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A081458
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Table T(m,n) = (3^m + 5^n)/2, for m, n = 0, 2, 4, 6, ... read by antidiagonals downwards.
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3
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1, 13, 5, 313, 17, 41, 7813, 317, 53, 365, 195313, 7817, 353, 377, 3281, 4882813, 195317, 7853, 677, 3293, 29525, 122070313, 4882817, 195353, 8177, 3593, 29537, 265721, 3051757813, 122070317, 4882853, 195677, 11093, 29837, 265733, 2391485, 76293945313, 3051757817, 122070353, 4883177, 198593, 37337, 266033, 2391497, 21523361
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OFFSET
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0,2
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COMMENTS
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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.
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
The values with m > n were missing in the original version. - M. F. Hasler, Jan 01 2013
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LINKS
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FORMULA
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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
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EXAMPLE
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The array (5^x+3^y)/2; x,y=0,2,4,... starts as follows:
[ 1 13 313 7813 195313 4882813 122070313 ...]
[ 5 17 317 7817 195317 4882817 122070317 ...]
[ 41 53 353 7853 195353 4882853 122070353 ...]
[ 365 377 677 8177 195677 4883177 122070677 ...]
[ 3281 3293 3593 11093 198593 4886093 122073593 ...]
[ 29525 29537 29837 37337 224837 4912337 122099837 ...]
[265721 265733 266033 273533 461033 5148533 122336033 ...]
[ ... ]
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MATHEMATICA
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Table[(5^(2*(n-k)) +3^(2*k))/2, {n, 0, 9}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 13 2019 *)
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PROG
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(PARI) matrix(7, 7, y, x, (3^(y*2-2) + 5^(x*2-2))/2) \\ M. F. Hasler, Jan 01 2013
(PARI) A081458(n, k) = (5^(2*(n-k)) +3^(2*k))/2;
(Magma) A081458:= func< n, k | (5^(2*(n-k)) +3^(2*k))/2 >;
(Sage)
def T(n, k): return (5^(2*(n-k)) +3^(2*k))/2
[[T(n, k) for k in (0..n)] for n in (0..9)] # G. C. Greubel, Aug 13 2019
(GAP) Flat(List([0..9], n-> List([0..n], k-> (5^(2*(n-k)) +3^(2*k))/2 ))); # G. C. Greubel, Aug 13 2019
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CROSSREFS
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Submatrix (even rows & cols) of A193770 (transposed). The values are listed in A193769 (subsequence of every other term). - M. F. Hasler, Jan 01 2013
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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