login
A095872
Square of the lower triangular matrix T[i,j] = 3j-2 for 1<=j<=i, read by rows.
3
1, 5, 16, 12, 44, 49, 22, 84, 119, 100, 35, 136, 210, 230, 169, 51, 200, 322, 390, 377, 256, 70, 176, 455, 580, 624, 560, 361, 70, 276, 455, 580, 624, 560, 361, 92, 364, 609, 800, 910, 912, 779, 484, 117, 464, 784, 1050, 1235, 1312, 1254, 1034, 625, 145, 576, 980, 1330, 1599
OFFSET
1,2
COMMENTS
Arranged by flush left columns (k=1,2,3...), (k=1) column = A000326, the pentagonal numbers (1, 5, 12, 22, 35...). The Octagonal pyramidal number triangle of A095871 is generated from A095872 by dividing the k-th row by the n-th term in the series 1, 4, 7, 10...(k starting with 1). Dividing the 3rd column of A095872 (49, 119, 210, 322, 455...) by 7 generates A059845: 7, 17, 30, 46, 65... Rightmost terms of each row of A095872 are A016778 (1, 16, 49, 100, 169...); i.e. squares of 1, 4, 7, 10... Row sums of A095872 are 1, 21, 105, 325, 780, 1596, 2926... Row sums of A095871 are the octagonal pyramidal numbers, A002414: 1, 9, 30, 70, 135, 231, 364...
[Editor's note: OEIS' "TABL" format (fmt=2) rather displays the transposed matrix as upper triangular matrix.]
FORMULA
a(k(k+1)/2) = (3k-2)^2 (diagonal elements: squares of the initial series), a(k(k-1)/2+1) = A000326(k) (1st column: pentagonal numbers). - M. F. Hasler, Apr 18 2009
EXAMPLE
Let M = the infinite lower triangular matrix in the format exemplified by a 3rd order matrix: [1 0 0 / 1 4 0 / 1 4 7]: i.e. for the n-th order matrix, each row has n terms in the series 1, 4, 7, 10... with the rest of the spaces filled in with zeros. Square the matrix and delete the zeros; then read by rows.
[1 0 0 / 1 4 0 / 1 4 7]^2 = [1 0 0 / 5 16 0 / 12 44 49]; then delete the zeros and read by rows: 1, 5, 16, 12, 44, 49...
PROG
(PARI) A095802(n)={ my( r=sqrtint(2*n)+1, T=matrix(r, r, i, j, if(j>=i, 3*j-2))^2); concat(vector(#T, i, vecextract(T[, i], 2^i-1)))[n] } \\ M. F. Hasler, Apr 18 2009
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Jun 10 2004
EXTENSIONS
Edited and extended by M. F. Hasler, Apr 18 2009
STATUS
approved