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A249111 Triangle of partial sums of rows in triangle A249095. 5
1, 1, 2, 3, 1, 2, 4, 5, 6, 1, 2, 5, 7, 10, 11, 12, 1, 2, 6, 9, 15, 18, 22, 23, 24, 1, 2, 7, 11, 21, 27, 37, 41, 46, 47, 48, 1, 2, 8, 13, 28, 38, 58, 68, 83, 88, 94, 95, 96, 1, 2, 9, 15, 36, 51, 86, 106, 141, 156, 177, 183, 190, 191, 192, 1, 2, 10, 17, 45, 66 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Length of row n = 2*n+1.

In the layout as given in the example, T(n,k) is the sum of the two elements to the left and to the right of the element just above, with the row continued to the left by 0's and to the right by the last element, cf. formula. - M. F. Hasler, Nov 17 2014

LINKS

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

FORMULA

T(n,0) = A249095(n,0) = 1; T(n,k) = T(n,k-1) + A249095(n,k), k <= n.

T(n+1,k+1) = T(n,k-1) + T(n,k+1), with T(n,k-1)=0 for k<1 and T(n,k+1)=T(n,k) for k>=2n (last element of the row). In particular, T(n,k)=k+1 if k<2n and T(n,k)=3*2^(n-1) if k>=2n. - M. F. Hasler, Nov 17 2014

EXAMPLE

The triangle begins:

. 0: 1

. 1: 1 2 3

. 2: 1 2 4 5 6

. 3: 1 2 5 7 10 11 12

. 4: 1 2 6 9 15 18 22 23 24

. 5: 1 2 7 11 21 27 37 41 46 47 48

. 6: 1 2 8 13 28 38 58 68 83 88 94 95 96

. 7: 1 2 9 15 36 51 86 106 141 156 177 183 190 191 192

. 8: 1 2 10 17 45 66 122 157 227 262 318 339 367 374 382 383 384 .

It can be seen that the elements (except for row 1) are sum of the neighbors to the upper left and upper right, with the table continued to the left with 0's and to the right with the last = largest element of each row. E.g., 1=0+1, 2=0+2, 4=1+3, 5=2+3 (=1+4 in the next row), 6=3+3 (in row 2), 7=2+5 etc. - M. F. Hasler, Nov 17 2014

PROG

(Haskell)

a249111 n k = a249111_tabf !! n !! k

a249111_row n = a249111_tabf !! n

a249111_tabf = map (scanl1 (+)) a249095_tabf

(PARI) T(n, k)=if(k<2, k+1, if(k>=2*n-2, 3<<(n-1), T(n-1, k-2)+T(n-1, k))) \\ M. F. Hasler, Nov 17 2014

CROSSREFS

Cf. A005408 (row lengths), A128543 (row sums), A248574 (central terms), A008949.

Sequence in context: A338900 A244567 A254112 * A166871 A275728 A081536

Adjacent sequences: A249108 A249109 A249110 * A249112 A249113 A249114

KEYWORD

nonn,tabf

AUTHOR

Reinhard Zumkeller, Nov 14 2014

STATUS

approved

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Last modified December 5 19:57 EST 2022. Contains 358588 sequences. (Running on oeis4.)