cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A001611 a(n) = Fibonacci(n) + 1.

Original entry on oeis.org

1, 2, 2, 3, 4, 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
Offset: 0

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Author

Keywords

Comments

a(0) = 1, a(1) = 2 then the largest number such that a triangle is constructible with three successive terms as sides. - Amarnath Murthy, Jun 03 2003
a(n+2) = A^(n)B(1), n>=0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g., 2=`0`, 3=`10`, 4=`110`, 6=`1110`, ..., in Wythoff code.
The first-difference sequence is the Fibonacci sequence (A000045). - Roland Schroeder (florola(AT)gmx.de), Aug 05 2010
2 and 3 are the only primes in this sequence.
a(n) is the number of 1 X n nonogram puzzles which can be solved uniquely. See A242876 for puzzle definition. - Lior Manor, Jan 23 2022

References

  • G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Programs

  • Haskell
    a001611 = (+ 1) . a000045
    a001611_list = 1 : 2 : map (subtract 1)
                           (zipWith (+) a001611_list $ tail a001611_list)
    -- Reinhard Zumkeller, Jul 30 2013
  • Magma
    [Fibonacci(n)+1: n in [1..37]]; // Bruno Berselli, Jul 26 2011
    
  • Maple
    A001611:=-(-1+2*z**2)/(z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation
    with(combinat): seq((fibonacci(n)+1), n=0..35);
  • Mathematica
    a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n-2] + a[n-1] - 1; Table[ a[n], {n, 0, 40} ]
    Fibonacci[Range[0,50]]+1  (* Harvey P. Dale, Mar 23 2011 *)
  • PARI
    a(n)=fibonacci(n)+1 \\ Charles R Greathouse IV, Jul 25 2011
    

Formula

G.f.: (1-2*x^2)/(1-2*x+x^3).
a(n) = 2*a(n-1) - a(n-3). - Tanya Khovanova, Jul 13 2007
a(0) = 1, a(1) = 2, a(n) = a(n - 2) + a(n - 1) - 1.
F(4*n) + 1 = F(2*n-1)*L(2*n+1); F(4*n+1) + 1 = F(2*n+1)*L(2*n); F(4*n+2) + 1 = F(2*n+2)*L(2*n); F(4*n+3) + 1 = F(2*n+1)*L(2*n+2) where F(n)=Fibonacci(n) and L(n)=Lucas(n). - R. K. Guy, Feb 27 2003
a(1) = 2; a(n+1)=floor(a(n)*(sqrt(5)+1)/2). - Roland Schroeder (florola(AT)gmx.de), Aug 05 2010
a(n) = Sum_{k=0..n+1} Fibonacci(k-3). - Ehren Metcalfe, Apr 15 2019
Product_{n>=1} (1 - (-1)^n/a(n)) = sin(3*Pi/10) (A019863). - Amiram Eldar, Nov 28 2024

A065220 a(n) = Fibonacci(n) - n.

Original entry on oeis.org

0, 0, -1, -1, -1, 0, 2, 6, 13, 25, 45, 78, 132, 220, 363, 595, 971, 1580, 2566, 4162, 6745, 10925, 17689, 28634, 46344, 75000, 121367, 196391, 317783, 514200, 832010, 1346238, 2178277, 3524545, 5702853, 9227430, 14930316, 24157780, 39088131, 63245947, 102334115, 165580100, 267914254
Offset: 0

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Author

Henry Bottomley, Oct 22 2001

Keywords

Comments

E(n) = Fib(n+4)-(n+4): cost of maximum height Huffman tree of size n for Fibonacci sequence (Fibonacci sequence is minimizing absolutely ordered sequence of Huffman tree). - Alex Vinokur (alexvn(AT)barak-online.net), Oct 26 2004

References

  • Vinokur A.B, Huffman trees and Fibonacci numbers, Kibernetika Issue 6 (1986) 9-12 (in Russian); English translation in Cybernetics 21, Issue 6 (1986), 692-696.

Crossrefs

Programs

  • GAP
    List([0..50], n-> Fibonacci(n) - n); # G. C. Greubel, Jul 09 2019
  • Haskell
    a065220 n = a065220_list !! n
    a065220_list = zipWith (-) a000045_list [0..]
    -- Reinhard Zumkeller, Nov 06 2012
    
  • Magma
    [Fibonacci(n) - n: n in [0..50]]; // G. C. Greubel, Jul 09 2019
    
  • Maple
    a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+a[n-2] od: seq(a[n]-n, n=0..42); # Zerinvary Lajos, Mar 20 2008
  • Mathematica
    lst={};Do[f=Fibonacci[n]-n;AppendTo[lst,f],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 21 2009 *)
    Table[Fibonacci[n]-n,{n,0,50}] (* or *) LinearRecurrence[{3,-2,-1,1},{0,0,-1,-1},50] (* Harvey P. Dale, May 29 2017 *)
  • PARI
    a(n) = { fibonacci(n) - n } \\ Harry J. Smith, Oct 14 2009
    
  • Sage
    [fibonacci(n) - n for n in (0..50)] # G. C. Greubel, Jul 09 2019
    

Formula

a(n) = A000045(n) - A001477(n) = A000126(n-3) - 2 = A001924(n-4) - 1.
a(n) = a(n-1) + a(n-2) + n - 3 = a(n-1) + A000071(n-2).
G.f.: x^2*(2x-1)/((1-x-x^2)*(1-x)^2).
a(n) = Sum_{i=0..n} (i - 2)*F(n-i) for F(n) the Fibonacci sequence A000045. - Greg Dresden, Jun 01 2022

A081659 a(n) = n + Fibonacci(n+1).

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 19, 28, 42, 64, 99, 155, 245, 390, 624, 1002, 1613, 2601, 4199, 6784, 10966, 17732, 28679, 46391, 75049, 121418, 196444, 317838, 514257, 832069, 1346299, 2178340, 3524610, 5702920, 9227499, 14930387, 24157853, 39088206
Offset: 0

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Author

Paul Barry, Mar 26 2003

Keywords

Comments

Row sums of triangle A135222. - Gary W. Adamson, Nov 23 2007
a(n) is the F(n+1)-th highest positive integer not equal to any a(k), 1 <= k <= n-1, where F(n) = Fibonacci numbers = A000045(n). - Jaroslav Krizek, Oct 28 2009

Crossrefs

Cf. A000045, A001611 (first differences), A002062, A135222.

Programs

Formula

a(n) = (sqrt(5)*(1+sqrt(5))^(n+1) - sqrt(5)*(1-sqrt(5))^(n+1))/(10*2^n) + n.
G.f.: (1-x-x^3)/((1-x-x^2)*(1-x)^2).
From Jaroslav Krizek, Oct 28 2009: (Start)
a(0) = 1, a(n) = a(n-1) + A000045(n-1) + 1 for n >= 1.
a(0) = 1, a(n) = a(n-1) + A000045(n+1) - A000045(n) + 1 for n >= 1.
a(0) = 1, a(1) = 2, a(2) = 4, a(n) = a(n-1) + a(n-2) - (n-3) n >= 3. (End)
E.g.f.: (1/10)*exp(-2*x/(1+sqrt(5)))*(5 - sqrt(5) + (5 + sqrt(5))*exp(sqrt(5)*x) + 10*exp((1/2)*(1+sqrt(5))*x)*x). - Stefano Spezia, Nov 20 2019

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

Original entry on oeis.org

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

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Author

Leonardo Sznajder, May 18 2009

Keywords

Examples

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

Crossrefs

Programs

Formula

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

Extensions

Edited and extended by Klaus Brockhaus, May 22 2009

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

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Author

Bruno Berselli, May 09 2012

Keywords

Crossrefs

Programs

  • Magma
    [Fibonacci(n)+n^3: n in [0..38]];
  • Mathematica
    Table[Fibonacci[n] + n^3, {n, 0, 38}]
  • PARI
    for(n=0, 38, print1(fibonacci(n)+n^3", "));
    

Formula

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

A069108 Primes of the form F(k)+k where F(k) is the k-th Fibonacci number.

Original entry on oeis.org

2, 3, 5, 7, 29, 43, 317839, 3524611, 39088207
Offset: 1

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Author

Benoit Cloitre, Apr 06 2002

Keywords

Crossrefs

Programs

  • Mathematica
    Select[Table[Fibonacci[k] + k, {k, 1, 40}], PrimeQ] (* Amiram Eldar, Jun 04 2022 *)

Formula

A002062 INTERSECT A000040. - R. J. Mathar, Apr 24 2017
a(n) = A002062(A175404(n)). - Amiram Eldar, Jun 04 2022

Extensions

a(10), a 169-digit number, has been certified prime with Primo. - Rick L. Shepherd, Apr 26 2002
a(11), a 343-digit number, has been certified prime with Primo. - Charles R Greathouse IV, Feb 15 2011

A210730 a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=a(1)=0.

Original entry on oeis.org

0, 0, 4, 9, 19, 35, 62, 106, 178, 295, 485, 793, 1292, 2100, 3408, 5525, 8951, 14495, 23466, 37982, 61470, 99475, 160969, 260469, 421464, 681960, 1103452, 1785441, 2888923, 4674395, 7563350, 12237778, 19801162, 32038975, 51840173, 83879185, 135719396
Offset: 0

Views

Author

Alex Ratushnyak, May 10 2012

Keywords

Comments

Deleting the 0's leaves row 4 of the convolution array A213579. - Clark Kimberling, Jun 20 2012

Crossrefs

Cf. A033818: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=0 (except first 2 terms and sign).
Cf. A002062: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=0 (except the first term and sign).
Cf. A065220: a(n)=a(n-1)+a(n-2)+n-3, a(0)=a(1)=0.
Cf. A001924: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=0 (except the first term).
Cf. A023548: a(n)=a(n-1)+a(n-2)+n, a(0)=a(1)=0 (except first 2 terms).
Cf. A023552: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=0 (except first 2 terms).
Cf. A210731: a(n)=a(n-1)+a(n-2)+n+3, a(0)=a(1)=0.

Programs

  • GAP
    F:=Fibonacci;; List([0..40], n-> F(n+3)+3*F(n+1)-n-5); # G. C. Greubel, Jul 08 2019
  • Magma
    I:=[0, 0, 4, 9]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-Self(n-3)+Self(n-4): n in [1..37]]; // Bruno Berselli, May 10 2012
    
  • Magma
    F:=Fibonacci; [F(n+3)+3*F(n+1)-n-5: n in [0..40]]; // G. C. Greubel, Jul 08 2019
    
  • Mathematica
    RecurrenceTable[{a[0]==a[1]==0, a[n]==a[n-1] +a[n-2] +n+2}, a, {n, 40}] (* Bruno Berselli, May 10 2012 *)
    LinearRecurrence[{3,-2,-1,1},{0,0,4,9},40] (* Harvey P. Dale, Jul 24 2013 *)
    With[{F=Fibonacci}, Table[F[n+3]+2*F[n+1]-n-5, {n, 40}]] (* G. C. Greubel, Jul 08 2019 *)
  • PARI
    concat(vector(2), Vec(x^2*(4-3*x)/((1-x)^2*(1-x-x^2)) + O(x^50))) \\ Colin Barker, Mar 11 2017
    
  • PARI
    vector(40, n, n--; f=fibonacci; f(n+3)+3*f(n+1)-n-5) \\ G. C. Greubel, Jul 08 2019
    
  • Sage
    f=fibonacci; [f(n+3)+3*f(n+1)-n-5 for n in (0..40)] # G. C. Greubel, Jul 08 2019
    

Formula

G.f.: x^2*(4-3*x)/((1-x)^2*(1-x-x^2)). - Bruno Berselli, May 10 2012
a(n) = A210677(n)-1. - Bruno Berselli, May 10 2012
a(0)=0, a(1)=0, a(2)=4, a(3)=9, a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). - Harvey P. Dale, Jul 24 2013
a(n) = -5 + (2^(-1-n)*((1-sqrt(5))^n*(-7+5*sqrt(5)) + (1+sqrt(5))^n*(7+5*sqrt(5)))) / sqrt(5) - n. - Colin Barker, Mar 11 2017
a(n) = Fibonacci(n+3) + 3*Fibonacci(n+1) - n - 5. - G. C. Greubel, Jul 08 2019

A210731 a(n) = a(n-1) + a(n-2) + n + 3 with n>1, a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 5, 11, 23, 42, 74, 126, 211, 349, 573, 936, 1524, 2476, 4017, 6511, 10547, 17078, 27646, 44746, 72415, 117185, 189625, 306836, 496488, 803352, 1299869, 2103251, 3403151, 5506434, 8909618, 14416086, 23325739, 37741861, 61067637, 98809536
Offset: 0

Views

Author

Alex Ratushnyak, May 10 2012

Keywords

Crossrefs

Cf. A033818: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=0 (except first 2 terms and sign).
Cf. A002062: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=0 (except the first term and sign).
Cf. A065220: a(n)=a(n-1)+a(n-2)+n-3, a(0)=a(1)=0.
Cf. A001924: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=0 (except the first term).
Cf. A023548: a(n)=a(n-1)+a(n-2)+n, a(0)=a(1)=0 (except first 2 terms).
Cf. A023552: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=0 (except first 2 terms).
Cf. A210730: a(n)=a(n-1)+a(n-2)+n+2, a(0)=a(1)=0.

Programs

  • GAP
    F:=Fibonacci;; List([0..40], n-> F(n+3)+4*F(n+1)-n-6); # G. C. Greubel, Jul 09 2019
  • Magma
    F:=Fibonacci; [F(n+3)+4*F(n+1)-n-6: n in [0..40]]; // G. C. Greubel, Jul 09 2019
    
  • Mathematica
    With[{F = Fibonacci}, Table[F[n+3]+4*F[n+1]-n-6, {n,0,40}]] (* G. C. Greubel, Jul 09 2019 *)
    nxt[{n_,a_,b_}]:={n+1,b,a+b+n+4}; NestList[nxt,{1,0,0},40][[;;,2]] (* or *) LinearRecurrence[{3,-2,-1,1},{0,0,5,11},40] (* Harvey P. Dale, Dec 30 2024 *)
  • PARI
    vector(40, n, n--; f=fibonacci; f(n+3)+4*f(n+1)-n-6) \\ G. C. Greubel, Jul 09 2019
    
  • Sage
    f=fibonacci; [f(n+3)+4*f(n+1)-n-6 for n in (0..40)] # G. C. Greubel, Jul 09 2019
    

Formula

From Colin Barker, Jun 29 2012: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: x^2*(5-4*x)/((1-x)^2*(1-x-x^2)). (End)
a(n) = Fibonacci(n+3) + 4*Fibonacci(n+1) - (n+6). - G. C. Greubel, Jul 09 2019

A037141 Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), for k >= -5, with F(-n)=(-1)^(n+1)*F(n);.

Original entry on oeis.org

5, 7, 11, 14, 18, 22, 27, 33, 41, 52, 68, 92, 129, 187, 279, 426, 662, 1042, 1655, 2645, 4245, 6832, 11016, 17784, 28733, 46447, 75107, 121478, 196506, 317902, 514323, 832137, 1346369, 2178412, 3524684, 5702996, 9227577, 14930467, 24157935, 39088290
Offset: 0

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Author

Keywords

Crossrefs

Programs

  • Mathematica
    CoefficientList[Series[(5-8x)/((1-x-x^2)(1-x)^2),{x,0,40}],x] (* or *) LinearRecurrence[{3,-2,-1,1},{5,7,11,14},40] (* Harvey P. Dale, Nov 06 2017 *)

Formula

a(n) = F(n-1)+(4+3*n); G.f.: (5-8*x)/((1-x-x^2)*(1-x)^2)

Extensions

More terms from Harvey P. Dale, Nov 06 2017

A173684 Semiprimes of the form Fibonacci(k) + k.

Original entry on oeis.org

10, 14, 65, 391, 1003, 2602, 10967, 2178341, 701408777, 86267571326, 591286729937, 4052739537943, 72723460248209, 117669030461063, 3416454622906783, 61305790721611673, 420196140727489759, 2427893228399975082557, 251728825683549488150424389
Offset: 1

Views

Author

Jonathan Vos Post, Jan 26 2011

Keywords

Comments

This is to A069108 as semiprimes are to primes. A002062(k) is semiprime for k = 5, 6, 10, 14, 16, 18, 21, 32, ...

Examples

			F(21) + 21 = 10967 = 11 * 997, thus 10967 is in the sequence.
		

Crossrefs

Programs

  • Magma
    IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [ a: n in [1..100] | IsSemiprime(a) where a is n+Fibonacci(n) ]; // Klaus Brockhaus, Jan 27 2011
  • Mathematica
    Select[Table[Fibonacci[n]+n,{n,200}],PrimeOmega[#]==2&] (* Harvey P. Dale, Nov 02 2011 *)

Formula

A002062 INTERSECTION A001358.
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