This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309852 #31 Mar 17 2025 04:59:45 %S A309852 2,1,2,1,1,2,1,3,3,2,1,4,9,3,2,1,7,27,11,3,2,1,11,81,36,13,3,2,1,18, %T A309852 243,119,45,15,5,2,1,29,729,393,161,54,25,5,2,1,47,2187,1298,573,207, %U A309852 125,27,5,2,1,76,6561,4287,2041,783,625,140,29,5,2 %N A309852 Array read by antidiagonals: ((z+sqrt(x))/2)^k + ((z-sqrt(x))/2)^k for columns k >= 0 and rows n >= 0, where x = 4*n+1 and y = floor(sqrt(x)) and z = y-1+(y mod 2). %C A309852 One of 4 related arrays (the others being A191347, A191348, and A309853) where the two halves of the main formula approach the integers shown and 0 respectively, and also with A309853 where rows represent various Fibonacci series a(n) = a(n-2)*b + a(n-1)*c where b and c are integers >= 0. %F A309852 For each row n>=0 let x = 4*n+1, y = floor(sqrt(x)), T(n,0)=2, and T(n,1)=y-1+(y % 2), then for each column k>=2: T(n, k-2)*((x-T(n, 1)^2)/4) + T(n, k-1)*T(n, 1). - _Charles L. Hohn_, Aug 23 2019 %e A309852 Array begins: %e A309852 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A309852 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... %e A309852 2, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, ... %e A309852 2, 3, 11, 36, 119, 393, 1298, 4287, 14159, 46764, 154451, ... %e A309852 2, 3, 13, 45, 161, 573, 2041, 7269, 25889, 92205, 328393, ... %e A309852 2, 3, 15, 54, 207, 783, 2970, 11259, 42687, 161838, 613575, ... %e A309852 2, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, ... %e A309852 2, 5, 27, 140, 727, 3775, 19602, 101785, 528527, 2744420, 14250627, ... %e A309852 2, 5, 29, 155, 833, 4475, 24041, 129155, 693857, 3727595, 20025689, ... %e A309852 2, 5, 31, 170, 943, 5225, 28954, 160445, 889087, 4926770, 27301111, ... %e A309852 2, 5, 33, 185, 1057, 6025, 34353, 195865, 1116737, 6367145, 36302673, ... %e A309852 ... %o A309852 (PARI) T(n, k) = my(x = 4*n+1, y = sqrtint(x), z = y-1+(y % 2)); round(((z+sqrt(x))/2)^k + ((z-sqrt(x))/2)^k); %o A309852 matrix(9,9, n, k, T(n-1,k-1)) \\ _Michel Marcus_, Aug 22 2019 %o A309852 (PARI) T(n, k) = my(x = 4*n+1, y = sqrtint(x), z=y-1+(y % 2)); v=if(k==0, 2, k==1, z, mapget(m2, n)*((x-z^2)/4) + mapget(m1, n)*z); mapput(m2, n, if(mapisdefined(m1, n), mapget(m1, n), 0)); mapput(m1, n, v); v; %o A309852 m1=Map(); m2=Map(); matrix(9, 9, n, k, T(n-1, k-1)) \\ _Charles L. Hohn_, Aug 26 2019 %Y A309852 Row 2 is A000032, row 3 (except the first term) is A000244, row 4 is A006497, row 5 is A206776, row 6 is A172012, row 7 (except the first term) is A000351, row 8 is A087130. %Y A309852 Cf. A191347, A191348, and A309853. %K A309852 nonn,tabl %O A309852 0,1 %A A309852 _Charles L. Hohn_, Aug 20 2019