A081458 Table T(m,n) = (3^m + 5^n)/2, for m, n = 0, 2, 4, 6, ... read by antidiagonals downwards.

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

Offset: 0

Author

Cino Hilliard, Apr 20 2003

Keywords

• easy
• nonn
• tabl

Comments

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

Examples

			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 ...]
[ ... ]
		

Links

• G. C. Greubel, Antidiagonal rows n = 1..100, flattened

Crossrefs

Submatrix (even rows & cols) of A193770 (transposed). The values are listed in A193769 (subsequence of every other term). - M. F. Hasler, Jan 01 2013

Programs

GAP

Flat(List([0..9], n-> List([0..n], k-> (5^(2*(n-k)) +3^(2*k))/2 ))); # G. C. Greubel, Aug 13 2019

Magma

A081458:= func< n,k | (5^(2*(n-k)) +3^(2*k))/2 >;
[A081458(n,k): k in [0..n], n in [0..9]]; // G. C. Greubel, Aug 13 2019

Mathematica

Table[(5^(2*(n-k)) +3^(2*k))/2, {n,0,9}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 13 2019 *)

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;
for(n=0,9, for(k=0,n, print1(A081458(n,k), ", "))) \\ G. C. Greubel, Aug 13 2019

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

Formula

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

Extensions

Edited and extended by M. F. Hasler, Jan 01 2013