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 A081458 #41 Aug 23 2024 21:08:19
%S A081458 1,13,5,313,17,41,7813,317,53,365,195313,7817,353,377,3281,4882813,
%T A081458 195317,7853,677,3293,29525,122070313,4882817,195353,8177,3593,29537,
%U A081458 265721,3051757813,122070317,4882853,195677,11093,29837,265733,2391485,76293945313,3051757817,122070353,4883177,198593,37337,266033,2391497,21523361
%N A081458 Table T(m,n) = (3^m + 5^n)/2, for m, n = 0, 2, 4, 6, ... read by antidiagonals downwards.
%C A081458 Except for the first term [in each row (Ed.)], these numbers always end in 3 and 7 and necessarily generate an odd number as the quotient upon a single division by 2. Indeed for even m,n 3^m+5^n can be written as (4-1)^m + (4+1)^n = 4h+1 + 4i+1 for some h,i. Then we add and get 4(h+i)+2. Divide by 2 to get 2(h+i) + 1 an odd number. We dispose of the endings > 1 being either 3 or 7 by noting that the ending digits of even powers of 3 are 1,9,1,9,... and ending digits of powers of 5 end in 5. Then when we add 1 and 5 we get 6 and 6/2 = 3. Similarly, 9 + 5 = 14. 14/2 = 7.
%C A081458 The terms are part of a Pythagorean triple: sqrt((a(n))^2 - (a(n)-1)^2) = 5^n. E.g., sqrt(313^2 - 312^2) = 5^2 since 313 = a(2). - _Gary W. Adamson_, Jun 27 2006
%C A081458 The values with m > n were missing in the original version. - _M. F. Hasler_, Jan 01 2013
%H A081458 G. C. Greubel, <a href="/A081458/b081458.txt">Antidiagonal rows n = 1..100, flattened</a>
%F A081458 Each row of the table obeys the recurrence relation a(n) = 26*a(n-1) - 25*a(n-2), n>1. Let M = the 2 X 2 matrix [13, 12; 12, 13]. Then T[1,n] = left term in M^n *[1,0]. - _Gary W. Adamson_, Jun 27 2006, edited by _M. F. Hasler_, Jan 01 2013
%e A081458 The array (5^x+3^y)/2; x,y=0,2,4,... starts as follows:
%e A081458 [ 1 13 313 7813 195313 4882813 122070313 ...]
%e A081458 [ 5 17 317 7817 195317 4882817 122070317 ...]
%e A081458 [ 41 53 353 7853 195353 4882853 122070353 ...]
%e A081458 [ 365 377 677 8177 195677 4883177 122070677 ...]
%e A081458 [ 3281 3293 3593 11093 198593 4886093 122073593 ...]
%e A081458 [ 29525 29537 29837 37337 224837 4912337 122099837 ...]
%e A081458 [265721 265733 266033 273533 461033 5148533 122336033 ...]
%e A081458 [ ... ]
%t A081458 Table[(5^(2*(n-k)) +3^(2*k))/2, {n,0,9}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 13 2019 *)
%o A081458 (PARI) matrix(7,7,y,x,(3^(y*2-2) + 5^(x*2-2))/2) \\ _M. F. Hasler_, Jan 01 2013
%o A081458 (PARI) A081458(n,k) = (5^(2*(n-k)) +3^(2*k))/2;
%o A081458 for(n=0,9, for(k=0,n, print1(A081458(n,k), ", "))) \\ _G. C. Greubel_, Aug 13 2019
%o A081458 (Magma) A081458:= func< n,k | (5^(2*(n-k)) +3^(2*k))/2 >;
%o A081458 [A081458(n,k): k in [0..n], n in [0..9]]; // _G. C. Greubel_, Aug 13 2019
%o A081458 (Sage)
%o A081458 def T(n, k): return (5^(2*(n-k)) +3^(2*k))/2
%o A081458 [[T(n, k) for k in (0..n)] for n in (0..9)] # _G. C. Greubel_, Aug 13 2019
%o A081458 (GAP) Flat(List([0..9], n-> List([0..n], k-> (5^(2*(n-k)) +3^(2*k))/2 ))); # _G. C. Greubel_, Aug 13 2019
%Y A081458 Submatrix (even rows & cols) of A193770 (transposed). The values are listed in A193769 (subsequence of every other term). - _M. F. Hasler_, Jan 01 2013
%K A081458 easy,nonn,tabl
%O A081458 0,2
%A A081458 _Cino Hilliard_, Apr 20 2003
%E A081458 Edited and extended by _M. F. Hasler_, Jan 01 2013