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 22 results. Next

A295999 Decimal expansion of the limiting ratio of terms in A296000.

Original entry on oeis.org

3, 6, 5, 1, 1, 8, 8, 3, 7, 4, 4, 4, 9, 6, 0, 8, 4, 1, 3, 6, 8, 8, 2, 1, 0, 4, 1, 1, 9, 2, 0, 1, 1, 2, 0, 2, 0, 1, 4, 8, 6, 0, 7, 8, 3, 7, 3, 3, 1, 4, 0, 8, 8, 2, 2, 6, 4, 0, 0, 7, 6, 8, 7, 0, 2, 2, 5, 9, 7, 3, 0, 4, 1, 6, 4, 1, 9, 2, 0, 4, 5, 4, 1, 5, 5, 5
Offset: 1

Views

Author

Clark Kimberling, Dec 04 2017

Keywords

Comments

A296000(n)/A296000(n-1) -> 3.65118837444960841368821041192011...
See A296000 for a guide to related sequences and limiting ratios.

Crossrefs

Cf. A296000.

Programs

  • Mathematica
    $RecursionLimit = Infinity;
mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 3; b[0] = 2; a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]  (* A296000 *)
t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
Take[RealDigits[Last[t], 10][[1]], 100]  (* A295999 *)

A296245 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n)^2, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 28, 66, 143, 273, 497, 870, 1488, 2502, 4159, 6857, 11241, 18354, 29884, 48562, 78807, 127769, 207017, 335270, 542816, 878662, 1422103, 2301441, 3724273, 6026555, 9751728, 15779244, 25531996, 41312329, 66845481, 108159035, 175005812, 283166216
Offset: 0

Views

Author

Clark Kimberling, Dec 10 2017

Keywords

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
*****
Guide to related sequences, each determined by a complementary equation and initial values (a(0),a(1); b(0),b(1),b(2)):
*****
Complementary equation a(n) = a(n-1) + a(n-2) + b(n)^2,
Initial values (1,2; 3,4,5): A296245
Initial values (1,3; 2,4,5): A296246
Initial values (1,4; 2,3,5): A296247
Initial values (2,3; 1,4,5): A296248
Initial values (2,4; 1,3,5): A296249
Initial values (3,4; 1,2,5): A296250
*****
Complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2,
Initial values (1,2; 3,4): A296251
Initial values (1,3; 2,4): A296252
Initial values (1,4; 2,3): A296253
Initial values (2,3; 1,4): A296254
Initial values (2,4; 1,3): A296255
Initial values (3,4; 1,2): A296256
*****
Complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)^2,
Initial values (1,2; 3): A296257
Initial values (1,3; 2): A296258
Initial values (2,3; 2): A296259
*****
Complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2),
Initial values (1,2; 3,4): A295367
Initial values (1,3; 2,4): A295363
Initial values (1,4; 2,3): A296262
Initial values (2,3; 1,4): A296263
Initial values (2,4; 1,3): A296264
Initial values (3,4; 1,2): A296265
*****
Complementary equation a(n) = a(n-1) + a(n-2) + b(n)*b(n-2),
Initial values (1,2; 3,4,5): A296266
Initial values (1,3; 2,4,5): A296267
Initial values (1,4; 2,3,5): A296268
Initial values (2,3; 1,4,5): A296269
Initial values (2,4; 1,3,5): A296270
Initial values (3,4; 1,2,5): A296271
*****
Complementary equation a(n) = a(n-1) + a(n-2) + b(n)*b(n-1),
Initial values (1,2; 3,4,5): A296272
Initial values (1,3; 2,4,5): A296273
Initial values (1,4; 2,3,5): A296274
Initial values (2,3; 1,4,5): A296275
Initial values (2,4; 1,3,5): A296276
Initial values (3,4; 1,2,5): A296277
*****
Complementary equation a(n) = a(n-1) + a(n-2) + b(n)*b(n-1)*b(n-2),
Initial values (1,2; 3,4,5): A296278
Initial values (1,3; 2,4,5): A296279
Initial values (1,4; 2,3,5): A296280
Initial values (2,3; 1,4,5): A296281
Initial values (2,4; 1,3,5): A296282
Initial values (3,4; 1,2,5): A296283
*****
Complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2),
Initial values (1,2; 3): A296284
Initial values (1,2; 4): A296285
Initial values (1,3; 2): A296286
Initial values (2,3; 1): A296287
*****
Complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1),
Initial values (1,2; 3,4): A296288
Initial values (1,3; 2,4): A296289
Initial values (1,4; 2,3): A296290
Initial values (2,3; 1,4): A296291
Initial values (2,4; 1,3): A296292
*****
Complementary equation a(n) = a(n-1) + a(n-2) + n*b(n),
Initial values (1,2; 3,4,5): A296293
Initial values (1,3; 2,4,5): A296294
Initial values (1,4; 2,3,5): A296295
Initial values (2,3; 1,4,5): A296296
Initial values (2,4; 1,3,5): A296297

Examples

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5;
a(2) = a(0) + a(1) + b(2)^2 = 28
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...)

Links

Crossrefs

Cf. A001622, A295862, A296000.

Programs

  • Mathematica
    a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]^2;
j = 1; While[j < 12, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}]  (* A296245 *)
Table[b[n], {n, 0, 20}] (* complement *)

Formula

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(2)^2 + f(n-2)*b(3)^2 + ... + f(2)*b(n-1)^2 + f(1)*b(n)^2, where f(n) = A000045(n), the n-th Fibonacci number.

A296220 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 10, 13, 16, 19, 22, 25, 29, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 82, 86, 91, 94, 97, 101, 106, 109, 113, 118, 121, 124, 128, 133, 136, 140, 145, 148, 151, 155, 160, 163, 167, 172, 175, 178, 182, 187, 190, 194, 199, 202, 205, 209, 214, 217, 221
Offset: 0

Views

Author

Clark Kimberling, Dec 08 2017

Keywords

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.
P. Majer proved that a(n)/n -> 4, that (a(n) - 4*n) is unbounded, and that a( ) is not a linear recurrence sequence; see the Math Overflow link and A297964. - Clark Kimberling, Feb 10 2018

Examples

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4;
a(2) = a(0)*b(1) + a(1)*b(0) = 10.
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, ...).

Links

Crossrefs

Cf. A296000.

Programs

  • Mathematica
    mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[0]*b[n - 1] + a[1]*b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 500}];  (* A296220 *)
Table[b[n], {n, 0, 20}]

A296001 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 10, 43, 185, 796, 3425, 14737, 63411, 272845, 1174000, 5051498, 21735632, 93524277, 402417118, 1731524071, 7450417675, 32057725596, 137938276110, 593522081260, 2553812262104, 10988566855385, 47281706383454, 203444160458068, 875381402033582
Offset: 0

Views

Author

Clark Kimberling, Dec 04 2017

Keywords

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> 4.302809183918588... (as in A296002). See A296000 for a guide to related sequences.

Examples

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that
a(2) = a(0)*b(1) + a(1)*b(0) = 10
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...)

Links

Crossrefs

Cf. A296000, A296002.

Programs

  • Mathematica
    mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 2; b[0] = 3; a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}];  (* A296001 *)
t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
Take[RealDigits[Last[t], 10][[1]], 100]  (* A296002 *)

Extensions

Conjectured g.f. removed by Alois P. Heinz, Jun 25 2018

A296002 Decimal expansion of the limiting ratio of terms in A296001.

Original entry on oeis.org

4, 3, 0, 2, 8, 0, 9, 1, 8, 3, 9, 1, 8, 5, 8, 8, 7, 3, 1, 1, 7, 8, 6, 3, 0, 0, 9, 8, 8, 2, 5, 5, 0, 4, 8, 3, 4, 9, 9, 1, 5, 4, 5, 9, 8, 6, 7, 0, 6, 2, 7, 0, 7, 7, 8, 0, 0, 1, 4, 3, 9, 9, 1, 2, 7, 7, 0, 4, 8, 2, 2, 7, 5, 9, 8, 8, 0, 0, 6, 0, 8, 8, 2, 6, 0, 8
Offset: 1

Views

Author

Clark Kimberling, Dec 04 2017

Keywords

Comments

A296001(n)/A296001(n-1) -> 4.302809183918588...
See A296000 for a guide to related sequences and limiting ratios.

Crossrefs

Cf. A296000, A296001.

Programs

  • Mathematica
    mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}];  (* A296001 *)
t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
Take[RealDigits[Last[t], 10][[1]], 100]  (* A296002 *)

A296003 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0), where a(0) = 2, a(1) = 4, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

2, 4, 10, 32, 94, 278, 824, 2440, 7228, 21408, 63406, 187800, 556234, 1647478, 4879574, 14452538, 42806168, 126785206, 375518042, 1112225982, 3294240212, 9757026674, 28898794076, 85593729210, 253515301048, 750872855508, 2223968505284, 6587048494582
Offset: 0

Views

Author

Clark Kimberling, Dec 07 2017

Keywords

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> 2.961844324... (as in A296004). See A296000 for a guide to related sequences.

Examples

			a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, so that
a(2) = a(0)*b(1) + a(1)*b(0) = 10
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...)

Links

Crossrefs

Cf. A296000, A296004.

Programs

  • Mathematica
    mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 2; a[1] = 4; b[0] = 1; a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}];  (* A296003 *)
t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
Take[RealDigits[Last[t], 10][[1]], 100]  (* A296004 *)

A296004 Decimal expansion of the limiting ratio of terms in A296003.

Original entry on oeis.org

2, 9, 6, 1, 8, 4, 4, 3, 2, 4, 2, 0, 3, 0, 8, 0, 2, 7, 4, 4, 1, 2, 5, 3, 8, 5, 8, 4, 2, 9, 3, 9, 3, 1, 3, 8, 1, 5, 9, 8, 5, 8, 8, 3, 7, 0, 9, 1, 9, 8, 4, 1, 4, 5, 0, 0, 4, 1, 0, 4, 8, 8, 0, 3, 3, 2, 3, 9, 6, 2, 0, 3, 9, 8, 9, 3, 3, 1, 3, 9, 4, 1, 6, 6, 2, 4
Offset: 1

Views

Author

Clark Kimberling, Dec 07 2017

Keywords

Comments

A296003(n)/A296003(n-1) -> 2.961844324.....
See A296000 for a guide to related sequences and limiting ratios.

Crossrefs

Cf. A296000, A296003.

Programs

  • Mathematica
    mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 2; a[1] = 4; b[0] = 1;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}];  (* A296003 *)
t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
Take[RealDigits[Last[t], 10][[1]], 100]  (* A296004 *)

A296005 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0), where a(0) = 2, a(1) = 3, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

2, 3, 11, 33, 104, 323, 1007, 3136, 9769, 30431, 94791, 295274, 919773, 2865082, 8924690, 27800290, 86597525, 269750118, 840267961, 2617423311, 8153238141, 25397226311, 79112015761, 246432856920, 767635009499, 2391172651130, 7448470401642, 23201884354901
Offset: 0

Views

Author

Clark Kimberling, Dec 07 2017

Keywords

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> 3.114986447390302... (as in A296006). See A296000 for a guide to related sequences.

Examples

			a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, so that
a(2) = a(0)*b(1) + a(1)*b(0) = 11
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, ...)

Links

Crossrefs

Cf. A296000, A296006.

Programs

  • Mathematica
    mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 2; a[1] = 3; b[0] = 1; a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}];  (* A296005 *)
t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
Take[RealDigits[Last[t], 10][[1]], 100]  (* A296006 *)

Extensions

Conjectured g.f. removed by Alois P. Heinz, Jun 25 2018

A296006 Decimal expansion of the limiting ratio of terms in A296005.

Original entry on oeis.org

3, 1, 1, 4, 9, 8, 6, 4, 4, 7, 3, 9, 0, 3, 0, 2, 2, 4, 0, 2, 1, 1, 6, 2, 7, 0, 8, 7, 5, 5, 5, 4, 0, 3, 2, 8, 8, 3, 6, 0, 0, 9, 2, 9, 4, 5, 6, 0, 5, 9, 4, 2, 0, 4, 1, 3, 8, 8, 0, 8, 6, 4, 2, 5, 5, 2, 0, 9, 6, 6, 0, 7, 0, 5, 7, 3, 1, 7, 7, 4, 7, 9, 9, 6, 6, 2
Offset: 1

Views

Author

Clark Kimberling, Dec 07 2017

Keywords

Comments

A296005(n)/A296005(n-1) -> 3.114986447390302...
See A296000 for a guide to related sequences and limiting ratios.

Crossrefs

Cf. A296000, A296005.

Programs

  • Mathematica
    mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 2; a[1] = 3; b[0] = 1;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}];  (* A296005 *)
t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
Take[RealDigits[Last[t], 10][[1]], 100]  (* A296006 *)

A296215 Solution of the complementary equation a(n) = a(1)*b(n-2) + a(2)*b(n-3) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 3, 6, 24, 87, 321, 1176, 4314, 15822, 58032, 212847, 780672, 2863317, 10501959, 38518662, 141277197, 518170812, 1900526031, 6970672818, 25566752964, 93772706622, 343935755925, 1261473710904, 4626782461218, 16969926331719, 62241612204120, 228287277978756
Offset: 0

Views

Author

Clark Kimberling, Dec 08 2017

Keywords

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

Examples

			a(0) =1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5
a(2) = a(1)*b(0) = 6
Complement: (b(n)) = (2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ...)

Links

Crossrefs

Cf. A296000, A296217.

Programs

  • Mathematica
    a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 1, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 200}];  (* A296215 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296216 *)
Showing 1-10 of 22 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.