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 1-6 of 6 results.

A245869 T(n,k)=Number of length n+2 0..k arrays with some pair in every consecutive three terms totalling exactly k.

Original entry on oeis.org

6, 19, 10, 36, 45, 16, 61, 100, 103, 26, 90, 193, 256, 239, 42, 127, 318, 549, 676, 553, 68, 168, 493, 960, 1629, 1764, 1281, 110, 217, 712, 1579, 3102, 4753, 4624, 2967, 178, 270, 993, 2368, 5515, 9726, 13961, 12100, 6873, 288, 331, 1330, 3433, 8840, 18505, 30900
Offset: 1

Views

Author

R. H. Hardin, Aug 04 2014

Keywords

Comments

Table starts
.....6.......19........36.........61..........90..........127..........168
....10.......45.......100........193.........318..........493..........712
....16......103.......256........549.........960.........1579.........2368
....26......239.......676.......1629........3102.........5515.........8840
....42......553......1764.......4753........9726........18505........31176
....68.....1281......4624......13961.......30900........63241.......113024
...110.....2967.....12100......40901.......97602.......214315.......404264
...178.....6873.....31684.....119953......309078.......729097......1455496
...288....15921.....82944.....351649......977664......2475985......5223552
...466....36881....217156....1031057.....3094038......8415217.....18775816
...754....85435....568516....3022933.....9789654.....28590415.....67437448
..1220...197911...1488400....8863117....30977796.....97151683....242306240
..1974...458463...3896676...25986061....98020170....330100459....870461352
..3194..1062035..10201636...76189749...310161870...1121650903...3127322696
..5168..2460217..26708224..223384017...981426624...3811203385..11235107264
..8362..5699123..69923044..654949861..3105480558..12950003383..40363689352
.13530.13202089.183060900.1920277409..9826505742..44002376953.145010699592
.21892.30582803.479259664.5630150189.31093507092.149514426895.520968428032

Examples

			Some solutions for n=6 k=4
..1....4....0....4....0....1....2....3....1....2....0....3....3....0....2....4
..4....2....1....1....4....2....4....1....0....3....1....0....3....4....1....3
..0....2....4....0....3....2....0....3....3....1....3....1....1....2....3....1
..4....0....0....4....0....1....4....0....1....1....1....3....0....0....4....3
..4....2....4....4....4....2....1....4....4....3....2....1....4....4....1....2
..0....2....0....0....3....2....0....0....0....3....2....2....4....3....0....2
..2....0....4....4....0....1....4....4....3....1....4....3....0....1....4....2
..4....4....0....1....1....2....3....0....1....4....0....1....0....2....1....2

Links

Crossrefs

Column 1 is A006355(n+4)
Column 3 is A206981(n+2)
Row 1 is A090381.

Formula

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1) +a(n-2) -a(n-4) -a(n-5)
k=3: a(n) = 2*a(n-1) +2*a(n-2) -a(n-3)
k=4: a(n) = 3*a(n-1) +a(n-2) -a(n-3) -5*a(n-4) -8*a(n-5) +3*a(n-6)
k=5: a(n) = 2*a(n-1) +4*a(n-2) -a(n-3)
k=6: a(n) = 3*a(n-1) +3*a(n-2) -a(n-3) -9*a(n-4) -24*a(n-5) +5*a(n-6)
k=7: a(n) = 2*a(n-1) +6*a(n-2) -a(n-3)
k=8: a(n) = 3*a(n-1) +5*a(n-2) -a(n-3) -13*a(n-4) -48*a(n-5) +7*a(n-6)
k=9: a(n) = 2*a(n-1) +8*a(n-2) -a(n-3)
Empirical for row n:
n=1: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4)
n=2: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5)
n=3: a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6)
n=4: a(n) = 3*a(n-1) -a(n-2) -5*a(n-3) +5*a(n-4) +a(n-5) -3*a(n-6) +a(n-7)
n=5: a(n) = 2*a(n-1) +2*a(n-2) -6*a(n-3) +6*a(n-5) -2*a(n-6) -2*a(n-7) +a(n-8)
n=6: a(n) = 3*a(n-1) -8*a(n-3) +6*a(n-4) +6*a(n-5) -8*a(n-6) +3*a(n-8) -a(n-9)
n=7: a(n) = 2*a(n-1) +3*a(n-2) -8*a(n-3) -2*a(n-4) +12*a(n-5) -2*a(n-6) -8*a(n-7) +3*a(n-8) +2*a(n-9) -a(n-10)
From Robert Israel, Aug 06 2024: (Start) For odd k, T(n,k) = 2 T(n-1,k) + (k-1) T(n-2,k) - T(n-3,k).
For even k, T(n,k) = 3 T(n-1,k) + (k-3) T(n-2,k) - T(n-3,k) + (2 k - 3) T(n-4,k) - k (k-2) T(n-5,k) + (k-1) T(n-6,k).
See links. (End)

A090385 Maximal number of vertices of polytope P_T associated with any tree having n nodes.

Original entry on oeis.org

7, 8, 11, 14, 15, 20, 21, 22, 25, 28, 31, 32, 34
Offset: 3

Views

Author

N. J. A. Sloane, Jan 30 2004

Keywords

Links

Crossrefs

Cf. A090381-A090384.

A245556 Irregular triangle read by rows: T(n,k) (n>=0, 0 <= k <= 2n) = number of triples (u,v,w) with entries in the range 0 to n which have some pair adding up to k.

Original entry on oeis.org

1, 4, 6, 4, 7, 12, 19, 12, 7, 10, 18, 28, 36, 28, 18, 10, 13, 24, 37, 48, 61, 48, 37, 24, 13, 16, 30, 46, 60, 76, 90, 76, 60, 46, 30, 16, 19, 36, 55, 72, 91, 108, 127, 108, 91, 72, 55, 36, 19, 22, 42, 64, 84, 106, 126, 148, 168, 148, 126, 106, 84, 64, 42, 22
Offset: 0

Views

Author

N. J. A. Sloane, Aug 04 2014

Keywords

Examples

			Triangle begins:
[1]
[4, 6, 4]
[7, 12, 19, 12, 7]
[10, 18, 28, 36, 28, 18, 10]
[13, 24, 37, 48, 61, 48, 37, 24, 13]
[16, 30, 46, 60, 76, 90, 76, 60, 46, 30, 16]
[19, 36, 55, 72, 91, 108, 127, 108, 91, 72, 55, 36, 19]
[22, 42, 64, 84, 106, 126, 148, 168, 148, 126, 106, 84, 64, 42, 22]
...
See A245557 for specific examples; also the Example section of A090381 for some of the T(10,10)= 331 triples with n=k=10.

Crossrefs

Rows are the partial sums of the rows of A245557.
Main "spine" of triangle is A090381.
Row sums are A005915.

Programs

  • Maple
    with(LinearAlgebra);
M:=10; A:=Array(0..M,0..2*M); B:=Array(0..M,0..2*M);
for n from 0 to M do
for i from 0 to n do for j from 0 to n do for k from 0 to n do
  s1:={i+j,i+k,j+k}; s1:=convert(s1,list); m1:=max(i,j,k);
    for r1 from 1 to nops(s1) do
       s:=s1[r1]; A[n,s] := A[n,s]+1;
       if (m1=n) then B[n,s] := B[n,s]+1; fi;
                              od:
od: od: od: od:
lprint("A245556");
for i from 0 to M do lprint([seq(A[i,j],j=0..2*i)]); od:
lprint("A245557");
for i from 0 to M do lprint([seq(B[i,j],j=0..2*i)]); od:

A090382 Number of minimal generators for toric ideal associated with path with n nodes.

Original entry on oeis.org

6, 32, 102, 259, 540, 1041, 1842, 3170, 5286
Offset: 3

Views

Author

N. J. A. Sloane, Jan 30 2004

Keywords

Links

Crossrefs

Cf. A090381-A090385.

A090383 Minimal number of vertices of polytope P_T associated with any binary tree having 2n+1 nodes.

Original entry on oeis.org

4, 7, 8, 8, 10, 11, 8, 12, 10, 8, 10
Offset: 1

Views

Author

N. J. A. Sloane, Jan 30 2004

Keywords

Links

Crossrefs

Cf. A090381-A090385.

A090384 Maximal number of vertices of polytope P_T associated with any binary tree having 2n+1 nodes.

Original entry on oeis.org

4, 7, 10, 13, 14, 13, 16, 17, 20, 19, 20
Offset: 1

Views

Author

N. J. A. Sloane, Jan 30 2004

Keywords

Links

Crossrefs

Cf. A090381, A090382, A090383, A090385.
Showing 1-6 of 6 results.

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

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