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A374258 Square array: T(n,k) = ((3^(n+1) + 1)^(k-1) + 2)/3, read by descending antidiagonals.

%I A374258 #56 Aug 02 2024 22:03:31
%S A374258 1,4,1,34,10,1,334,262,28,1,3334,7318,2242,82,1,33334,204886,183790,
%T A374258 19846,244,1,333334,5736790,15070726,4842262,177634,730,1,3333334,
%U A374258 160630102,1235799478,1181511766,129672334,1595782,2188,1,33333334,4497642838,101335557142,288288870742,94660803334,3491569558,14353282,6562,1
%N A374258 Square array: T(n,k) = ((3^(n+1) + 1)^(k-1) + 2)/3, read by descending antidiagonals.
%C A374258 This sequence gives the matrix M in the definition of A365450. Similar to A266577.
%C A374258 Conjecture: For each natural number n, the digits of the product of any (n+1) not necessarily distinct terms of the n-th row in the base (3^(n+1)+1) numeral system appear in nondecreasing order.
%C A374258 Proof of the conjecture. Let b := 3^(n+1)+1. The product of any (n+1) terms of the n-th row has the form p/(b-1), where p is the product of (n+1) numbers of the form b^k+2. Let p = (dm, ..., d1, d0)_b, and we have d0+d1+...+dm = 3^(n+1) = b-1. Then p/(b-1) = (dm, ..., d2+...+dm, 1+d1+...+dm)_b, which do form a nondecreasing sequence. - _Max Alekseyev_, Jul 03 2024
%C A374258 The preceding result is similar to the property of the nondecreasing products mentioned in A237424. Specifically; the first row of this array is A093137, which is a subsequence of A237424.  - _Ahmad J. Masad_, Jul 30 2024
%C A374258 More generally: Let r and n be positive integers and S be a sequence of all numbers of the form (b^c(1)+b^c(2)+...+b^c(r)+1)/(r+1), where c(1),...,c(r) are nonnegative integers. Then in the numeral system base b := (r+1)^(n+1)+1, the digits of the product of any n+1 (possibly equal) terms of S appear in nondecreasing order. Proof is similar. - _Ahmad J. Masad_, Jul 30 2024; edited by _Max Alekseyev_, Aug 01 2024
%H A374258 Michel Marcus, <a href="/A374258/b374258.txt">Table of n, a(n) for n = 1..820</a>
%e A374258 The array begins:
%e A374258   1    4    34   334   3334
%e A374258   1   10   262  7318
%e A374258   1   28  2242
%e A374258   1   82
%e A374258   1
%e A374258 Example of the conjecture: Take 5 terms from the 4th row and find their product in base 244 numeral system (since 3^(4+1)+1=244) as follows: 82,19846 twice and 4842262 twice, the product is equal to 82*19846*19846*4842262*4842262 = 757279838666167487626528 = (1, 3, 7, 19, 31, 55, 91, 115, 163, 195, 212)_244 which is in agreement with the conjecture since the digits in 244 base numeral system are in nondecreasing order.
%e A374258 Example of the general property: Take r=3 and n=4, then b=4^5+1=1025. The sequence S is the sequence of the numbers of the form (1025^b(1)+1025^b(2)+1025^b(3)+1)/4. Let's multiply 5 terms of the sequence S, say ((1025^0+1025^0+1025^1+1)/4)*(((1025^0+1025^1+1025^1+1)/4)^2)*(((1025^3+1025^4+1025^4+1)/4)^2) = 257*513^2*552175667969^2 = 20621601208620337073958261562113 = (16,112,308,488,580,680,832,936,964,984,1013)_1025. The digits of the product in base 1025 are in nondecreasing order.
%t A374258 T[n_, k_] := ((3^(n+1) + 1)^(k-1) + 2)/3; Table[T[k, n-k+1], {n, 1, 9}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Jul 02 2024 *)
%o A374258 (PARI) T(n,k) = ((3^(n+1) + 1)^(k-1) + 2)/3 \\ _Andrew Howroyd_, Jul 01 2024
%Y A374258 Cf. A093137, A266577, A365450, A237424.
%K A374258 nonn,tabl
%O A374258 1,2
%A A374258 _Ahmad J. Masad_, Jul 01 2024