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.

Previous Showing 41-49 of 49 results.

A251705 4-step Fibonacci sequence starting with 1, 1, 1, 0.

Original entry on oeis.org

1, 1, 1, 0, 3, 5, 9, 17, 34, 65, 125, 241, 465, 896, 1727, 3329, 6417, 12369, 23842, 45957, 88585, 170753, 329137, 634432, 1222907, 2357229, 4543705, 8758273, 16882114, 32541321, 62725413, 120907121, 233055969, 449229824, 865918327, 1669111241
Offset: 0

Views

Author

Arie Bos, Dec 07 2014

Keywords

Links

Crossrefs

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251655, A251656, A251703, A251704.

Programs

  • J
    NB. see A251655 for the program and apply it to 1,1,1,0.
  • Mathematica
    LinearRecurrence[Table[1, {4}], {1, 1, 1, 0}, 36] (* Michael De Vlieger, Dec 09 2014 *)

Formula

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: (-1+3*x^3+x^2)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025

A073937 a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-4), a(0)=4, a(1)=1, a(2)=-1, a(3)=1.

Original entry on oeis.org

4, 1, -1, 1, 7, 6, -1, 1, 15, 19, 4, 1, 31, 53, 27, 6, 63, 137, 107, 39, 132, 337, 351, 185, 303, 806, 1039, 721, 791, 1915, 2884, 2481, 2303, 4621, 7683, 7846, 7087, 11545, 19987, 23375, 22020, 30177, 51519, 66737, 67415, 82374, 133215, 184993, 201567, 232163, 348804
Offset: 0

Views

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 13 2002

Keywords

Comments

From Kai Wang, Nov 03 2020: (Start)
Let f(x) = x^4 - x^3 - x^2 - x - 1 and {x1,x2,x3,x4} be the roots of f(x). Then a(n) = (x1*x2*x3)^n + (x1*x2*x4)^n + (x1*x3*x4)^n + (x2*x3*x4)^n.
Let g(y) = y^4 - y^3 + y^2 - y - 1 and {y1,y2,y3,y4} be the roots of g(y). Then a(n) = y1^n + y2^n + y3^n + y4^n. (End)

Links

Crossrefs

Cf. A000078, A001630, A073817, A074058, A100329.

Programs

  • Mathematica
    CoefficientList[Series[(4-3*x+2*x^2-x^3)/(1-x+x^2-x^3-x^4), {x, 0, 50}], x]
LinearRecurrence[{1,-1,1,1},{4,1,-1,1},60] (* Harvey P. Dale, Sep 05 2021 *)
  • PARI
    polsym(y^4 - y^3 + y^2 - y - 1, 55) \\ Joerg Arndt, Nov 07 2020

Formula

G.f.: (4 - 3*x + 2*x^2 - x^3)/(1 - x + x^2 - x^3 - x^4).
From Kai Wang, Nov 03 2020: (Start)
For n >= 1, a(n) = Sum_{j1,j2,j3,j4>=0; j1+2*j2+3*j3+4*j4=n} (-1)^j2*n*(j1+j2+j3+j4-1)!/(j1!*j2!*j3!*j4!).
For n > 1, a(n) = (-1)^n*(4*A100329(n+1) + 3*A100329(n) + 2*A100329(n-1) + A100329(n-2)). (End)
From Peter Bala, Jan 19 2023: (Start)
a(n) = (-1)^n*A074058(n).
a(n) = trace of M^n, where M is the 4 X 4 matrix [[0, 0, 0, -1], [-1, 0, 0, 1], [0, -1, 0, 1], [0, 0, -1, 1]].
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^k) for positive integers n and r and all primes p. See Zarelua. (End)

A253333 Primes in the 7th-order Fibonacci numbers A060455.

Original entry on oeis.org

7, 13, 97, 193, 769, 1531, 3049, 6073, 12097, 24097, 95617, 379399, 2998753, 187339729, 373174033, 2949551617, 184265983633, 731152932481, 88025699967469825543, 175344042716296888429, 4979552865927484193343796114081304399449
Offset: 1

Views

Author

Robert Price, Dec 30 2014

Keywords

Comments

a(22) is too large to display here. It has 53 digits and is the 180th term in A060455.

Links

Crossrefs

Cf. A001590, A001631, A100683, A231574, A231575, A232543, A214899, A020992, A233554, A214727, A234696, A141523, A235862, A214825, A235873, A001630, A241660, A247027, A000288, A247561, A000322, A248920, A000383, A247192, A060455, A253318.

Programs

  • Mathematica
    a={1,1,1,1,1,1,1}; step=7; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,sum]]; a=RotateLeft[a]; a[[7]]=sum]; lst
With[{c=PadRight[{},7,1]},Select[LinearRecurrence[c,c,150],PrimeQ]] (* Harvey P. Dale, May 08 2015 *)
  • PARI
    lista(nn) = {gf = ( -1+x^2+2*x^3+3*x^4+4*x^5+5*x^6 ) / ( -1+x+x^2+x^3+x^4+x^5+x^6+x^7 ); for (n=0, nn, if (isprime(p=polcoeff(gf+O(x^(n+1)), n)), print1(p, ", ")););} \\ Michel Marcus, Jan 11 2015

A254413 Primes in the 8th-order Fibonacci numbers A123526.

Original entry on oeis.org

29, 113, 449, 226241, 14307889, 113783041, 1820091580429249, 233322881089059894782836851617, 29566627412209231076314948970028097, 59243719929958343565697204780596496129, 7507351981539044730893385057192143660843521
Offset: 1

Views

Author

Robert Price, Jan 30 2015

Keywords

Comments

a(12) is too large to display here. It has 46 digits and is the 158th term in A123526.

Crossrefs

Cf. A001590, A001631, A100683, A231574, A231575, A232543, A214899, A020992, A233554, A214727, A234696, A141523, A235862, A214825, A235873, A001630, A241660, A247027, A000288, A247561, A000322, A248920, A000383, A247192, A060455, A253318, A079262, A253705, A123526, A254412.

Programs

  • Mathematica
    a={1,1,1,1,1,1,1,1}; step=8; lst={}; For[n=step+1,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,sum]]; a=RotateLeft[a]; a[[step]]=sum]; lst
Select[With[{lr=PadRight[{},8,1]},LinearRecurrence[lr,lr,200]],PrimeQ] (* Harvey P. Dale, Dec 03 2022 *)

A382478 Number of palindromic binary strings of length n having no 4-runs of 1's.

Original entry on oeis.org

1, 2, 2, 4, 3, 7, 6, 14, 12, 27, 23, 52, 44, 100, 85, 193, 164, 372, 316, 717, 609, 1382, 1174, 2664, 2263, 5135, 4362, 9898, 8408, 19079, 16207, 36776, 31240, 70888, 60217, 136641, 116072, 263384, 223736, 507689, 431265, 978602, 831290, 1886316, 1602363, 3635991, 3088654, 7008598, 5953572
Offset: 0

Views

Author

R. J. Mathar, Mar 28 2025

Keywords

Links

Crossrefs

Cf. A001630 (bisection), A001631 (bisection).
Cf. A123231 (2-runs), A001590 (3-runs), A251653 (5-runs), A382479 (6-runs).

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-(x^2+1)*(x^5+2*x+1) / (-1+x^2+x^4+x^6+x^8))); // Vincenzo Librandi, May 19 2025
  • Mathematica
    LinearRecurrence[{0,1,0,1,0,1,0,1},{1,2,2,4,3,7,6,14},50] (* Vincenzo Librandi, May 19 2025 *)

Formula

G.f.: -(x^2+1)*(x^5+2*x+1)/(-1+x^2+x^4+x^6+x^8).

A248921 Primes in the pentanacci numbers sequence A000322.

Original entry on oeis.org

5, 17, 977, 28697, 56417, 1428864769, 2809074173, 21344178433, 626815657409, 18407729752001, 2317881588988297338942875602391948125494800020122167809, 136507010958920295813169620935932629930648432530102206331972221346174230852977164801
Offset: 1

Views

Author

Robert Price, Oct 16 2014

Keywords

Comments

a(13) is too large to display here. It has 132 digits and is the 450th term in A000322.

Links

Crossrefs

Cf. A001590, A001631, A100683, A231574, A231575, A232543, A214899, A020992, A233554, A214727, A234696, A141523, A235862, A214825, A235873, A001630, A241660, A247027, A000288, A247561, A000322, A248920.

Programs

  • Mathematica
    a={1,1,1,1,1}; For[n=5, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[sum]]; a=RotateLeft[a]; a[[5]]=sum]
Select[With[{c={1,1,1,1,1}},LinearRecurrence[c,c,300]],PrimeQ] (* Harvey P. Dale, Nov 30 2019 *)

A253706 Primes in the 8th-order Fibonacci numbers A079262.

Original entry on oeis.org

2, 509, 128257, 133294824621464999938178340471931877, 4596852049500861351052672455121859744010232939954169259264638023409631672658340253083284317818242062413
Offset: 1

Views

Author

Robert Price, Jan 09 2015

Keywords

Comments

a(6) is too large to display here. It has 395 digits and is the 1322nd term in A079262.

Crossrefs

Cf. A001590, A001631, A100683, A231574, A231575, A232543, A214899, A020992, A233554, A214727, A234696, A141523, A235862, A214825, A235873, A001630, A241660, A247027, A000288, A247561, A000322, A248920, A000383, A247192, A060455, A253318, A079262, A253705.

Programs

  • Mathematica
    a={0,0,0,0,0,0,0,1}; step=8; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,sum]]; a=RotateLeft[a]; a[[step]]=sum]; lst
  • PARI
    lista(nn) = {gf = x^7/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8); for (n=0, nn, if (isprime(p=polcoeff(gf+O(x^(n+1)), n)), print1(p, ", ")););} \\ Michel Marcus, Jan 12 2015

A247028 Primes in tetranacci sequence A001631.

Original entry on oeis.org

2, 7, 193, 19079, 1823013184807, 324494495853101147203936847, 16085434555484907108254435283952049, 255525859571903290673264616283734506003204622439226993660213169027169
Offset: 1

Views

Author

Robert Price, Sep 09 2014

Keywords

Comments

a(9) is too large to display here. It has 160 digits and is the 564th term in A001631.

Crossrefs

Cf. A001590, A001631, A100683, A231574, A231575, A232543, A214899, A020992, A233554. A214727, A234696, A141523, A235862,A214825, A235873, A001630, A241660, A247027.

Programs

  • Mathematica
    a={0,0,1,0}; For[n=4, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[sum]]; a=RotateLeft[a]; a[[4]]=sum]

A247946 Primes in the tetranacci sequence A000288.

Original entry on oeis.org

7, 13, 181, 349, 673, 1297, 34513, 90799453, 175021573, 4657290577, 17304140641, 1131469145856472270556751793, 1544310310927991136025089626209, 1442398599584422734286432395814518441223501, 18598135820391234761502881488353916158281807617671450769
Offset: 1

Views

Author

Robert Price, Sep 27 2014

Keywords

Comments

a(16) is too large to display here. It has 63 digits and is the 221st term in A000288.

Crossrefs

Cf. A001590, A001631, A100683, A231574, A231575, A232543, A214899, A020992, A233554, A214727, A234696, A141523, A235862, A214825, A235873, A001630, A241660, A247027, A000288, A247561.

Programs

  • Mathematica
    a={1,1,1,1}; For[n=4, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[sum]]; a=RotateLeft[a]; a[[4]]=sum]
Select[LinearRecurrence[{1,1,1,1},{1,1,1,1},300],PrimeQ] (* Harvey P. Dale, Jan 15 2015 *)
Previous Showing 41-49 of 49 results.

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