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A095872 Square of the lower triangular matrix T[i,j] = 3j-2 for 1<=j<=i, read by rows.

%I A095872 #13 Jan 25 2021 10:50:18
%S A095872 1,5,16,12,44,49,22,84,119,100,35,136,210,230,169,51,200,322,390,377,
%T A095872 256,70,176,455,580,624,560,361,70,276,455,580,624,560,361,92,364,609,
%U A095872 800,910,912,779,484,117,464,784,1050,1235,1312,1254,1034,625,145,576,980,1330,1599
%N A095872 Square of the lower triangular matrix T[i,j] = 3j-2 for 1<=j<=i, read by rows.
%C A095872 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...
%C A095872 [Editor's note: OEIS' "TABL" format (fmt=2) rather displays the transposed matrix as upper triangular matrix.]
%F A095872 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
%e A095872 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.
%e A095872 [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...
%o A095872 (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
%Y A095872 Cf. A095871, A000326, A059845, A002414, A016778.
%K A095872 nonn,tabl
%O A095872 1,2
%A A095872 _Gary W. Adamson_, Jun 10 2004
%E A095872 Edited and extended by _M. F. Hasler_, Apr 18 2009