A157725 -id:A157725 - cp's OEIS frontend

A018910 Pisot sequence L(4,5).

4, 5, 7, 10, 15, 23, 36, 57, 91, 146, 235, 379, 612, 989, 1599, 2586, 4183, 6767, 10948, 17713, 28659, 46370, 75027, 121395, 196420, 317813, 514231, 832042, 1346271, 2178311, 3524580, 5702889, 9227467, 14930354, 24157819, 39088171, 63245988, 102334157, 165580143

Offset: 0

R. K. Guy

Colin Barker, Table of n, a(n) for n = 0..1000
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1).
Index entries for Pisot sequences

See A008776 for definitions of Pisot sequences.

LinearRecurrence[{2, 0, -1}, {4, 5, 7}, 40] (* Jean-François Alcover, Dec 12 2016 *)

pisotL(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]));
  a
}
pisotL(50, 4, 5) \\ Colin Barker, Aug 07 2016

a(n) = Fib(n+3)+2 = A020743(n-2) = A157725(n+3); a(n) = 2a(n-1) - a(n-3).

G.f.: -(-4+3*x+3*x^2)/(x-1)/(x^2+x-1) = -2/(x-1)+(-x-2)/(x^2+x-1) . - R. J. Mathar, Nov 23 2007

A214781 a(n) = smallest k>=2 such that n divides Fibonacci(k-1)+2.

Original entry on oeis.org

2, 4, 2, 4, 5, 10, 6, 0, 10, 7, 0, 10, 12, 22, 8, 0, 16, 10, 11, 28, 0, 0, 9, 0, 48, 40, 34, 22, 0, 34, 0, 0, 0, 16, 28, 10, 36, 0, 18, 0, 18, 0, 17, 0, 34, 22, 14, 0, 54, 148, 16, 40, 52, 34, 0, 0, 11, 0, 0, 34, 28, 0, 0, 0, 68, 0, 21, 16

Offset: 1

Art DuPre, Aug 03 2012

nonn

0 is inserted if no such k exists.

			n=1 divides F(0)+2=2. n=2 divides F(0)+2=2. n=3 divides F(1)+2=3. n=4 divides F(3)+2=4.

Cf. A157725, A001177, A214782.

a(n) = {k = 2; while (((fibonacci(k-1)+2) % n), k++; if (k > 6*n+2 , return(0));); return (k);} \\ Michel Marcus, May 30 2013

A298187 T(n,k)=Number of nXk 0..1 arrays with every element equal to 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 2, 0, 0, 3, 2, 2, 3, 0, 0, 5, 3, 3, 3, 5, 0, 0, 8, 5, 4, 4, 5, 8, 0, 0, 13, 8, 6, 5, 6, 8, 13, 0, 0, 21, 13, 9, 7, 7, 9, 13, 21, 0, 0, 34, 21, 14, 10, 10, 10, 14, 21, 34, 0, 0, 55, 34, 22, 15, 14, 14, 15, 22, 34, 55, 0, 0, 89, 55, 35, 23, 20, 19, 20, 23, 35, 55

Offset: 1

R. H. Hardin, Jan 14 2018

Table starts
.0..0..0..0..0..0..0..0..0...0...0...0...0....0....0....0....0.....0.....0
.0..1..1..2..3..5..8.13.21..34..55..89.144..233..377..610..987..1597..2584
.0..1..1..2..3..5..8.13.21..34..55..89.144..233..377..610..987..1597..2584
.0..2..2..3..4..6..9.14.22..35..56..90.145..234..378..611..988..1598..2585
.0..3..3..4..5..7.10.15.23..36..57..91.146..235..379..612..989..1599..2586
.0..5..5..6..7.10.14.20.29..44..68.106.166..262..416..663.1059..1695..2718
.0..8..8..9.10.14.19.27.38..58..90.142.225..362..587..959.1572..2587..4270
.0.13.13.14.15.20.27.41.58..91.145.240.398..680.1157.2003.3476..6073.10620
.0.21.21.22.23.29.38.58.81.127.203.341.574.1019.1784.3212.5779.10480.18971

			Some solutions for n=7 k=4
..0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..0..0. .0..0..0..0
..0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..0..0. .0..0..0..0
..0..0..0..0. .0..0..1..1. .0..0..0..0. .1..1..1..1. .0..0..0..0
..0..0..0..0. .0..0..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1
..0..0..0..0. .0..0..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1
..0..0..0..0. .0..0..1..1. .1..1..1..1. .0..0..0..0. .0..0..0..0
..0..0..0..0. .0..0..1..1. .1..1..1..1. .0..0..0..0. .0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..760

Column 2 is A000045(n-1).
Column 3 is A000045(n-1).
Column 4 is A001611(n-1).
Column 5 is A157725(n-1).

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +a(n-2)
k=3: a(n) = a(n-1) +a(n-2)
k=4: a(n) = 2*a(n-1) -a(n-3) for n>4
k=5: a(n) = 2*a(n-1) -a(n-3) for n>4
k=6: a(n) = 3*a(n-1) -2*a(n-2) -a(n-3) +2*a(n-4) -2*a(n-5) +a(n-7) for n>8
k=7: a(n) = 3*a(n-1) -2*a(n-2) -a(n-3) +2*a(n-4) -2*a(n-5) +a(n-6) -a(n-7) +a(n-8) -a(n-10) +a(n-11) for n>12

A020743 Pisot sequence L(7,10).

Original entry on oeis.org

7, 10, 15, 23, 36, 57, 91, 146, 235, 379, 612, 989, 1599, 2586, 4183, 6767, 10948, 17713, 28659, 46370, 75027, 121395, 196420, 317813, 514231, 832042, 1346271, 2178311, 3524580, 5702889, 9227467, 14930354, 24157819, 39088171, 63245988, 102334157, 165580143

Offset: 0

David W. Wilson

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for Pisot sequences
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Subsequence of A018910. See A008776 for definitions of Pisot sequences.

Cf. A020717.

LinearRecurrence[{2,0,-1},{7,10,15},40] (* Harvey P. Dale, Jun 10 2022 *)

pisotL(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]));
  a
}
pisotL(50, 7, 10) \\ Colin Barker, Aug 07 2016

a(n) = Fib(n+5)+2 = A157725(n+5). a(n) = 2a(n-1) - a(n-3).

A158569 a(n) = Sum_{i=1..F(n)} F(i), where F = A000045, Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 2, 4, 12, 54, 609, 28656, 14930351, 365435296161, 4660046610375530308, 1454489111232772683678306641952, 5789092068864820527338372482892113982249794889764, 7191684930184179482016276395611672639105248126232175323349533708710427892956420

Offset: 0

Ctibor O. Zizka, Mar 21 2009

a(14) has 79 digits. - Emeric Deutsch, Apr 05 2009

Alois P. Heinz, Table of n, a(n) for n = 0..19

Cf. A000045, A000071, A157725.

with(combinat): a := proc (n) options operator, arrow: add(fibonacci(i), i = 1 .. fibonacci(n)) end proc: seq(a(n), n = 0 .. 14); # Emeric Deutsch, Apr 05 2009
# second Maple program:
F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= n-> F(F(n)+2)-1:
seq(a(n), n=0..14);  # Alois P. Heinz, Apr 14 2025

Total/@Table[Sum[Fibonacci[Range[i]],{i,{Fibonacci[n]}}],{n,14}] (* Harvey P. Dale, Aug 26 2013 *)

a(n) = A000071(A157725(n)). - Alois P. Heinz, Apr 14 2025

More terms from Emeric Deutsch, Apr 05 2009

A226649 Fibonacci shuffles: a(2n) = A000071(n) and a(2n+1) = A001611(n+2).

Original entry on oeis.org

0, 2, 0, 3, 1, 4, 2, 6, 4, 9, 7, 14, 12, 22, 20, 35, 33, 56, 54, 90, 88, 145, 143, 234, 232, 378, 376, 611, 609, 988, 986, 1598, 1596, 2585, 2583, 4182, 4180, 6766, 6764, 10947, 10945, 17712, 17710, 28658, 28656, 46369, 46367, 75026, 75024, 121394, 121392, 196419, 196417, 317812, 317810

Offset: 0

V. T. Jayabalaji, Jun 14 2013

a(2*n+1) = a(2*n) + A157725(n); a(2*n) = a(2*n-1) - 2 for n > 0. - Reinhard Zumkeller, Jul 30 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (-1,1,1,1,1).

Cf. A000071, A001611, A096748.

import Data.List (transpose)
a226649 n = a226649_list !! n
a226649_list = concat $ transpose [a000071_list, drop 2 a001611_list]
-- Reinhard Zumkeller, Jul 30 2013

LinearRecurrence[{-1,1,1,1,1},{0,2,0,3,1},60] (* Harvey P. Dale, Sep 12 2018 *)

G.f. -x*(2+x^2+2*x^3+2*x) / ( (1+x)*(x^4+x^2-1) ). - R. J. Mathar, Jul 15 2013
a(n) + a(n+1) = A096748(n+2). - R. J. Mathar, Jul 15 2013
a(2n-1) - 1 = a(2n) + 1 = fib(n+1) = A000045(n+1) for n > 0. - T. D. Noe, Jul 23 2013

A364005 Numbers whose Wythoff representation (A189921, A317208) is palindromic.

Original entry on oeis.org

0, 1, 2, 5, 7, 10, 13, 15, 23, 28, 34, 36, 52, 57, 65, 75, 81, 89, 91, 117, 128, 146, 159, 175, 185, 198, 204, 217, 233, 235, 277, 295, 327, 369, 379, 400, 426, 442, 463, 473, 494, 520, 526, 547, 573, 589, 610, 612, 680, 709, 761, 829, 848, 916, 945, 989, 1023

Offset: 1

Amiram Eldar, Jul 01 2023

Includes all the odd-indexed Fibonacci numbers (A001519), since the Wythoff representation of Fibonacci(1) is 1 and the Wythoff representation of Fibonacci(2*n+1), for n >= 1, is n 0's.
A157725(n) = Fibonacci(n) + 2 is a term for n >= 4, since its Wythoff representation is n-4 1's between 2 0's.
A232970 is a subsequence since the Wythoff representation of A232970(n) = (Fibonacci(3*n+1) + 1)/2 is n 0's and n-1 1's interleaved.

			The first 10 terms are:
   n  a(n)  A317208(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     2              2
   4     5             22
   5     7            212
   6    10           2112
   7    13            222
   8    15          21112
   9    23         211112
  10    28          21212

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001519, A157725, A189921, A232970, A317208.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341, A352507.

z[n_] := Floor[(n + 1)*GoldenRatio] - n - 1; h[n_] := z[n] - z[n - 1]; w[n_] := Module[{m = n, zm = 0, hm, s = {}}, While[zm != 1, hm = h[m]; AppendTo[s, hm]; If[hm == 1, zm = z[m], zm = z[z[m]]]; m = zm]; s]; w[0] = {0}; Select[Range[0, 1000], PalindromeQ[w[#]] &]

A217762 Square array T, read by antidiagonals: T(n,k) = F(n) + 2*F(k) where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

0, 2, 1, 2, 3, 1, 4, 3, 3, 2, 6, 5, 3, 4, 3, 10, 7, 5, 4, 5, 5, 16, 11, 7, 6, 5, 7, 8, 26, 17, 11, 8, 7, 7, 10, 13, 42, 27, 17, 12, 9, 9, 10, 15, 21, 68, 43, 27, 18, 13, 11, 12, 15, 23, 34, 110, 69, 43, 28, 19, 15, 14, 17, 23, 36, 55, 178, 111, 69, 44, 29, 21

Offset: 0

Philippe Deléham, Apr 07 2013

			Square array begins:
...0....2....2....4....6...10...16...26...42...
...1....3....3....5....7...11...17...27...43...
...1....3....3....5....7...11...17...27...43...
...2....4....4....6....8...12...18...28...44...
...3....5....5....7....9...13...19...29...45...
...5....7....7....9...11...15...21...31...47...
...8...10...10...12...14...18...24...34...50...
..13...15...15...17...19...23...29...39...55...
..21...23...23...25...27...31...37...47...63...
..34...36...36...38...40...44...50...60...76...
..55...57...57...59...61...65...71...81...97...
..89...91...91...93...95...99..105..115..131...
.144..146..146..148..150..154..160..170..186...
...

Cf. A000045

T(n,0) = A000045(n).
T(1,k) = A001588(k).
T(n,1) = T(n,2) = A157725(n).
T(n,3) = A157727(n).
T(n,n)= A022086(n) = 3*A000045(n).
T(n+1,n) = A000032(n+1) = A000204(n+1).
T(n+2,n) = A000285(n).
T(n+3,n) = A013655(n+1) = A001060(n+1).
T(n+4,n) = A021120(n).
T(n+5,n) = A022088(n+2) = 5*A000045(n+2).
T(n+6,n) = A022097(n+2).
T(n+7,n) = A022122(n+2).
T(n+8,n) = 3*A013655(n+2).
T(n+9,n) = A097657(n+2).
T(n+10,n) = A022118(n+4).
T(n,n+1) = A000045(n+3).
T(n,n+2) = A013655(n+1) = A001060(n+1).
T(n,n+3) = A000032(n+3).
T(n,n+4) = A022095(n+2).
T(n,n+5) = A022120(n+2).
T(n,n+6) = A022136(n+2).
T(n,n+7) = A022098(n+4).
T(n,n+8) = A022380(n+4).
T(n,n+9) = A206419(n+6).
Sum(T(n-k,k), 0<=k<=n) = 3*A000071(n+2).

cp's OEIS Frontend

A018910 Pisot sequence L(4,5).

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A214781 a(n) = smallest k>=2 such that n divides Fibonacci(k-1)+2.

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A298187 T(n,k)=Number of nXk 0..1 arrays with every element equal to 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero.

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A020743 Pisot sequence L(7,10).

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A158569 a(n) = Sum_{i=1..F(n)} F(i), where F = A000045, Fibonacci numbers.

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Maple

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A226649 Fibonacci shuffles: a(2n) = A000071(n) and a(2n+1) = A001611(n+2).

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Haskell

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A364005 Numbers whose Wythoff representation (A189921, A317208) is palindromic.

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Mathematica

A217762 Square array T, read by antidiagonals: T(n,k) = F(n) + 2*F(k) where F(n) is the n-th Fibonacci number.

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