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: Casey Mongoven

Casey Mongoven's wiki page.

Casey Mongoven has authored 95 sequences. Here are the ten most recent ones:

A241953 Number of possible representations of n as a sum of distinct positive integers from the Fibonacci-type sequences 2,1,3,4,7,11,... and 0,3,3,6,9,15,... (A000032 and A022086).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 5, 6, 6, 7, 8, 8, 9, 11, 10, 13, 13, 14, 16, 17, 16, 19, 21, 19, 24, 24, 25, 27, 30, 28, 32, 34, 33, 38, 37, 39, 42, 45, 42, 49, 48, 48, 55, 54, 55, 59, 63, 60, 68, 66, 68, 74, 74, 76, 81, 82, 81, 91, 86, 89, 97, 96, 97, 105, 104, 104, 114, 110, 113, 120, 120, 123, 130, 128, 131, 140, 137, 141, 149, 146
Offset: 1

Views

Author

Casey Mongoven, May 03 2014

Keywords

Examples

			a(10) = 6 because 10 can be represented in 6 possible ways as a sum of integers in the set {1,2,3,4,6,7,9,11,15,...}: 9+1, 7+3, 7+2+1, 6+4, 6+3+1, 4+3+2+1.

Links

  • D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322.

Crossrefs

Cf. A000032, A022086, A067595.

A241952 Number of possible representations of n as a sum of distinct positive integers from the Fibonacci-type sequences 2,1,3,4,7,11,... and 0,2,2,4,6,10,16,... (A000032 and A118658).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 4, 6, 6, 6, 8, 8, 7, 10, 11, 11, 12, 14, 15, 15, 17, 17, 17, 19, 21, 22, 24, 25, 26, 28, 29, 30, 31, 34, 35, 36, 40, 40, 39, 43, 44, 44, 47, 50, 52, 53, 57, 58, 58, 61, 63, 65, 68, 70, 73, 76, 76, 80, 81, 82, 86, 88, 92, 93, 95, 99, 99, 101, 104, 105, 108, 111, 115, 118, 119, 124, 126, 127, 133, 134, 137, 142, 143, 149
Offset: 1

Views

Author

Casey Mongoven, May 03 2014

Keywords

Examples

			a(10) = 6 because 10 can be represented in 6 possible ways as a sum of integers in the set {1,2,3,4,6,7,10,11,16,...}: 10, 7+3, 7+2+1, 6+4, 6+3+1, 4+3+2+1.

Links

  • D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322.

Crossrefs

Cf. A118658, A000032, A067595.

A241951 Number of possible representations of n as a sum of distinct positive integers from the Fibonacci and Lucas sequences 0,1,1,2,3,5,8,13,... and 2,1,3,4,7,11,... (A000045 and A000032).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 12, 15, 15, 16, 19, 19, 21, 22, 24, 26, 26, 28, 31, 31, 33, 35, 37, 40, 40, 44, 45, 46, 51, 51, 54, 57, 58, 61, 62, 65, 70, 69, 72, 76, 76, 81, 81, 86, 90, 89, 95, 97, 100, 105, 105, 110, 114, 114, 121, 121, 126, 133, 131, 138, 139, 142, 149, 147, 154, 160, 159, 165, 167
Offset: 0

Views

Author

Casey Mongoven, May 03 2014

Keywords

Examples

			a(10) = 6 because 10 can be represented in 6 possible ways as a sum of integers in the set {1,2,3,4,5,7,8,11,13,...}: 8+2, 7+3, 7+2+1, 5+4+1, 5+3+2, 4+3+2+1.

Links

Crossrefs

Cf. A000045, A000032, A000119.

Extensions

a(0)=1 from Alois P. Heinz, Sep 16 2015

A241950 Number of possible representations of n as a sum of distinct positive integers from the Fibonacci-type sequences 0,2,2,4,6,10,16,... and 0,3,3,6,9,15,... (A118658 and A022086).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 1, 3, 2, 2, 3, 3, 2, 5, 4, 3, 5, 6, 4, 6, 6, 4, 7, 8, 7, 7, 10, 8, 10, 11, 9, 10, 12, 12, 11, 13, 11, 14, 14, 15, 15, 16, 17, 19, 18, 17, 20, 19, 20, 22, 22, 20, 26, 25, 23, 27, 27, 25, 29, 30, 24, 31, 30, 29, 31, 34, 32, 35, 39, 34, 39, 39, 39, 39, 42, 39, 44, 44, 43, 47, 47, 48, 51, 51, 48, 56, 52, 53, 55, 56, 54, 61, 62, 56, 66
Offset: 0

Views

Author

Casey Mongoven, May 03 2014

Keywords

Examples

			a(9) = 3 because 9 can be represented in 3 possible ways as a sum of integers in the set {2,3,4,6,9,10,15,16,...}: 9, 6+3, 4+3+2.

Links

Crossrefs

Cf. A022086, A118658, A000119.

Extensions

a(0)=1 from Alois P. Heinz, Sep 16 2015

A241949 Number of possible representations of n as a sum of distinct positive integers from the Fibonacci-type sequences 0,1,1,2,3,5,8,13,... and 0,3,3,6,9,15,... (A000045 and A022086).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 4, 5, 4, 6, 5, 5, 8, 8, 9, 10, 10, 10, 11, 12, 13, 15, 18, 16, 17, 19, 17, 22, 24, 22, 26, 26, 24, 29, 28, 30, 34, 35, 35, 35, 38, 38, 41, 46, 43, 46, 52, 46, 52, 54, 51, 59, 60, 58, 63, 63, 64, 67, 68, 71, 71, 80, 78, 76, 85, 80, 84, 96, 87, 94, 102, 93, 102, 102, 101, 111, 114, 115, 115, 117, 121, 119, 129
Offset: 0

Views

Author

Casey Mongoven, May 03 2014

Keywords

Examples

			a(10) = 4 because 10 can be represented in 4 possible ways as a sum of integers in the set {1,2,3,5,6,8,9,13,15,...}: 9+1, 8+2, 6+3+1, 5+3+2.

Links

Crossrefs

Cf. A022086, A000045, A000119.

Extensions

a(0)=1 from Alois P. Heinz, Sep 16 2015

A241948 Number of possible representations of n as a sum of distinct positive integers from the Fibonacci-type sequences 0,1,1,2,3,5,8,13,... and 0,2,2,4,6,10,16,... (A000045 and A118658).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 8, 10, 11, 12, 14, 14, 16, 18, 18, 21, 22, 23, 26, 26, 29, 31, 32, 35, 36, 39, 41, 41, 46, 47, 49, 53, 52, 57, 60, 60, 65, 66, 70, 74, 73, 79, 81, 84, 89, 88, 94, 97, 97, 105, 105, 109, 115, 113, 121, 124, 125, 132, 132, 139, 143, 141, 151, 152, 157, 164, 161, 171, 175
Offset: 0

Views

Author

Casey Mongoven, May 03 2014

Keywords

Examples

			a(10) = 7 because 10 can be represented in 7 possible ways as a sum of integers in the set {1,2,3,4,5,6,8,10,13,16,...}: 10, 8+2, 6+4, 6+3+1, 5+4+1, 5+3+2, 4+3+2+1.

Links

Crossrefs

Cf. A118658, A000045, A000119.

Extensions

a(0)=1 from Alois P. Heinz, Sep 16 2015

A227539 Signature sequence of Soldner's constant (A070769).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 5, 1, 4, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 8, 1, 4, 7, 3, 6, 9, 2, 5, 8, 1, 4, 7, 10, 3, 6, 9, 2, 5, 8, 11, 1, 4, 7, 10, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 13, 3, 6, 9, 12, 2, 5, 8, 11, 14, 1, 4, 7, 10, 13, 3, 6, 9, 12, 15, 2, 5, 8
Offset: 1

Views

Author

Casey Mongoven, Jul 16 2013

Keywords

Comments

Arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x; the sequence of j's is the signature of 1/x.
The plot looks surprisingly regular. - T. D. Noe, Jul 23 2013
Where does this first differ from A133334? - R. J. Mathar, Jul 30 2013

References

  • Clark Kimberling, Fractal Sequences and Interspersions, Ars Combinatoria 45 (1997) 157-168.

Links

Crossrefs

Cf. A070769, A099803.

Programs

  • Mathematica
    x = FindRoot[LogIntegral[x] == 0, {x, 2}, WorkingPrecision -> 105][[1,2]]; Take[Transpose[Sort[Flatten[Table[{i + j*x, i}, {i, 30}, {j, 20}], 1], #1[[1]] < #2[[1]] &]][[2]], 100] (* T. D. Noe, Jul 23 2013 *)

A223489 a(n) = number of missing residues in the Lucas sequence mod the n-th prime number.

Original entry on oeis.org

0, 0, 1, 0, 4, 1, 1, 7, 4, 19, 12, 9, 22, 10, 32, 9, 22, 33, 16, 27, 17, 30, 20, 65, 17, 66, 24, 74, 61, 73, 30, 49, 37, 106, 77, 114, 33, 40, 40, 49, 67, 119, 72, 49, 49, 183, 181, 54, 56, 149, 205, 90, 138, 94, 61, 178, 149, 102, 73, 254, 70, 81, 264, 117, 69
Offset: 1

Views

Author

Casey Mongoven, Mar 20 2013

Keywords

Comments

The Lucas numbers mod n for any n are periodic - see A106291 for period lengths.

Examples

			The 5th prime number is 11. The Lucas sequence mod 11 is {2,1,3,4,7,0,7,7,3,10,2,1,3,...} - a periodic sequence. There are 4 residues which do not occur in this sequence, namely {5,6,8,9}. So a(5) = 4.

References

  • V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.

Links

Crossrefs

Cf. A137751.

Programs

  • Mathematica
    pisano[n_] := Module[{a = {2, 1}, a0, k = 0, s}, If[n == 1, 1, a0 = a; Reap[While[k++; s = Mod[Plus @@ a, n]; Sow[s]; a[[1]] = a[[2]]; a[[2]] = s; a != a0]][[2, 1]]]]; Join[{2}, Table[u = Union[pisano[n]]; mx = Max[u]; Length[Complement[Range[0,mx], u]], {n, Prime[Range[2, 100]]}]] (* T. D. Noe, Mar 22 2013 *)

A223488 Number of distinct residues in the Lucas sequence mod the n-th prime.

Original entry on oeis.org

2, 3, 4, 7, 7, 12, 16, 12, 19, 10, 19, 28, 19, 33, 15, 44, 37, 28, 51, 44, 56, 49, 63, 24, 80, 35, 79, 33, 48, 40, 97, 82, 100, 33, 72, 37, 124, 123, 127, 124, 112, 62, 119, 144, 148, 16, 30, 169, 171, 80, 28, 149, 103, 157, 196, 85, 120, 169, 204, 27, 213, 212
Offset: 1

Views

Author

Casey Mongoven, Mar 20 2013

Keywords

Comments

The Lucas numbers mod n for any n are periodic; see A106291 for period lengths.

Examples

			The 5th prime number is 11. The Lucas sequence mod 11 is {2,1,3,4,7,0,7,7,3,10,2,1,3,...} - a periodic sequence. There are 7 distinct residues in this sequence, namely {0,1,2,3,4,7,10}. So a(5) = 7.

References

  • V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.

Links

Crossrefs

Cf. A137750.

Programs

  • Mathematica
    pisano[n_] := Module[{a = {2, 1}, a0, k = 0, s}, If[n == 1, 1, a0 = a; Reap[While[k++; s = Mod[Plus @@ a, n]; Sow[s]; a[[1]] = a[[2]]; a[[2]] = s; a != a0]][[2, 1]]]]; Join[{2}, Table[u = Union[pisano[n]]; Length[u], {n, Prime[Range[2, 100]]}]] (* T. D. Noe, Mar 22 2013 *)

A223487 Number of missing residues in Lucas sequence mod n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 4, 2, 1, 0, 8, 5, 1, 7, 7, 10, 8, 8, 4, 10, 13, 2, 0, 8, 19, 16, 12, 10, 16, 14, 22, 21, 9, 25, 15, 30, 22, 16, 10, 24, 28, 25, 32, 31, 12, 26, 20, 16, 9, 25, 39, 28, 28, 38, 22, 42, 33, 41, 30, 22, 49, 32, 16, 42, 36, 44, 27, 55
Offset: 1

Views

Author

Casey Mongoven, Mar 20 2013

Keywords

Comments

The Lucas numbers mod n for any n are periodic - see A106291 for period lengths.

Links

Crossrefs

Cf. A118965.

Programs

  • Mathematica
    pisano[n_] := Module[{a = {2, 1}, a0, k = 0, s, t}, If[n == 1, 1, a0 = a; t = a; While[k++; s = Mod[Plus @@ a, n]; AppendTo[t, s]; a[[1]] = a[[2]]; a[[2]] = s; a != a0]; t]]; Join[{0, 0}, Table[u = Union[pisano[n]]; mx = Max[u]; Length[Complement[Range[0, mx], u]], {n, 3, 100}]] (* T. D. Noe, Mar 22 2013 *)

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