A001611 -id:A001611 - cp's OEIS frontend

A160536 a(n) = Fibonacci(n) + n^2.

0, 2, 5, 11, 19, 30, 44, 62, 85, 115, 155, 210, 288, 402, 573, 835, 1243, 1886, 2908, 4542, 7165, 11387, 18195, 29186, 46944, 75650, 122069, 197147, 318595, 515070, 832940, 1347230, 2179333, 3525667, 5704043, 9228690, 14931648, 24159186, 39089613, 63247507

Offset: 0

Leonardo Sznajder, May 18 2009

			a(6) = Fibonacci(6) + 6^2 = 8 + 36 = 44.

Bruno Berselli, Table of n, a(n) for n = 0..1000 (terms 0..285 from Vincenzo Librandi).
Math Marathon (MM95) (In Russian) [Vladimir Joseph Stephan Orlovsky, May 12 2010]
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Equals A000045 + A000290.

Cf. A001611, A002062, A212272.

[ Fibonacci(n)+n^2: n in [0..40] ]; // Klaus Brockhaus, May 22 2009

Table[Fibonacci[n]+n^2,{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)

a(n)=fibonacci(n)+n^2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = a(n-4) - a(n-3) - 2*a(n-2) + 3*a(n-1) - 2 for n > 3; a(0)=0, a(1)=2, a(2)=5, a(3)=11. - Klaus Brockhaus, May 22 2009

G.f.: x*(2-3*x+x^2-2*x^3) / ((1-x)^3*(1-x-x^2)). - Klaus Brockhaus, May 22 2009

Edited and extended by Klaus Brockhaus, May 22 2009

A187890 a(1) = 0, a(2) = 4, a(n) = a(n-1) + a(n-2) - 1.

Original entry on oeis.org

0, 4, 3, 6, 8, 13, 20, 32, 51, 82, 132, 213, 344, 556, 899, 1454, 2352, 3805, 6156, 9960, 16115, 26074, 42188, 68261, 110448, 178708, 289155, 467862, 757016, 1224877, 1981892, 3206768, 5188659, 8395426, 13584084, 21979509, 35563592, 57543100

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 15 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000071, A001611, A001612, A374438.

Join[{a = 0, b = 4}, Table[c = a+b-1; a=b; b=c, {n, 100}]]
LinearRecurrence[{2, 0, -1}, {0, 4, 3}, 40] (* Harvey P. Dale, Sep 25 2013 *)
CoefficientList[Series[(-x (-4 + 5 x))/((x - 1) (x^2 + x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 26 2013 *)

G.f.: -x^2*(-4+5*x) / ( (x-1)*(x^2+x-1) ). - R. J. Mathar, Mar 15 2011
a(n) = 1+A001060(n-2), n>2. - R. J. Mathar, Mar 15 2011
a(n) - a(n-1) = A013655(n-4). - R. J. Mathar, Jun 19 2021
If we start the sequence 1, 3, 6, ... and set the offset to 0, then the sequence has the generating function (1 + x - 3*x^3)/(x^3 - 2*x + 1) and gives the row sums of A374438. - Peter Luschny, Jul 22 2024

Definition adapted to offset by Georg Fischer, Jun 19 2021

A232896 a(n) = 3a(n-1) - 2a(n-2) - a(n-3) + a(n-4), where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 8.

Original entry on oeis.org

1, 3, 5, 8, 12, 18, 27, 41, 63, 98, 154, 244, 389, 623, 1001, 1612, 2600, 4198, 6783, 10965, 17731, 28678, 46390, 75048, 121417, 196443, 317837, 514256, 832068, 1346298, 2178339, 3524609, 5702919, 9227498, 14930386, 24157852, 39088205, 63246023, 102334193

Offset: 1

Clark Kimberling, Dec 02 2013

Conjecture: a(n) is the position of 2*n-1, for n >= 1, in the sequence S = A232895 of positive integers generated by these rules: 1 and 2 are in S; if x is in S then x + 2 and 2*x are in S, where duplicates are deleted as they occur.

			a(5) = 3*a(4) - 2*a(3) - a(4) + a(5) = 3*8 - 2*5 - 3 + 1 = 12.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A232895.

I:=[1,3,5,8]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Mar 18 2015

a[1] = 1; a[2] = 3; a[3] = 5; a[4] = 8; a[n_] := a[n] = 3*a[n - 1] - 2*a[n - 2] - a[n - 3] + a[n - 4]; t = Table[a[n], {n, 1, 100}]
CoefficientList[Series[(1 - 2 x^2) / ((1 - x)^2 (1 - x - x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 18 2015 *)
LinearRecurrence[{3, -2, -1, 1}, {1, 3, 5, 8}, 39] (* Robert G. Wilson v, Jul 23 2018 *)

Vec(x*(1-2*x^2)/((1-x)^2*(1-x-x^2)) + O(x^50)) \\ Michel Marcus, Mar 18 2015

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4), where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 8.
a(n) = n-1 + A000045(n+1). - Tom Edgar, Mar 09 2015
G.f.: x*(1-2*x^2)/((1-x)^2*(1-x-x^2)). - Vincenzo Librandi, Mar 18 2015
a(n) = -1 + (2^(-1-n)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))) / sqrt(5) + n. - Colin Barker, Mar 11 2017
a(n) = Sum_{k=1..n} A001611(k-1). - Ehren Metcalfe, Apr 15 2019

A256968 Let b(n) = Product_{i=1..n} p_i/(p_i - 1), p_i = i-th prime; a(n) = minimum k such that b(k) >= n.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 9, 14, 22, 35, 55, 89, 142, 230, 373, 609, 996, 1637, 2698, 4461, 7398, 12301, 20503, 34253, 57348, 96198, 161659, 272124, 458789, 774616, 1309627, 2216968, 3757384, 6375166, 10828012, 18409028, 31326514, 53354259, 90945529, 155142139

Offset: 0

N. J. A. Sloane, Apr 17 2015

nonn

A001611 is similar but different.

Equal to A005579 except for n = 2 and n = 3. The following argument shows that they are equal for n > 3. First note that b(k+1) > b(k). Next, Product_{i=1..k} p_i is 2 times an odd number, i.e., it is not divisible by 4. Similarly since p_i - 1 is even for i > 1, Product_{i=1..k} (p_i - 1) is divisible by 2^(k-1), i.e., it is divisible by 4 for k >= 3. Thus b(k) is not an integer for k >= 3. Since b(3) = 15/4 > 3, this means that a(n) = A005579(n) for n > 3 - Chai Wah Wu, Apr 17 2015

			The sequence b(n) for n >= 0 begins 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, 323323/55296, 676039/110592, 2800733/442368, 86822723/13271040, 3212440751/477757440, 131710070791/19110297600, 5663533044013/802632499200, ... = A060753/A038110. So a(3) = 2.

Popular Computing (Calabasas, CA), Problem 182 (Suggested by Victor Meally), Annotated and scanned copy of page 10 of Vol. 5 (No. 53, Aug 1977).

Cf. A005579, A060753, A038110, A001611.

from sympy import prime
A256968_list, count, bn, bd = [0,0], 2, 1, 1
for k in range(1,10**4):
    p = prime(k)
    bn *= p
    bd *= p-1
    while bn >= count*bd:
        A256968_list.append(k)
count += 1 # Chai Wah Wu, Apr 17 2015; corrected by Max Alekseyev, Jan 26 2025

More terms from Chai Wah Wu, Apr 17 2015
a(32)-a(33) from Chai Wah Wu, Apr 19 2015
a(0)-a(1) corrected and a(34)-a(39) copied over from A005579 by Max Alekseyev, Jan 26 2025

A298917 T(n,k) = Number of n X k 0..1 arrays with every element equal to 0, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 5, 3, 3, 3, 5, 1, 1, 8, 5, 4, 4, 5, 8, 1, 1, 13, 8, 6, 7, 6, 8, 13, 1, 1, 21, 13, 9, 9, 9, 9, 13, 21, 1, 1, 34, 21, 14, 15, 14, 15, 14, 21, 34, 1, 1, 55, 34, 22, 26, 24, 24, 26, 22, 34, 55, 1, 1, 89, 55, 35, 46, 44, 40, 44, 46, 35, 55

Offset: 1

R. H. Hardin, Jan 29 2018

Table starts
.1..1..1..1..1..1...1...1...1....1....1.....1.....1......1......1.......1
.1..1..1..2..3..5...8..13..21...34...55....89...144....233....377.....610
.1..1..1..2..3..5...8..13..21...34...55....89...144....233....377.....610
.1..2..2..3..4..6...9..14..22...35...56....90...145....234....378.....611
.1..3..3..4..7..9..15..26..46...84..151...276...506....929...1708....3138
.1..5..5..6..9.14..24..44..81..156..306...602..1192...2370...4720....9415
.1..8..8..9.15.24..40..76.141..277..570..1171..2441...5157..10913...23193
.1.13.13.14.26.44..76.168.359..792.1895..4521.10886..26818..66131..163463
.1.21.21.22.46.81.141.359.873.2145.5971.16568.45898.131372.376833.1078872

			All solutions for n=5, k=4
..0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..0..0
..0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..0..0
..1..1..1..1. .0..0..1..1. .0..0..0..0. .0..0..0..0
..1..1..1..1. .0..0..1..1. .1..1..1..1. .0..0..0..0
..1..1..1..1. .0..0..1..1. .1..1..1..1. .0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..612

Columns 2 and 3 are A000045(n-1).

Column 4 is A001611(n-1).

Empirical for column k:
k=1: a(n) = a(n-1).
k=2: a(n) = a(n-1) +a(n-2) for n>3.
k=3: a(n) = a(n-1) +a(n-2) for n>3.
k=4: a(n) = 2*a(n-1) -a(n-3).
k=5: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-5) -a(n-6) -a(n-7) -a(n-8) for n>9.
k=6: a(n) = 3*a(n-1) -2*a(n-2) +a(n-3) -2*a(n-4) -a(n-7) +2*a(n-8) for n>9.
k=7: [order 14] for n>15.

A212272 a(n) = Fibonacci(n) + n^3.

Original entry on oeis.org

0, 2, 9, 29, 67, 130, 224, 356, 533, 763, 1055, 1420, 1872, 2430, 3121, 3985, 5083, 6510, 8416, 11040, 14765, 20207, 28359, 40824, 60192, 90650, 138969, 216101, 339763, 538618, 859040, 1376060, 2211077, 3560515, 5742191, 9270340, 14977008, 24208470, 39143041

Offset: 0

Bruno Berselli, May 09 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

Cf. A000045, A000578; A001611, A002062, A160536.

Magma
```
[Fibonacci(n)+n^3: n in [0..38]];
```
Mathematica
```
Table[Fibonacci[n] + n^3, {n, 0, 38}]
```

for(n=0, 38, print1(fibonacci(n)+n^3", "));

G.f.: x*(2-x+2*x^2-9*x^3)/((1-x-x^2)*(1-x)^4).

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

A020717 Pisot sequences L(6,9), E(6,9).

Original entry on oeis.org

6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156, 165580142

Offset: 0

David W. Wilson

Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.

Colin Barker, Table of n, a(n) for n = 0..1000
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for Pisot sequences
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Subsequence of A001611, A048577.
See A008776 for definitions of Pisot sequences.
Pairwise sums of A018910.

Table[Fibonacci[n + 5] + 1, {n, 0, 36}] (* Michael De Vlieger, Jul 27 2016 *)

pisotE(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1/2));
  a
}
pisotE(50, 6, 9) \\ Colin Barker, Jul 27 2016

a(n) = Fibonacci(n+5)+1 = A001611(n+5).
a(n) = 2*a(n-1) - a(n-3).
a(n) = A020706(n+1). - R. J. Mathar, Oct 25 2008

A061489 Numbers that are Fibonacci numbers plus or minus 1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 08 2001

Only the first four terms are Fibonacci numbers per se. - Alonso del Arte, Oct 05 2017

Harry J. Smith, Table of n, a(n) for n=1..500
Ron Knott, The Mathematical Magic of the Fibonacci Numbers.
Index entries for linear recurrences with constant coefficients, signature (-1,1,1,1,1).

Cf. A000071, A001611 (the union of those two sequences forms this sequence).

Union[Table[Fibonacci[i] - 1, {i, 30}], Table[Fibonacci[j] + 1, {j, 0, 30}]]
Union[Flatten[# + {1, -1} &/@ Fibonacci[Range[30]]]] (* Harvey P. Dale, Dec 26 2015 *)

{ t="b061489.txt"; for (n=1, 4, write(t, n, " ", n - 1) ); g=3; h=2; for (n=3, 250, f=g + h; h=g; g=f; write(t, 2*n - 1, " ", f - 1); write(t, 2*n, " ", f + 1) ) } \\ Harry J. Smith, Jul 23 2009

a(n) = -a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n+5), for n >= 10. - Amiram Eldar, Jun 24 2023

More terms from Robert G. Wilson v, Nov 12 2001

A063726 a(n) = gcd(1 + Fibonacci(n+1), 1 + Fibonacci(n)).

Original entry on oeis.org

1, 2, 1, 1, 2, 3, 1, 2, 1, 7, 2, 5, 1, 18, 1, 13, 2, 47, 1, 34, 1, 123, 2, 89, 1, 322, 1, 233, 2, 843, 1, 610, 1, 2207, 2, 1597, 1, 5778, 1, 4181, 2, 15127, 1, 10946, 1, 39603, 2, 28657, 1, 103682, 1, 75025, 2, 271443, 1, 196418, 1, 710647, 2, 514229, 1, 1860498, 1

Offset: 0

Jason Earls, Aug 11 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A000045, A001611.

List([0..65],n->Gcd(1+Fibonacci(n+1),1+Fibonacci(n))); # Muniru A Asiru, Oct 09 2018

[GCD(1 + Fibonacci(n+1), 1 + Fibonacci(n)): n in [0..50]]; // G. C. Greubel, Oct 08 2018

with(combinat): seq(gcd(1+fibonacci(n+1),1+fibonacci(n)),n=0..65); # Muniru A Asiru, Oct 09 2018

Table[GCD[Fibonacci[n],Fibonacci[n+1]+1],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2010 *)

j=[]; for(n=0,75,j=concat(j,gcd(1+fibonacci(n+1),1+fibonacci(n) ))); j

{ g=0; f=1; for (n=0, 1000, write("b063726.txt", n, " ", gcd(1 + f, 1 + g)); h=g; g=f; f+=h ) } \\ Harry J. Smith, Aug 28 2009

Conjectures from Colin Barker, Jan 30 2018: (Start)
G.f.: (1 + 2*x + x^2 + x^3 - x^4 - 3*x^5 - 3*x^6 - 3*x^7 - 5*x^8 - x^9 + x^10 + 3*x^11 + 2*x^12 + x^13) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)*(1 + x^2 - x^4)*(1 - x^2 - x^4)).
a(n) = 3*a(n-4) + a(n-6) - a(n-8) - 3*a(n-10) + a(n-14) for n>13.
(End)

cp's OEIS Frontend

A160536 a(n) = Fibonacci(n) + n^2.

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A187890 a(1) = 0, a(2) = 4, a(n) = a(n-1) + a(n-2) - 1.

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A232896 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4), where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 8.

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A256968 Let b(n) = Product_{i=1..n} p_i/(p_i - 1), p_i = i-th prime; a(n) = minimum k such that b(k) >= n.

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A298917 T(n,k) = Number of n X k 0..1 arrays with every element equal to 0, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero.

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A212272 a(n) = Fibonacci(n) + n^3.

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Magma

Mathematica

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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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A020717 Pisot sequences L(6,9), E(6,9).

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A232896 a(n) = 3a(n-1) - 2a(n-2) - a(n-3) + a(n-4), where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 8.