = Sum_{k=0..n} floor(sqrt(k))^5.

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A363497 a(n) = Sum_{k=0..n} floor(sqrt(k))^3.

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A363498 a(n) = Sum_{k=0..n} floor(sqrt(k))^4.

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A363753 a(n) = Sum_{k=0..n} (-1)^kF(k-1)F(k)*F(k+1)/2, where F(n) is the Fibonacci number A000045(n).

A363754 a(n) = Sum_{k=0..n} F(2k-1)F(2k)F(2k+1)/2, where F(n) is the Fibonacci number A000045(n).