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 1-7 of 7 results.

A169693 A169690 union A169691.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 12, 13, 18, 20, 21, 29, 32, 33, 34, 47, 52, 54, 55, 76, 84, 87, 88, 89, 123, 136, 141, 143, 144, 199, 220, 228, 231, 232, 233, 322, 356, 369, 374, 376, 377, 521, 576, 597, 605, 608, 609, 610, 843, 932, 966, 979, 984, 986, 987, 1364, 1508, 1563, 1584
Offset: 1

Views

Author

N. J. A. Sloane, Apr 14 2010

Keywords

A169692 Numbers that are in neither A169690 nor A169691.

Original entry on oeis.org

6, 9, 10, 14, 15, 16, 17, 19, 22, 23, 24, 25, 26, 27, 28, 30, 31, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 85, 86, 90, 91, 92, 93, 94, 95, 96, 97, 98
Offset: 1

Views

Author

N. J. A. Sloane, Apr 14 2010

Keywords

Crossrefs

Cf. A169690, A169691.

A169677 The first of a pair of sequences A and B with property that all the differences |a_i - b_j| are distinct - for precise definition see Comments lines.

Original entry on oeis.org

0, 1, 7, 18, 35, 59, 88, 125, 178, 233, 285, 344, 352, 442, 557, 675, 796, 797, 957, 1011, 1220, 1411, 1564, 1579, 1888, 2120, 2152, 2503, 2829, 2953, 3393, 3464, 3593, 3724, 4237, 4956, 5310, 5388, 5968, 6478, 6756, 7344, 7698, 8004, 8182
Offset: 1

Views

Author

R. K. Guy and N. J. A. Sloane, Mar 27 2010

Keywords

Comments

Consider pairs of sequences A = a_1 a_2 a_3 a_4 ... and B = b_1 b_2 b_3 ... such that
1: All the terms are nonnegative integers
2: The terms of A are strictly increasing
3: The terms of B are strictly increasing
4: All the numbers |a_i - b_j| are distinct
5: The terms are computed in the following order: a(1), b(1), a(2), b(2), ..., b(n-1), a(n), b(n), a(n+1), ... and always the smallest value is chosen that satisfies constraints 1-4.
Computed by Alois P. Heinz and Wouter Meeussen, Mar 27 2010

Links

Crossrefs

Cf. A169678, A169679, A169680, A169690-A169693.

Programs

  • Maple
    # Maple program from Alois P. Heinz:
ab:=proc() false end: ab(0):=true:
a:= proc(n) option remember;
local ok,i,k,s;
if n=1 then 0
else b(n-1);
for k from a(n-1)+1 do
ok:=true;
for i from 1 to n-1 do
if ab(abs(k-b(i))) then ok:= false; break fi
od;
if ok then s:={};
for i from 1 to n-1 do
s:= s union {abs(k-b(i))};
od
fi;
if ok and nops(s)=n-1 then break fi
od;
for i from 1 to n-1 do
ab(abs(k-b(i))):=true
od;
k
fi
end;
b:= proc(n) option remember;
local ok,i,k,s;
if n=1 then 0
else a(n);
for k from b(n-1)+1 do
ok:=true;
for i from 1 to n do
if ab(abs(k-a(i))) then ok:= false; break fi
od;
if ok then s:={};
for i from 1 to n do
s:= s union {abs(k-a(i))};
od
fi;
if ok and nops(s)=n then break fi
od;
for i from 1 to n do
ab(abs(k-a(i))):=true
od;
k
fi
end;
seq(a(n), n=1..80);
seq(b(n), n=1..80);
  • Mathematica
    ClearAll[ab, a, b]; ab[] = False; ab[0] = True; a[n] := a[n] = Module[{ ok, i, k, s}, If[ n == 1 , 0, b[n-1]; For[ k = a[n-1] + 1 , True, k++, ok = True; For[ i = 1 , i <= n-1, i++, If[ ab[Abs[k - b[i]]] , ok = False; Break[] ]]; If[ ok , s = {}; For[ i=1 , i <= n-1 , i++, s = s ~Union~ {Abs[k - b[i]]};]]; If[ ok && (Length[s] == n-1) , Break[] ]]; For[ i=1 , i <= n-1 , i++, ab[Abs[k - b[i]]] = True]; k]]; b[n_] := b[n] = Module[{ ok, i, k, s}, If[ n == 1 , 0, a[n]; For[ k = b[n-1] + 1 , True, k++, ok = True; For[ i=1 , i <= n, i++, If[ ab[Abs[k - a[i]]] , ok = False; Break[] ]]; If[ ok , s = {}; For[ i=1 , i <= n , i++, s = s ~Union~ {Abs[k - a[i]]};]]; If[ ok && Length[s] == n , Break[] ]]; For[ i=1 , i <= n, i++, ab[Abs[k - a[i]]] := True]; k]]; Table[a[n], {n, 1, 45}] (* Jean-François Alcover, Aug 13 2012, translated from Alois P. Heinz's Maple program *)

Extensions

Comments clarified by Zak Seidov and Alois P. Heinz, Apr 13 2010.

A169678 The second of a pair of sequences A and B with property that all the differences |a_i - b_j| are distinct - for precise definition see Comments lines in A169677.

Original entry on oeis.org

0, 3, 12, 26, 45, 72, 105, 149, 199, 255, 316, 392, 401, 502, 596, 733, 865, 891, 1086, 1119, 1311, 1330, 1646, 1773, 2011, 2324, 2371, 2554, 2692, 3055, 3258, 3820, 3960, 4063, 4606, 5126, 5515, 5535, 6228, 6233, 7134, 7515, 7861, 8619
Offset: 1

Views

Author

R. K. Guy and N. J. A. Sloane, Mar 27 2010

Keywords

Comments

Computed by Alois P. Heinz and Wouter Meeussen, Mar 27 2010

Links

Crossrefs

Cf. A169677, A169679, A169680, A169690-A169693.

A169691 Let T be the sequence Fibonacci(2n+1), n>=0 (cf. A001519); sequence lists the differences T(j)-T(i) for i

Original entry on oeis.org

1, 3, 4, 8, 11, 12, 21, 29, 32, 33, 55, 76, 84, 87, 88, 144, 199, 220, 228, 231, 232, 377, 521, 576, 597, 605, 608, 609, 987, 1364, 1508, 1563, 1584, 1592, 1595, 1596, 2584, 3571, 3948, 4092, 4147, 4168, 4176, 4179, 4180, 6765, 9349, 10336, 10713, 10857, 10912, 10933
Offset: 1

Views

Author

N. J. A. Sloane, Apr 14 2010

Keywords

Comments

See Comments in A169691.

Crossrefs

Cf. A000045, A001519, A169690, A169692, A169693, A169677-A169680.

Programs

  • Mathematica
    Module[{nn=31,fbs},fbs=Fibonacci[Range[1,nn,2]];Sort[Flatten[Table[ fbs[[n]]- Take[fbs,n-1],{n,nn/2}]]]] (* Harvey P. Dale, Aug 30 2015 *)

A175567 (n!)^2 modulo n(n+1)/2.

Original entry on oeis.org

0, 1, 0, 6, 0, 15, 0, 0, 0, 45, 0, 66, 0, 0, 0, 120, 0, 153, 0, 0, 0, 231, 0, 0, 0, 0, 0, 378, 0, 435, 0, 0, 0, 0, 0, 630, 0, 0, 0, 780, 0, 861, 0, 0, 0, 1035, 0, 0, 0, 0, 0, 1326, 0, 0, 0, 0, 0, 1653, 0, 1770, 0, 0, 0, 0, 0, 2145, 0, 0, 0, 2415, 0, 2556, 0, 0, 0, 0, 0, 3003, 0, 0, 0, 3321
Offset: 1

Views

Author

John W. Layman, Jul 12 2010

Keywords

Comments

It appears that if n is one less than an odd prime then (n!)^2 modulo n(n+1)/2 is n(n-1)/2 else 0. This result appears to hold for any even power of n!. See A119690 for similar results related to odd powers of n!.

Links

Crossrefs

Cf. A119690, A169690, A175553.

Programs

  • Mathematica
    Table[Mod[(n!)^2, (n^2 + n)/2], {n, 100}] (* Vincenzo Librandi, Jul 10 2014 *)
Table[PowerMod[n!,2,(n(n+1))/2],{n,100}] (* Harvey P. Dale, Aug 27 2016 *)
  • PARI
    a(n) = (n!)^2 % (n*(n+1)/2); \\ Michel Marcus, Jul 09 2014

A169694 Numbers of the form Fibonacci(i) + Fibonacci(j), where i and j have opposite parity and |i-j| > 1.

Original entry on oeis.org

2, 4, 5, 6, 9, 10, 13, 14, 16, 22, 23, 26, 34, 35, 37, 42, 56, 57, 60, 68, 89, 90, 92, 97, 110, 145, 146, 149, 157, 178, 233, 234, 236, 241, 254, 288, 378, 379, 382, 390, 411, 466, 610, 611, 613, 618, 631, 665, 754, 988, 989, 992, 1000, 1021, 1076, 1220, 1597, 1598, 1600
Offset: 1

Views

Author

N. J. A. Sloane, Apr 14 2010

Keywords

Comments

By Zeckendorf's theorem, a number has at most one representation in this form (cf. A035517).

Examples

			5 = Fibonacci(0)+Fibonacci(5).

Crossrefs

Cf. A000045, A035517, A169689, A169690.
Showing 1-7 of 7 results.

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

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