A090162 -id:A090162 - cp's OEIS frontend

A100022 Number of digits in A090162(n).

1, 4, 29, 205, 1412, 9688, 66416, 455237, 3120256

Offset: 1

Eric W. Weisstein, Nov 23 2003 and Wolfdieter Lang, Dec 01 2003

f[n_] := Floor[ Log[10, Binomial[ Fibonacci[2n]Fibonacci[2n + 1], Fibonacci[2n - 1]Fibonacci[2n] - 1]] + 1]; Do[ Print[ f[n]], {n, 9}] (* Robert G. Wilson v, Nov 23 2004 *)

a(n+1)/a(n) -> Phi^4 = 6.8541... - Robert G. Wilson v, Nov 23 2004

a(7)-a(9) from Robert G. Wilson v, Nov 23 2004

A114182 F(4n) - 2n - 1 where F(n) = Fibonacci numbers. Also, the floor of the log base phi of sequence A090162 (phi = (1+Sqrt(5))/2).

Original entry on oeis.org

0, 16, 137, 978, 6754, 46355, 317796, 2178292, 14930333, 102334134, 701408710, 4807526951, 32951280072, 225851433688, 1548008755889, 10610209857690, 72723460248106, 498454011879227, 3416454622906668, 23416728348467644

Offset: 1

Greg Huber, Feb 04 2006

Index entries for linear recurrences with constant coefficients, signature (9,-16,9,-1).

Cf. A033888.

Table[Fibonacci[4n]-2n-1,{n,20}] (* or *) LinearRecurrence[{9,-16,9,-1},{0,16,137,978},20] (* Harvey P. Dale, May 28 2015 *)

G.f. x^2*(16-7*x+x^2) / ( (x^2-7*x+1)*(x-1)^2 ). - R. J. Mathar, Oct 19 2012

a(0)=0, a(1)=16, a(2)=137, a(3)=978, a(n)=9*a(n-1)-16*a(n-2)+ 9*a(n-3)- a(n-4). - Harvey P. Dale, May 28 2015

Corrected by R. J. Mathar, Oct 19 2012

A114184 Binomial coefficient binomial(F(n+1)F(n),F(n)F(n-1)-1), where F(n) = Fibonacci number. A090162 is the bisection (shifted by 1) of this sequence.

Original entry on oeis.org

0, 1, 6, 3003, 23206929840, 61218182743304701891431482520

Offset: 1

Greg Huber, Feb 04 2006

nonn

			a(3) = binomial(3*2,2*1-1) = binomial(6,1) = 6.

Cf. A001906, A090162.

A003015 Numbers that occur 5 or more times in Pascal's triangle.

Original entry on oeis.org

1, 120, 210, 1540, 3003, 7140, 11628, 24310, 61218182743304701891431482520

Offset: 1

N. J. A. Sloane

nonn

The subject of a recent thread on sci.math. Apparently it has been known for many years that there are infinitely many numbers that occur at least 6 times in Pascal's triangle, namely the solutions to binomial(n,m-1) = binomial(n-1,m) given by n = F_{2k}*F_{2k+1}; m = F_{2k-1}*F_{2k} where F_i is the i-th Fibonacci number. The first of these outside the range of the existing database entry is {104 choose 39} = {103 choose 40} = 61218182743304701891431482520. - Christopher E. Thompson, Mar 09 2001
It may be that there are no terms that appear exactly 5 times in Pascal's triangle, in which case the title could be changed to "Numbers that occur 6 or more times in Pascal's triangle". - N. J. A. Sloane, Nov 24 2004
No other terms below 33*10^16 (David W. Wilson).
61218182743304701891431482520 really is the next term. Weger shows this and I checked it. - T. D. Noe, Nov 15 2004
Blokhuis et al. show that there are no other solutions less than 10^60, nor within the first 10^6 rows of Pascal's triangle other than those given by the parametric solution mentioned above. - Christopher E. Thompson, Jan 19 2018
See the b-file of A090162 for the explicit numbers produced by the parametric formula. - Jeppe Stig Nielsen, Aug 23 2020

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 93, #47.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Aart Blokhuis, Andries Brouwer, and Benne de Weger, Binomial collisions and near collisions, INTEGERS, Volume 17, Article A64, 2017 (also available as arXiv:1707.06893 [math.NT]).
Jean-Marie de Koninck, Nicolas Doyon, and William Verreault, Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture, Integers (2021) Vol. 21, #A34.
B. M. M. de Weger, Equal binomial coefficients: some elementary considerations, Econometric Institute Research Papers, No. EI 9536-/B, 1995.
B. M. M. de Weger, Equal binomial coefficients: some elementary considerations, Journal of Number Theory, Volume 63, Issue 2, April 1997, Pages 373-386.
Zoe Griffiths, My MegaFavNumber: 61,218,182,743,304,701,891,431,482,520, YouTube video (2020).
R. K. Guy and V. Klee, Monthly research problems, 1969-1971, Amer. Math. Monthly, 78 (1971), 1113-1122.
D. A. Lind, The quadratic field Q(sqrt(5)) and a certain diophantine equation, Fibonacci Quart. 6 (3) (1968), 86-93.
Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Terence Tao, and Joni Teräväinen, Singmaster's conjecture in the interior of Pascal's triangle, arXiv:2106.03335 [math.NT], 2021.
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 69.
David Singmaster, Repeated binomial coefficients and Fibonacci numbers, Fibonacci Quarterly 13 (1975) 295-298.
Eric Weisstein's World of Mathematics, Pascal's Triangle
Tomohiro Yamada, Necessary conditions for binomial collisions, arXiv:2002.07043 [math.NT], 2020.

Cf. A003016, A059233.
Cf. A182237, A098565 (subsequence).
Cf. A090162 (easy subsequence).

A089508 Solution to a binomial problem together with companion sequence A081016(n-1).

Original entry on oeis.org

1, 14, 103, 713, 4894, 33551, 229969, 1576238, 10803703, 74049689, 507544126, 3478759199, 23843770273, 163427632718, 1120149658759, 7677619978601, 52623190191454, 360684711361583, 2472169789339633, 16944503814015854

Offset: 1

Wolfdieter Lang, Dec 01 2003

a(n) and b(n) := A081016(n-1) are the solutions to the Diophantine equation binomial(a,b) = binomial(a+1,b-1). The first few binomials are given by A090162(n).

			n = 2: a(2) = 14, b(2) = A081016(1) = 6 satisfy binomial(14,6) = 3003 = binomial(15,5). 3003 = A090162(2).

A. I. Shirshov: On the equation binomial(n,m)=binomial(n+1,m-1), pp. 83-86, in: Kvant Selecta: Algebra and Analysis, I, ed. S. Tabachnikov, Am.Math.Soc., 1999.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Wikipedia, Singmaster's conjecture
Index entries for linear recurrences with constant coefficients, signature (8, -8, 1).

Equals A081018 - 1.

[Fibonacci(2*n)*Fibonacci(2*n+1) - 1: n in [1..30]]; // G. C. Greubel, Dec 18 2017

Rest[CoefficientList[Series[x*(1 + 6*x - x^2)/((1 - x)*(1 - 7*x + x^2)), {x, 0, 50}], x]] (* G. C. Greubel, Dec 18 2017 *)

x='x+O('x^30); Vec(x*(1 + 6*x - x^2)/((1 - x)*(1 - 7*x + x^2))) \\ G. C. Greubel, Dec 18 2017

G.f.: x*(1+6*x-x^2)/((1-x)*(1-7*x+x^2)).

a(n) = A081018(n) - 1 = F(2*n)*F(2*n+1) - 1, n>=1; with F(n) := A000045(n) (Fibonacci).

A114185 a(n) = Fibonacci(2*n) - n - 1.

Original entry on oeis.org

0, 4, 16, 49, 137, 369, 978, 2574, 6754, 17699, 46355, 121379, 317796, 832024, 2178292, 5702869, 14930333, 39088149, 102334134, 267914274, 701408710, 1836311879, 4807526951, 12586268999, 32951280072, 86267571244, 225851433688, 591286729849, 1548008755889

Offset: 2

Greg Huber, Feb 04 2006

Related to the log base phi of sequence A090162.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Cf. A001906, A090162.

[Fibonacci(2*n)-n-1: n in [2..40]]; // Vincenzo Librandi, Sep 03 2017

g:=(-1+3*z)/(1-3*z+z^2): gser:=series(g, z=0, 43): seq(abs(coeff(gser, z, n)-n), n=3..20); # Zerinvary Lajos, Mar 22 2009

Rest[Table[Fibonacci[2 n] - n - 1, {n, 30}]] (* Vincenzo Librandi, Sep 03 2017 *)

More terms from Vincenzo Librandi, Sep 03 2017

cp's OEIS Frontend

A100022 Number of digits in A090162(n).

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A114182 F(4n) - 2n - 1 where F(n) = Fibonacci numbers. Also, the floor of the log base phi of sequence A090162 (phi = (1+Sqrt(5))/2).

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A114184 Binomial coefficient binomial(F(n+1)*F(n),F(n)*F(n-1)-1), where F(n) = Fibonacci number. A090162 is the bisection (shifted by 1) of this sequence.

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A003015 Numbers that occur 5 or more times in Pascal's triangle.

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A089508 Solution to a binomial problem together with companion sequence A081016(n-1).

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A114185 a(n) = Fibonacci(2*n) - n - 1.

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A114184 Binomial coefficient binomial(F(n+1)F(n),F(n)F(n-1)-1), where F(n) = Fibonacci number. A090162 is the bisection (shifted by 1) of this sequence.