Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #15 Mar 12 2024 09:53:45
%S 1,3,9,7,15,25,13,23,35,49,21,33,47,63,81,31,45,61,79,99,121,43,59,77,
%T 97,119,143,169,57,75,95,117,141,167,195,225,73,93,115,139,165,193,
%U 223,255,289,91,113,137,163,191,221,253,287,323,361,111,135,161,189,219,251,285,321,359,399,441
%N Odd numbers obtained by transposing the left half of A176271 into rows of a triangle: T(n,k) = A176271(n - 1 + k, k), 1 <= k <= n.
%H Reinhard Zumkeller, <a href="/A214661/b214661.txt">Rows n = 1..150 of triangle, flattened</a>
%F T(n, k) = (n+k)^2 - 3*n - k + 1.
%F T(n,k) = A176271(n+k-1, k).
%F T(n, k) = A214604(n,k) - 2*A025581(n,k).
%F T(n, k) = 2*A000290(A094727(n,k)) - A214604(n,k).
%F T(2*n-1, n) = A214675() (main diagonal).
%F T(n,1) = A002061(n).
%F T(n,n) = A016754(n-1).
%F Sum_{k=1..n} T(n, k) = A051673(n) (row sums).
%e . Take the first n elements of the n-th diagonal (northwest to
%e . southeast) of the triangle on the left side
%e . and write this as n-th row on the triangle of the right side.
%e . 1: 1 1
%e . 2: 3 _ 3 9
%e . 3: 7 9 __ 7 15 25
%e . 4: 13 15 __ __ 13 23 35 49
%e . 5: 21 23 25 __ __ 21 33 47 63 ..
%e . 6: 31 33 35 __ __ __ 31 45 61 .. .. ..
%e . 7: 43 45 47 49 __ __ __ 43 59 .. .. .. .. ..
%e . 8: 57 59 61 63 __ __ __ __ 57 .. .. .. .. .. .. .. .
%t Table[(n+k)^2-3*n-k+1, {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 10 2024 *)
%o (Haskell)
%o import Data.List (transpose)
%o a214661 n k = a214661_tabl !! (n-1) !! (k-1)
%o a214661_row n = a214661_tabl !! (n-1)
%o a214661_tabl = zipWith take [1..] $ transpose $ map reverse a176271_tabl
%o (Magma) [(n+k)^2-3*n-k+1: k in [1..n], n in [1..15]]; // _G. C. Greubel_, Mar 10 2024
%o (SageMath) flatten([[(n+k)^2-3*n-k+1 for k in range(1,n+1)] for n in range(1,16)]) // _G. C. Greubel_, Mar 10 2024
%Y Cf. A000290, A002061, A016754, A025581, A094727, A176271, A214604.
%Y Cf. A051673 (row sums), A214675 (main diagonal).
%K nonn,tabl
%O 1,2
%A _Reinhard Zumkeller_, Jul 25 2012