A276472 Modified Pascal's triangle read by rows: T(n,k) = T(n-1,k) + T(n-1,k-1), 1<k<n. T(n,1) = T(n-1,1) + T(n,2), n>2. T(n,n) = T(n,n-1) + T(n-1,n-1), n>1. T(1,1) = 1, T(2,1) = 1. n>=1.

1, 1, 2, 4, 3, 5, 11, 7, 8, 13, 29, 18, 15, 21, 34, 76, 47, 33, 36, 55, 89, 199, 123, 80, 69, 91, 144, 233, 521, 322, 203, 149, 160, 235, 377, 610, 1364, 843, 525, 352, 309, 395, 612, 987, 1597, 3571, 2207, 1368, 877, 661, 704, 1007, 1599, 2584, 4181

Offset

1,3

Comments

The recurrence relations for the border terms are the only way in which this differs from Pascal's triangle.

Column T(2n,n+1) appears to be divisible by 4 for n>=2; T(2n-1,n) divisible by 3 for n>=2; T(2n,n-2) divisible by 2 for n>=3.

The symmetry of T(n,k) can be observed in a hexagonal arrangement (see the links).

Consider T(n,k) mod 3 = q. Terms with q = 0 show reflection symmetry with respect to the central column T(2n-1,n), while q = 1 and q = 2 are mirror images of each other (see the link).

Links

Yuriy Sibirmovsky, T(n,k), read by rows as a linear sequence a(j) for j = 1..5050

Yuriy Sibirmovsky, Symmetrical hexagonal arrangement for initial terms of T(n,k)

Yuriy Sibirmovsky, T(n,k) compared with Pascal's triangle

Yuriy Sibirmovsky, Illustration for T(n,k) mod 3

Formula

Conjectures:

Relations with other sequences:

T(n+1,1) = A002878(n-1), n>=1.

T(n,n) = A001519(n) = A122367(n-1), n>=1.

T(n+1,2) = A005248(n-1), n>=1.

T(n+1,n) = A001906(n) = A088305(n), n>=1.

T(2n-1,n) = 3*A054441(n-1), n>=2. [the central column].

Sum_{k=1..n} T(n,k) = 3*A105693(n-1), n>=2. [row sums].

Sum_{k=1..n} T(n,k)-T(n,1)-T(n,n) = 3*A258109(n), n>=2.

T(2n,n+1) - T(2n,n) = A026671(n), n>=1.

T(2n,n-1) - T(2n,n) = 2*A026726(n-1), n>=2.

T(n,ceiling(n/2)) - T(n-1,floor(n/2)) = 2*A026732(n-3), n>=3.

T(2n+1,2n) = 3*A004187(n), n>=1.

T(2n+1,2) = 3*A049685(n-1), n>=1.

T(2n+1,2n) + T(2n+1,2) = 3*A033891(n-1), n>=1.

T(2n+1,3) = 5*A206351(n), n>=1.

T(2n+1,2n)/3 - T(2n+1,3)/5 = 4*A092521(n-1), n>=2.

T(2n,1) = 1 + 5*A081018(n-1), n>=1.

T(2n,2) = 2 + 5*A049684(n-1), n>=1.

T(2n+1,2) = 3 + 5*A058038(n-1), n>=1.

T(2n,3) = 3 + 5*A081016(n-2), n>=2.

T(2n+1,1) = 4 + 5*A003482(n-1), n>=1.

T(3n,1) = 4*A049629(n-1), n>=1.

T(3n,1) = 4 + 8*A119032(n), n>=1.

T(3n+1,3) = 8*A133273(n), n>=1.

T(3n+2,3n+2) = 2 + 32*A049664(n), n>=1.

T(3n,3n-2) = 4 + 32*A049664(n-1), n>=1.

T(3n+2,2) = 2 + 16*A049683(n), n>=1.

T(3n+2,2) = 2*A023039(n), n>=1.

T(2n-1,2n-1) = A033889(n-1), n>=1.

T(3n-1,3n-1) = 2*A007805(n-1), n>=1.

T(5n-1,1) = 11*A097842(n-1), n>=1.

T(4n+5,3) - T(4n+1,3) = 15*A000045(8n+1), n>=1.

T(5n+4,3) - T(5n-1,3) = 11*A000204(10n-2), n>=1.

Relations between left and right sides:

T(n,1) = T(n,n) - T(n-2,n-2), n>=3.

T(n,2) = T(n,n-1) - T(n-2,n-3), n>=4.

T(n,1) + T(n,n) = 3*T(n,n-1), n>=2.

Example

Triangle T(n,k) begins:
n\k 1    2    3    4   5    6    7    8    9
1   1
2   1    2
3   4    3    5
4   11   7    8    13
5   29   18   15   21   34
6   76   47   33   36   55   89
7   199  123  80   69   91   144 233
8   521  322  203  149  160  235 377  610
9   1364 843  525  352  309  395 612  987  1597
...
In another format:
__________________1__________________
_______________1_____2_______________
____________4_____3_____5____________
________11_____7_____8_____13________
____29_____18_____15____21_____34____
_76_____47____33_____36____55_____89_

Mathematica

Nm=12;
T=Table[0, {n, 1, Nm}, {k, 1, n}];
T[[1, 1]]=1;
T[[2, 1]]=1;
T[[2, 2]]=2;
Do[T[[n, 1]]=T[[n-1, 1]]+T[[n, 2]];
T[[n, n]]=T[[n-1, n-1]]+T[[n, n-1]];
If[k!=1&&k!=n, T[[n, k]]=T[[n-1, k]]+T[[n-1, k-1]]], {n, 3, Nm}, {k, 1, n}];
{Row[#, "\t"]}&/@T//Grid

Prog

(PARI) T(n, k) = if (k==1, if (n==1, 1, if (n==2, 1, T(n-1, 1) + T(n, 2))), if (k<n,  T(n-1, k) + T(n-1, k-1), if (k==n,  T(n, n-1) + T(n-1, n-1), 0)));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", "); ); print()); \\ Michel Marcus, Sep 14 2016

Crossrefs

Cf. A000045, A000204, A001519, A001906, A002878, A005248, A054441, A088305, A122367, A258109, A026671, A026726, A026732, A004187, A049685, A033891, A206351, A092521, A081018, A049684, A058038, A081016, A003482, A049629, A133273, A049664, A049683, A119032, A023039, A007805, A033889.

Sequence in context: A100826 A338161 A232799 * A245702 A297706 A093416

Adjacent sequences: A276469 A276470 A276471 * A276473 A276474 A276475

Keyword

nonn,tabl

Author

Yuriy Sibirmovsky, Sep 12 2016

Status

approved