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Table by antidiagonals of n-Stohr sequences: T(n,k) is least positive integer not the sum of at most n distinct terms in the n-th row from T(n,1) through to T(n,k-1).
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%I #10 May 03 2022 11:03:37

%S 1,2,1,3,2,1,4,4,2,1,5,7,4,2,1,6,10,8,4,2,1,7,13,15,8,4,2,1,8,16,22,

%T 16,8,4,2,1,9,19,29,31,16,8,4,2,1,10,22,36,46,32,16,8,4,2,1,11,25,43,

%U 61,63,32,16,8,4,2,1,12,28,50,76,94,64,32,16,8,4,2,1,13,31,57,91,125,127,64,32,16,8,4,2,1

%N Table by antidiagonals of n-Stohr sequences: T(n,k) is least positive integer not the sum of at most n distinct terms in the n-th row from T(n,1) through to T(n,k-1).

%H G. C. Greubel, <a href="/A062001/b062001.txt">Antidiagonals n = 1..50, flattened</a>

%F If k <= n+1 then A(n, k) = 2^(k-1), while if k > n+1, A(n, k) = (2^n - 1)*(k - n) + 1 (array).

%F T(n, k) = A(k, n-k+1) (antidiagonals).

%F T(2*n-1, n) = A000079(n-1), n >= 1.

%F T(2*n, n) = A000079(n), n >= 1.

%F T(2*n+1, n) = A000225(n+1), n >= 1.

%F T(2*n+2, n) = A033484(n), n >= 1.

%F T(2*n+3, n) = A036563(n+3), n >= 1.

%F T(2*n+4, n) = A048487(n), n >= 1.

%F From _G. C. Greubel_, May 03 2022: (Start)

%F T(n, k) = (2^k - 1)*(n-2*k+1) + 1 for k < n/2, otherwise 2^(n-k).

%F T(2*n+5, n) = A048488(n), n >= 1.

%F T(2*n+6, n) = A048489(n), n >= 1.

%F T(2*n+7, n) = A048490(n), n >= 1.

%F T(2*n+8, n) = A048491(n), n >= 1.

%F T(2*n+9, n) = A139634(n), n >= 1.

%F T(2*n+10, n) = A139635(n), n >= 1.

%F T(2*n+11, n) = A139697(n), n >= 1. (End)

%e Array begins as:

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, ... A000027;

%e 1, 2, 4, 7, 10, 13, 16, 19, 22, ... A033627;

%e 1, 2, 4, 8, 15, 22, 29, 36, 43, ... A026474;

%e 1, 2, 4, 8, 16, 31, 46, 61, 76, ... A051039;

%e 1, 2, 4, 8, 16, 32, 63, 94, 125, ... A051040;

%e 1, 2, 4, 8, 16, 32, 64, 127, 190, ... ;

%e 1, 2, 4, 8, 16, 32, 64, 128, 255, ... ;

%e 1, 2, 4, 8, 16, 32, 64, 128, 256, ... ;

%e 1, 2, 4, 8, 16, 32, 64, 128, 256, ... ;

%e Antidiagonal triangle begins as:

%e 1;

%e 2, 1;

%e 3, 2, 1;

%e 4, 4, 2, 1;

%e 5, 7, 4, 2, 1;

%e 6, 10, 8, 4, 2, 1;

%e 7, 13, 15, 8, 4, 2, 1;

%e 8, 16, 22, 16, 8, 4, 2, 1;

%e 9, 19, 29, 31, 16, 8, 4, 2, 1;

%e 10, 22, 36, 46, 32, 16, 8, 4, 2, 1;

%e 11, 25, 43, 61, 63, 32, 16, 8, 4, 2, 1;

%e 12, 28, 50, 76, 94, 64, 32, 16, 8, 4, 2, 1;

%e 13, 31, 57, 91, 125, 127, 64, 32, 16, 8, 4, 2, 1;

%t T[n_, k_]:= If[k<n/2, (2^k -1)*(n-2*k+1) +1, 2^(n-k)];

%t Table[T[n, k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, May 03 2022 *)

%o (SageMath)

%o def A062001(n,k):

%o if (k<n/2): return (2^k -1)*(n-2*k+1) +1

%o else: return 2^(n-k)

%o flatten([[A062001(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, May 03 2022

%Y Rows include A000027, A033627, A026474, A051039, A051040.

%Y Diagonals include A000079, A000225, A033484, A036563, A048487.

%Y Cf. A048488, A048489, A048490, A048491, A139634, A139635, A139697.

%Y A048483 can be seen as half this table.

%K nonn,tabl

%O 1,2

%A _Henry Bottomley_, May 29 2001