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.

User: Philippe Deléham

Philippe Deléham's wiki page.

Philippe Deléham has authored 1178 sequences. Here are the ten most recent ones:

A370179 a(n) = floor(x*a(n-1)) for n > 0 where x = (9 + 3*sqrt(13))/2, a(0) = 1.

Original entry on oeis.org

1, 9, 89, 881, 8729, 86489, 856961, 8491049, 84132089, 833608241, 8259662969, 81839440889, 810891934721, 8034582380489, 79609268836889, 788794660956401, 7815635368139609, 77439870261864089, 767299550670033281, 7602654788387076329, 75329589051513986489
Offset: 0

Views

Author

Philippe Deléham, Apr 24 2024

Keywords

Links

Crossrefs

Cf. A057092, A090655 (x value), A370174.

Programs

  • Mathematica
    LinearRecurrence[{10,0,-9},{1,9,89},21] (* Stefano Spezia, Apr 24 2024 *)

Formula

a(n) = 10*a(n-1) - 9*a(n-3) for n > 2, a(0) = 1, a(1) = 9, a(2) = 89.
a(n) = 9*a(n-1) + 9*a(n-2) - 1.
G.f.: (1-x-x^2)/((1-x)*(1-9*x-9*x^2)).
a(n) = Sum_{k = 0..n} A370174(n,k)*8^k.
a(n) = (16*A057092(n) + 8*A057092(n-1) + 1)/17.

A370178 a(n) = floor(x*a(n-1)) for n > 0 where x = 4 + 2*sqrt(6), a(0) = 1.

Original entry on oeis.org

1, 8, 71, 631, 5615, 49967, 444655, 3956975, 35213039, 313360111, 2788585199, 24815562479, 220833181423, 1965189951215, 17488185061103, 155627000098543, 1384921481277167, 12324387851005679, 109674474658262767, 975990900074147567, 8685322997859282671
Offset: 0

Views

Author

Philippe Deléham, Apr 02 2024

Keywords

Links

Crossrefs

Cf. A057091, A090654 (x value), A370174.

Programs

  • Mathematica
    LinearRecurrence[{9,0,-8},{1,8,71},21] (* James C. McMahon, Apr 21 2024 *)

Formula

a(n) = 9*a(n-1) - 8*a(n-3) for n>2, a(0) = 1, a(1) = 8, a(2) = 71.
a(n) = 8*a(n-1) + 8*a(n-2) - 1.
G.f.: (1-x-x^2)/((1-x)*(1-8*x-8*x^2)).
a(n) = Sum_{k=0..n} A370174(n,k)*7^k.
a(n) = (7*(8-3*sqrt(6))*(4-2*sqrt(6))^n + 7*(8+3*sqrt(6))*(4+2*sqrt(6))^n + 8)/120.
a(n) = (14*A057091(n) + 7*A057091(n-1) + 1)/15.

A370177 a(n) = floor(x*a(n-1)) for n > 0 where x = (7 + sqrt(77))/2, a(0) = 1.

Original entry on oeis.org

1, 7, 55, 433, 3415, 26935, 212449, 1675687, 13216951, 104248465, 822257911, 6485544631, 51154617793, 403481136967, 3182450283319, 25101519942001, 197987791577239, 1561625180634679, 12317290805483425, 97152411902826727, 766287918958171063, 6044082316026984529
Offset: 0

Views

Author

Philippe Deléham, Mar 29 2024

Keywords

Comments

x = (7+sqrt(77))/2 = A092290 = 7.88748219...

Crossrefs

Cf. A057090, A092290, A370174.

Formula

a(n) = 8*a(n-1) - 7*a(n-3) for n > 2, a(0) = 1, a(1) = 7, a(2) = 55.
G.f.: (1-x-x^2)/(1-8*x+7*x^3).
a(n) = 7*a(n-1) + 7*a(n-2) - 1.
a(n) = (12*A057090(n) + 6*A057090(n-1) + 1)/13.
a(n) = (6*(77+8*sqrt(77))*((7+sqrt(77))/2)^n + 6*(77-8*sqrt(77))*((7-sqrt(77))/2)^n + 77)/1001.
a(n) = Sum_{k = 0..n} A370174(n,k)*6^k.

A370175 a(n) = floor(x*a(n-1)) for n > 0 where x = (5+3*sqrt(5))/2, a(0) = 1.

Original entry on oeis.org

1, 5, 29, 169, 989, 5789, 33889, 198389, 1161389, 6798889, 39801389, 233001389, 1364013889, 7985076389, 46745451389, 273652638889, 1601990451389, 9378215451389, 54901029513889, 321396224826389, 1881486271701389, 11014412482638889, 64479493771701389
Offset: 0

Views

Author

Philippe Deléham, Mar 18 2024

Keywords

Comments

x = A090550 = 1 + 3*phi = 5.854101966..., where phi is the golden ratio.

Examples

			a(0) = 1, a(1) = floor(x) = 5 where x = (5+3*sqrt(5))/2.
a(2) = floor(5*x) = 29, a(3) = floor(29*x) = 169.

Links

Crossrefs

Cf. A057088, A090550, A370174.

Programs

  • Mathematica
    NestList[Floor[#*(5 + 3*Sqrt[5])/2] &, 1, 30] (* or *)
LinearRecurrence[{6, 0, -5}, {1, 5, 29}, 30] (* Paolo Xausa, May 25 2024 *)

Formula

a(n) = 6*a(n-1) - 5*a(n-3), a(0) = 1, a(1) = 5, a(2) = 29.
a(n) = 5*a(n-1) + 5*a(n-2) - 1.
a(n) = (4*(5-2*sqrt(5))*((5-3*sqrt(5))/2)^n + 4*(5+2*sqrt(5))*((5+3*sqrt(5))/2)^n + 5)/45.
G.f.: (1 - x - x^2)/(1 - 6*x + 5*x^3).
a(n) = Sum_{k = 0..n} A370174(n,k)*4^k.
a(n) = (8*A057088(n) + 4*A057088(n-1) + 1)/9.

A370176 a(n) = floor(x*a(n-1)) for n > 0 where x = 3+sqrt(15), a(0) = 1.

Original entry on oeis.org

1, 6, 41, 281, 1931, 13271, 91211, 626891, 4308611, 29613011, 203529731, 1398856451, 9614317091, 66079041251, 454160150051, 3121435147811, 21453571787171, 147450041609891, 1013421680382371, 6965230331953571, 47871912074015651, 329022854435815331, 2261368599058985891
Offset: 0

Views

Author

Philippe Deléham, Mar 19 2024

Keywords

Comments

x = A092294 = 3+sqrt(15) = 6.872983346...

Examples

			a(0) = 1;
a(1) = floor(x) = 6 where x = 3+sqrt(15);
a(2) = floor(6*x) = 41;
a(3) = floor(41*x) = 281.

Links

Crossrefs

Cf. A057089, A092294, A370174.

Programs

  • Mathematica
    NestList[Floor[(Sqrt[15]+3)*#] &, 1, 25] (* or *)
LinearRecurrence[{7, 0, -6}, {1, 6, 41}, 25] (* Paolo Xausa, Mar 31 2024 *)

Formula

a(n) = 7*a(n-1) - 6*a(n-3), a(0) = 1, a(1) = 6, a(2) = 41.
a(n) = 6*a(n-1) + 6*a(n-2) - 1.
a(n) = ((30-7*sqrt(15))*(3-sqrt(15))^n + (30+7*sqrt(15))*(3+sqrt(15))^n + 6)/66.
G.f.: (1-x-x^2)/(1-7*x+6*x^3).
a(n) = Sum_{k = 0..n} A370174(n,k)*5^k.
a(n) = (10*A057089(n) + 5*A057089(n-1) + 1)/11.

A370174 Triangle read by rows: Riordan array (1/(1 - x), x*(1 + x)/(1 - x - x^2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 11, 15, 7, 1, 1, 19, 37, 28, 9, 1, 1, 32, 82, 87, 45, 11, 1, 1, 53, 170, 234, 169, 66, 13, 1, 1, 87, 337, 573, 535, 291, 91, 15, 1, 1, 142, 647, 1314, 1511, 1061, 461, 120, 17, 1, 1, 231, 1213, 2871, 3933, 3398, 1904, 687, 153, 19, 1
Offset: 0

Views

Author

Philippe Deléham, Feb 29 2024

Keywords

Examples

			Triangle T(n,k) begins:
      k=0   1   2   3   4   5    6
  n=0:  1;
  n=1:  1,  1;
  n=2:  1,  3,  1;
  n=3:  1,  6,  5,  1;
  n=4:  1, 11, 15,  7,  1;
  n=5:  1, 19, 37, 28,  9,  1;
  n=6:  1, 32, 82, 87, 45, 11,  1;
  ...
87 = 28 + 37 + 7 + 15.

Crossrefs

Cf. A000012 (column k=0), A000384, A001911, A005408.
Cf. A057960 (row sums), A196472, A218988.

Formula

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(n,0) = 1, T(n,k) = 0 if k > n.
Sum_{k = 0..n} T(n,k)* x^k = A000012(n), A057960(n), A196472(n+1), A218988(n-1) for x = 0, 1, 2, 3 respectively.

A370173 Riordan array (1-x-x^2, x*(1+x)).

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 0, -2, 1, 1, 0, -1, -2, 2, 1, 0, 0, -3, -1, 3, 1, 0, 0, -1, -5, 1, 4, 1, 0, 0, 0, -4, -6, 4, 5, 1, 0, 0, 0, -1, -9, -5, 8, 6, 1, 0, 0, 0, 0, -5, -15, -1, 13, 7, 1, 0, 0, 0, 0, -1, -14, -20, 7, 19, 8, 1, 0, 0, 0, 0, 0, -6, -29, -21, 20, 26, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Feb 27 2024

Keywords

Comments

Triangle T(n,k) read by rows : matrix product of A155112*A130595.
Triangle T(n,k), read by rows, given by [-1, 2, -1/2, -1/2, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Examples

			Triangle T(n,k) begins:
  1;
 -1,  1;
 -1,  0,  1;
  0, -2,  1,  1;
  0, -1, -2,  2, 1;
  0,  0, -3, -1, 3, 1;
...

Crossrefs

Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.
Cf. A084938, A130595, A155112.

Programs

  • Python
    from functools import cache
@cache
def T(n, k):
    if k > n: return 0
    if n == 0: return 1
    if k == 0: return -1 if n == 1 or n == 2 else 0
    return T(n-1, k-1) + T(n-2, k-1)
for n in range(9):
    print([T(n, k) for k in range(n+1)])  # Peter Luschny, Feb 28 2024

Formula

T(n,k) = T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = -1, T(n,0) = 0 for n>2, T(n,k) = 0 if k>n.
T(n,k) = Sum_{j = k..n} A155112(n,j)*A130595(j,k).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A155020(n), A155116(n), A155117(n), A155119(n), A155127(n), A155130(n), A155132(n), A155144(n), A155157(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.

A369490 a(n) = 3^(n+1) + 2*(-2)^(n+1).

Original entry on oeis.org

-1, 17, 11, 113, 179, 857, 1931, 7073, 18659, 61097, 173051, 539633, 1577939, 4815737, 14283371, 43177793, 128878019, 387944777, 1161212891, 3488881553, 10456158899, 31389448217, 94126401611, 282463090913, 847221500579, 2542000046057
Offset: 0

Views

Author

Philippe Deléham, Jan 24 2024

Keywords

Links

Crossrefs

Cf. A000079, A000244, A003063, A015441, A085279.

Programs

  • Mathematica
    LinearRecurrence[{1,6},{-1,17},26] (* James C. McMahon, Jan 30 2024 *)
  • Python
    def A369490(n): return 3**(n+1)+(1<Chai Wah Wu, Feb 25 2024

Formula

a(n) = a(n-1) + 6*a(n-2); a(0) = -1, a(1) = 17.
G.f.: (18*x-1)/((1+2*x)*(1-3*x)).
a(2*n) = A003063(2*n+2).
a(2*n+1) = A085279(2*n+3).
a(n) = 18*A015441(n) - A015441(n+1).

A369404 a(n) = 3*2^n + 5*(-1)^n.

Original entry on oeis.org

8, 1, 17, 19, 53, 91, 197, 379, 773, 1531, 3077, 6139, 12293, 24571, 49157, 98299, 196613, 393211, 786437, 1572859, 3145733, 6291451, 12582917, 25165819, 50331653, 100663291, 201326597, 402653179, 805306373, 1610612731, 3221225477, 6442450939, 12884901893
Offset: 0

Views

Author

Philippe Deléham, Jan 22 2024

Keywords

Links

Crossrefs

Cf. A000079, A001045, A014551, A033999.

Programs

  • Mathematica
    LinearRecurrence[{1,2},{8,1},33] (* James C. McMahon, Jan 31 2024 *)
  • Python
    def A369404(n): return (3<Chai Wah Wu, Feb 25 2024

Formula

a(n) = 3*A000079(n) + 5*A033999(n).
a(n) = 4*A014551(n) - 3*A001045(n).
a(n) = a(n-1) + 2*a(n-2).
G.f.: (8 - 7*x)/((1 + x)*(1 - 2*x)).

A369153 Numbers k such that gcd(2*k^7+1, 3*k^3+2) > 1.

Original entry on oeis.org

435, 1598, 2761, 3924, 5087, 6250, 7413, 8576, 9739, 10902, 12065, 13228, 14391, 15554, 16717, 17880, 19043, 20206, 21369, 22532, 23695, 24858, 26021, 27184, 28347, 29510, 30673, 31836, 32999, 34162, 35325, 36488, 37651, 38814, 39977, 41140, 42303, 43466
Offset: 0

Views

Author

Philippe Deléham, Jan 15 2024

Keywords

Comments

This GCD is 1163 if k == 435 (mod 1163), or 1 otherwise.

Examples

			a(0) = 435, 2*435^7+1 = 5894606169966093751 and 3*435^3+2 = 246938627, gcd(5894606169966093751, 246938627) = 1163.

Links

Programs

Formula

a(n) = 435 + 1163*n.
a(n) = 2*a(n-1) - a(n-2).
G.f.: (435 + 728*x)/(1 - x)^2.

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

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