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 14 results. Next

A342603 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 6*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 7, 1, 13, 7, 43, 1, 19, 13, 85, 7, 85, 43, 259, 1, 25, 19, 127, 13, 163, 85, 517, 7, 127, 85, 553, 43, 517, 259, 1555, 1, 31, 25, 169, 19, 241, 127, 775, 13, 241, 163, 1063, 85, 1027, 517, 3109, 7, 169, 127, 847, 85, 1063, 553, 3361, 43, 775, 517, 3361, 259, 3109, 1555, 9331, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2021

Keywords

Links

Crossrefs

Cf. A002487, A003464, A016921, A116528, A178243, A342633, A342634, A342635, A342636, A342637, A342638.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 6*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Mar 17 2021
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 6 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 64}]
nmax = 64; CoefficientList[Series[x Product[(1 + x^(2^k) + 6 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

Formula

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 6*x^(2^(k+1))).
a(2^n-1) = (6^n - 1)/5 = A003464(n); a(2^n) = 1; a(2^n+1) = 6*n + 1 = A016921(n). - Alois P. Heinz, Mar 17 2021

A342633 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 3*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 4, 1, 7, 4, 13, 1, 10, 7, 25, 4, 25, 13, 40, 1, 13, 10, 37, 7, 46, 25, 79, 4, 37, 25, 88, 13, 79, 40, 121, 1, 16, 13, 49, 10, 67, 37, 118, 7, 67, 46, 163, 25, 154, 79, 241, 4, 49, 37, 136, 25, 163, 88, 277, 13, 118, 79, 277, 40, 241, 121, 364, 1, 19, 16, 61, 13, 88, 49, 157
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2021

Keywords

Links

Crossrefs

Cf. A002487, A116528, A178243, A342603, A342634, A342635, A342636, A342637, A342638.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 3*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..71);  # Alois P. Heinz, Mar 17 2021
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 3 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 71}]
nmax = 71; CoefficientList[Series[x Product[(1 + x^(2^k) + 3 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

Formula

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 3*x^(2^(k+1))).

A342634 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 4*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 5, 1, 9, 5, 21, 1, 13, 9, 41, 5, 41, 21, 85, 1, 17, 13, 61, 9, 77, 41, 169, 5, 61, 41, 185, 21, 169, 85, 341, 1, 21, 17, 81, 13, 113, 61, 253, 9, 113, 77, 349, 41, 333, 169, 681, 5, 81, 61, 285, 41, 349, 185, 761, 21, 253, 169, 761, 85, 681, 341, 1365, 1, 25, 21, 101, 17
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2021

Keywords

Links

Crossrefs

Cf. A002487, A116528, A178243, A342603, A342633, A342635, A342636, A342637, A342638.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 4*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..68);  # Alois P. Heinz, Mar 17 2021
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 4 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 68}]
nmax = 68; CoefficientList[Series[x Product[(1 + x^(2^k) + 4 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

Formula

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 4*x^(2^(k+1))).
a(n) == 1 (mod 4) for n >= 1. - Hugo Pfoertner, Mar 17 2021

A342635 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 5*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 6, 1, 11, 6, 31, 1, 16, 11, 61, 6, 61, 31, 156, 1, 21, 16, 91, 11, 116, 61, 311, 6, 91, 61, 336, 31, 311, 156, 781, 1, 26, 21, 121, 16, 171, 91, 466, 11, 171, 116, 641, 61, 616, 311, 1561, 6, 121, 91, 516, 61, 641, 336, 1711, 31, 466, 311, 1711, 156, 1561, 781, 3906, 1, 31, 26
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2021

Keywords

Links

Crossrefs

Cf. A002487, A116528, A178243, A342603, A342633, A342634, A342636, A342637, A342638.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 5*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..66);  # Alois P. Heinz, Mar 17 2021
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 5 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 66}]
nmax = 66; CoefficientList[Series[x Product[(1 + x^(2^k) + 5 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

Formula

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 5*x^(2^(k+1))).
a(n) == 1 (mod 5) for n >= 1. - Hugo Pfoertner, Mar 17 2021

A342636 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 7*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 8, 1, 15, 8, 57, 1, 22, 15, 113, 8, 113, 57, 400, 1, 29, 22, 169, 15, 218, 113, 799, 8, 169, 113, 848, 57, 799, 400, 2801, 1, 36, 29, 225, 22, 323, 169, 1198, 15, 323, 218, 1639, 113, 1590, 799, 5601, 8, 225, 169, 1296, 113, 1639, 848, 5993, 57, 1198, 799, 5993, 400, 5601, 2801
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2021

Keywords

Links

Crossrefs

Cf. A002487, A116528, A178243, A342603, A342633, A342634, A342635, A342637, A342638.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 7*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..62);  # Alois P. Heinz, Mar 17 2021
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 7 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 62}]
nmax = 62; CoefficientList[Series[x Product[(1 + x^(2^k) + 7 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

Formula

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 7*x^(2^(k+1))).

A342637 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 8*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 9, 1, 17, 9, 73, 1, 25, 17, 145, 9, 145, 73, 585, 1, 33, 25, 217, 17, 281, 145, 1169, 9, 217, 145, 1233, 73, 1169, 585, 4681, 1, 41, 33, 289, 25, 417, 217, 1753, 17, 417, 281, 2393, 145, 2329, 1169, 9361, 9, 289, 217, 1881, 145, 2393, 1233, 9937, 73, 1753, 1169, 9937, 585
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2021

Keywords

Links

Crossrefs

Cf. A002487, A116528, A178243, A342603, A342633, A342634, A342635, A342636, A342638.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 8*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 17 2021
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 8 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 60}]
nmax = 60; CoefficientList[Series[x Product[(1 + x^(2^k) + 8 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

Formula

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 8*x^(2^(k+1))).
a(n) == 1 (mod 8) for n >= 1. - Hugo Pfoertner, Mar 17 2021

A342638 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 9*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 10, 1, 19, 10, 91, 1, 28, 19, 181, 10, 181, 91, 820, 1, 37, 28, 271, 19, 352, 181, 1639, 10, 271, 181, 1720, 91, 1639, 820, 7381, 1, 46, 37, 361, 28, 523, 271, 2458, 19, 523, 352, 3349, 181, 3268, 1639, 14761, 10, 361, 271, 2620, 181, 3349, 1720, 15571, 91, 2458, 1639, 15571, 820
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2021

Keywords

Links

Crossrefs

Cf. A002487, A116528, A178243, A342603, A342633, A342634, A342635, A342636, A342637.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 9*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 17 2021
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 9 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 60}]
nmax = 60; CoefficientList[Series[x Product[(1 + x^(2^k) + 9 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

Formula

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 9*x^(2^(k+1))).

A116529 a(2*n + 1) = a(n), a(2*n + 2) = 2*a(n) + a(n-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 7, 5, 12, 1, 7, 4, 9, 3, 10, 7, 17, 2, 11, 7, 16, 5, 17, 12, 29, 1, 14, 7, 15, 4, 15, 9, 22, 3, 15, 10, 23, 7, 24, 17, 41, 2, 21, 11, 24, 7, 25, 16, 39, 5, 26, 17, 39, 12, 41, 29, 70, 1, 31, 14, 29, 7, 28, 15
Offset: 0

Views

Author

Roger L. Bagula, Mar 15 2006

Keywords

Links

Crossrefs

Cf. A116528, A116552, A116553, A116554, A116555.

Programs

  • Maple
    gg:= 1:
for iter from 1 to 7 do
  gg:= convert(series(1+(x^4+2*x^2+x)*eval(gg,x=x^2), x, 2^iter+1),polynom)
od:
seq(coeff(gg,x,n),n=0..2^7); # Robert Israel, Nov 13 2017
  • Mathematica
    b[0] := 0; b[1] := 1;
b[n_?EvenQ] := b[n] = b[n/2];
b[n_?OddQ] := b[n] = 2*b[(n - 1)/2] + b[(n - 3)/2];
Table[b[n], {n, 1, 70}]
  • PARI
    \\ See links.

Formula

From G. C. Greubel, Oct 30 2016: (Start)
a(2*n + 1) = a(n), n>=1.
a(2*n + 2) = 2*a(n) + a(n-1), n>=1. (End)
G.f. g(x) satisfies g(x) = 1 + (x^4+2*x^2+x)*g(x^2). - Robert Israel, Nov 13 2017

Extensions

New name using formula, Joerg Arndt, Dec 17 2022

A116552 a(2*n+1) = 3*a(n), a(2*n+2) = 4*a(n) + a(n-1).

Original entry on oeis.org

1, 3, 4, 9, 13, 12, 19, 27, 40, 39, 61, 36, 61, 57, 88, 81, 127, 120, 187, 117, 196, 183, 283, 108, 205, 183, 280, 171, 289, 264, 409, 243, 412, 381, 589, 360, 607, 561, 868, 351, 655, 588, 901, 549, 928, 849, 1315, 324, 715, 615, 928, 549, 937, 840, 1303, 513
Offset: 0

Views

Author

Roger L. Bagula, Mar 15 2006

Keywords

Links

Crossrefs

Cf. A116528, A116529, A116553, A116554, A116555.

Programs

  • Mathematica
    b[0] := 0; b[1] := 1;
b[n_?EvenQ] := b[n] = 3*b[n/2];
b[n_?OddQ] := b[n] = 4*b[(n - 1)/2] + b[(n - 3)/2];
Table[b[n], {n, 1, 70}]

Formula

From G. C. Greubel, Oct 30 2016: (Start)
a(2*n+1) = 3*a(n).
a(2*n+2) = 4*a(n) + a(n-1). (End)

Extensions

New name using formula, Joerg Arndt, Dec 17 2022

A116553 a(2*n+1) = 5*a(n), a(2*n+2) = 6*a(n) + a(n-1).

Original entry on oeis.org

1, 5, 6, 25, 31, 30, 41, 125, 156, 155, 211, 150, 211, 205, 276, 625, 791, 780, 1061, 775, 1086, 1055, 1421, 750, 1111, 1055, 1416, 1025, 1441, 1380, 1861, 3125, 4026, 3955, 5371, 3900, 5471, 5305, 7146, 3875, 5711, 5430, 7291, 5275, 7416, 7105, 9581, 3750
Offset: 0

Views

Author

Roger L. Bagula, Mar 15 2006

Keywords

Links

Crossrefs

Cf. A116528, A116529, A116552, A116554, A116555.

Programs

  • Mathematica
    b[0] := 0; b[1] := 1;
b[n_?EvenQ] := b[n] = 5*b[n/2];
b[n_?OddQ] := b[n] = 6*b[(n - 1)/2] + b[(n - 3)/2];
Table[b[n], {n, 1, 70}]

Formula

From G. C. Greubel, Oct 30 2016: (Start)
a(2*n+1) = 5*a(n).
a(2*n+2) = 6*a(n) + a(n-1). (End)

Extensions

New name using formula, Joerg Arndt, Dec 17 2022
Showing 1-10 of 14 results. Next

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

Content is available under The OEIS End-User License Agreement.