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 1-5 of 5 results.

A004922 a(n) = floor(n*phi^7), where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, 377, 406, 435, 464, 493, 522, 551, 580, 609, 638, 667, 696, 725, 754, 783, 812, 841, 871, 900, 929, 958, 987, 1016, 1045, 1074, 1103, 1132, 1161, 1190, 1219, 1248, 1277, 1306, 1335, 1364
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A004919, A004920, A004921, A004923, A004924, A004925, A004926.
Cf. A004927, A004928, A004929, A004930, A004931, A004932, A004933.
Cf. A004934, A004935, A004976, A066096, A090909.
Cf. A001622, A004962 (ceiling).

Programs

  • Magma
    [Floor(n*((1 + Sqrt(5))/2)^7): n in [0..50]]; // Vincenzo Librandi, Jul 22 2015
  • Mathematica
    Table[Floor[n ((1 + Sqrt[5])/2)^7], {n, 0, 50}] (* Vincenzo Librandi, Jul 22 2015 *)
  • Python
    from sympy import sqrt
phi = (1 + sqrt(5))/2
for n in range(0,101): print(int(n*phi**7), end=', ') # Karl V. Keller, Jr., Jul 22 2015

A249079 a(n) = 29*n + floor( n/29 ) + 0^( 1-floor( (14+(n mod 29))/29 ) ).

Original entry on oeis.org

0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, 377, 406, 436, 465, 494, 523, 552, 581, 610, 639, 668, 697, 726, 755, 784, 813, 842, 871, 900, 929, 958, 987, 1016, 1045, 1074, 1103, 1132, 1161, 1190, 1219, 1248, 1278, 1307, 1336
Offset: 0

Views

Author

Karl V. Keller, Jr., Oct 20 2014

Keywords

Comments

This is an approximation to A004942 (Nearest integer to n*phi^7, where phi is the golden ratio, A001622).

Examples

			n= 0, 29*n+floor(0.0) +0^(1-floor(0.48))=    0 +0 +0 =    0 (n/29=0,0^1=0).
n=14, 29*n+floor(0.48)+0^(1-floor(0.97))=  406 +0 +0 =  406 (0^1=0).
n=15, 29*n+floor(0.52)+0^(1-floor(1.0)) =  435 +0 +1 =  436 (0^0=1).
n=28, 29*n+floor(0.97)+0^(1-floor(1.45))=  812 +0 +1 =  813 (0^0=1).
n=29, 29*n+floor(1.0) +0^(1-floor(0.48))=  841 +1 +0 =  842 (n/29*1,0^1=0).
n=43, 29*n+floor(1.48)+0^(1-floor(0.97))= 1247 +1 +0 = 1248 (0^1=0).
n=44, 29*n+floor(1.52)+0^(1-floor(1.0)) = 1276 +1 +1 = 1278 (0^0=1).
n=58, 29*n+floor(2.0) +0^(1-floor(0.48))= 1682 +2 +0 = 1684 (n/29*2,0^1=0).
n=85, 29*n+floor(2.93)+0^(1-floor(1.41))= 2465 +2 +1 = 2468 (0^0=1).
n=86, 29*n+floor(2.97)+0^(1-floor(1.45))= 2494 +2 +1 = 2497 (0^0=1).
n=87, 29*n+floor(3.0) +0^(1-floor(0.48))= 2523 +3 +0 = 2526 (n/29*3,0^0=0).

Links

Crossrefs

Cf. A001622 (phi), A195819 (29*n).
Cf. A004942 (round(n*phi^7)), A004922 (floor(n*phi^7)), A004962 (ceiling(n*phi^7)).

Programs

  • Magma
    [29*n + Floor(n/29) + 0^(1-Floor((14+(n mod 29))/29)) : n in [0..50]]; // Vincenzo Librandi, Nov 05 2014
  • PARI
    a(n) = 29*n + n\29 + 0^(1 - (14+(n % 29))\29); \\ Michel Marcus, Oct 25 2014
  • Python
    for n in range(101):
    print(29*n+n//29+0**(1-(14+n%29)//29), end=', ')
  • Python
    def A249079(n):
    a, b = divmod(n,29)
    return 29*n+a+int(b>=15) # Chai Wah Wu, Jul 27 2022

A248739 a(n) = 29*n + ceiling(n/29).

Original entry on oeis.org

0, 30, 59, 88, 117, 146, 175, 204, 233, 262, 291, 320, 349, 378, 407, 436, 465, 494, 523, 552, 581, 610, 639, 668, 697, 726, 755, 784, 813, 842, 872, 901, 930, 959, 988, 1017, 1046, 1075, 1104, 1133, 1162, 1191, 1220, 1249, 1278, 1307, 1336, 1365, 1394, 1423
Offset: 0

Views

Author

Karl V. Keller, Jr., Oct 13 2014

Keywords

Comments

This is an approximation to A004962 (ceiling of n*phi^7, where phi is the golden ratio, A001622).

Examples

			For n = 10, 29n + ceiling(n/29) = 290 + ceiling(0.3) = 290 + 1 = 291.

Links

Crossrefs

Cf. A001622 (phi), A195819 (29*n).
Cf. A004922 (floor(n*phi^7)), A004962 (ceiling(n*phi^7)), A004942 (round(n*phi^7)).

Programs

  • Magma
    [29*n + Ceiling(n/29): n in [0..60]]; // Vincenzo Librandi, Oct 13 2014
  • Maple
    A248739:=n->29*n+ceil(n/29): seq(A248739(n), n=0..50); # Wesley Ivan Hurt, Oct 14 2014
  • Mathematica
    Table[29 n + Ceiling[n/29], {n, 0, 60}] (* Vincenzo Librandi, Oct 13 2014 *)
  • Python
    from math import *
for n in range(0,101):
  print(n, (29*n+ceil(n/29.0)))

Formula

a(n) = 29*n + ceiling(n/29).
a(n) = A004962(n) for n < 871. - Joerg Arndt, Oct 18 2014

A248786 a(n) = 29*n + floor(n/29) + 0^n - 0^(n mod 29).

Original entry on oeis.org

0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, 377, 406, 435, 464, 493, 522, 551, 580, 609, 638, 667, 696, 725, 754, 783, 812, 841, 871, 900, 929, 958, 987, 1016, 1045, 1074, 1103, 1132, 1161, 1190, 1219, 1248
Offset: 0

Views

Author

Karl V. Keller, Jr., Oct 14 2014

Keywords

Comments

This is an approximation to A004922 (floor of n*phi^7, where phi is the golden ratio, A001622).
The "+ 0^n - 0^(n mod 29)" corrects a(n), for n=0 and multiples of 29. (See examples below.)

Examples

			For n = 0,  29*n + floor(0.0)  + 0^0  - 0^(0) =   0  + 0  + 1  - 1 = 0 (n=29*0).
For n = 28, 29*n + floor(0.97) + 0^28 - 0^(28)= 812  + 0  + 0  - 0 = 812.
For n = 29, 29*n + floor(1.0)  + 0^29 - 0^(0) = 841  + 1  + 0  - 1 = 841 (n=29*1).
For n = 31, 29*n + floor(1.1)  + 0^31 - 0^(2) = 899  + 1  + 0  - 0 = 900.
For n = 87, 29*n + floor(3.0)  + 0^87 - 0^(0) = 2523 + 3  + 0  - 1 = 2525 (n=29*3).

Links

Crossrefs

Cf. A001622 (phi), A195819 (29*n).
Cf. A004922 (floor(n*phi^7)), A004962 (ceiling(n*phi^7)), A004942 (round(n*phi^7)).

Programs

  • Magma
    [(29*n+Floor(n/29))+ 0^n-0^(n mod 29): n in [0..60]]; // Vincenzo Librandi, Oct 14 2014
  • PARI
    a(n) = 29*n+ n\29 + 0^n - 0^(n % 29); \\ Michel Marcus, Oct 14 2014
  • Python
    from math import *
from decimal import *
getcontext().prec = 100
for n in range(0,101):
  print(n, (29*n+floor(n/29.0))+ 0**n-0**(n%29))
  • Python
    def A248786(n):
    a, b = divmod(n,29)
    return 29*n+a-int(not b) if n else 0 # Chai Wah Wu, Jul 27 2022

A256278 a(0)=1, a(1)=2, a(n) = 31*a(n-1) - 29*a(n-2).

Original entry on oeis.org

1, 2, 33, 965, 28958, 869713, 26121321, 784539274, 23563199185, 707707535789, 21255600833094, 638400107288033, 19173990901769297, 575880114843495250, 17296237823997043137, 519482849213446974997, 15602377428720941973934, 468608697663159238917041
Offset: 0

Views

Author

Karl V. Keller, Jr., Jun 02 2015

Keywords

Comments

The sequence A084330 is a(0)=0, a(1)=1, a(n)=31a(n-1)-29a(n-2), and the ratio A084330(n+1)/a(n) converges to phi^7 (~29.034441853748633...), where phi is the golden ratio (A001622).
The continued fraction for phi^7 is {29,{29}}, and 29 occurs in the following approximations for n*phi^7: A248786 (29*n+floor(n/29)+0^n-0^(n mod 29)) for A004922 (floor(n*phi^7)), A249079 (29*n+floor(n/29)+0^(1-floor((14+(n mod 29))/29))) for A004942 (round(n*phi^7)), and A248739 (29*n+ceiling(n/29)) for A004962 (ceiling(n*phi^7)).

Examples

			For n=3, 31*a(2)-29*a(1) = 31*(33)-29*(2) = 1023-58 = 965.

Links

Crossrefs

Cf. A001622, A195819 (29*n), A084330, A004922, A004942, A004962, A248786, A249079, A248739.

Programs

  • Magma
    I:=[1,2]; [n le 2 select I[n] else 31*Self(n-1)-29*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 03 2015
  • Maple
    a:= n-> (<<0|1>, <-29|31>>^n. <<1, 2>>)[1,1]:
seq(a(n), n=0..23);  # Alois P. Heinz, Dec 22 2023
  • Mathematica
    LinearRecurrence[{31, -29}, {1, 2}, 50] (* or *) CoefficientList[Series[(1 - 29 x)/(29 x^2 - 31 x + 1), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 03 2015 *)
  • Python
    print(1, end=', ')
print(2, end=', ')
an = [1,2]
for n in range(2,26):
  print(31*an[n-1]-29*an[n-2], end=', ')
  an.append(31*an[n-1]-29*an[n-2])

Formula

G.f.: (1-29*x)/(29*x^2-31*x+1). - Vincenzo Librandi, Jun 03 2015
E.g.f.: exp(31*x/2)*(65*cosh(13*sqrt(5)*x/2) - 27*sqrt(5)*sinh(13*sqrt(5)*x/2))/65. - Stefano Spezia, Aug 31 2025
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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