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Square array T(n, k), n >= 0, k > 0, read by antidiagonals, where T(0, k) = A001906(k) for k > 0 and where T(n, k) = n - A130312(n) + A000045(2k + A072649(n)) for n > 0, k > 0.
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%I #44 Apr 21 2024 22:11:44

%S 1,2,3,4,5,8,6,9,13,21,7,14,22,34,55,10,15,35,56,89,144,11,23,36,90,

%T 145,233,377,12,24,57,91,234,378,610,987,16,25,58,146,235,611,988,

%U 1597,2584,17,37,59,147,379,612,1598,2585,4181,6765,18,38,92,148,380,989

%N Square array T(n, k), n >= 0, k > 0, read by antidiagonals, where T(0, k) = A001906(k) for k > 0 and where T(n, k) = n - A130312(n) + A000045(2k + A072649(n)) for n > 0, k > 0.

%C Each positive integer occurs exactly once, so this sequence is a permutation of the natural numbers.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F T(0, k) = A001906(k) for k > 0.

%F T(n, k) = n - A130312(n) + A000045(2k + A072649(n)) for n > 0, k > 0.

%e Square array begins:

%e 1, 3, 8, 21, 55, 144, 377, 987, ...

%e 2, 5, 13, 34, 89, 233, 610, 1597, ...

%e 4, 9, 22, 56, 145, 378, 988, 2585, ...

%e 6, 14, 35, 90, 234, 611, 1598, 4182, ...

%e 7, 15, 36, 91, 235, 612, 1599, 4183, ...

%e 10, 23, 57, 146, 379, 989, 2586, 6767, ...

%e 11, 24, 58, 147, 380, 990, 2587, 6768, ...

%e 12, 25, 59, 148, 381, 991, 2588, 6769, ...

%e 16, 37, 92, 236, 613, 1600, 4184, 10949, ...

%o (PARI) b1(n)=local(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2) \\ A072649

%o T(n, k)=if(n==0, fibonacci(2*k), n - fibonacci(b1(n)) + fibonacci(2*k + b1(n)))

%Y Cf. A000045, A001906, A072649, A130312.

%K nonn,tabl

%O 1,2

%A _Mikhail Kurkov_, Jul 20 2022 [verification needed]