A193770 -id:A193770 - cp's OEIS frontend

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

Cino Hilliard, Apr 20 2003

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

			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 ...]
[ ... ]

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

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

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

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

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

matrix(7,7,y,x,(3^(y*2-2) + 5^(x*2-2))/2) \\ M. F. Hasler, Jan 01 2013

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

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

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

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

A193769 Numbers of the form (3^a+5^b)/2 ; a,b >= 0 (with repetition).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 13, 14, 14, 16, 17, 26, 41, 43, 53, 63, 64, 67, 76, 103, 122, 124, 134, 184, 313, 314, 317, 326, 353, 365, 367, 377, 427, 434, 677, 1094, 1096, 1106, 1156, 1406, 1563, 1564, 1567, 1576, 1603, 1684, 1927, 2656

Offset: 1

M. F. Hasler, Jan 01 2013

nonn

Elements of the table A193770 (motivated by the less fundamental A081458), sorted by increasing size.

The value 14 = (3^1+5^2)/2 = (3^3+5^0)/2, therefore it occurs twice.

Cf. A193770 (the table), A081458 (the subtable obtained for even a,b).

select( t->t<3000, vecsort(concat(Vec( matrix(9,6,x,y,(3^(x-1)+5^(y-1))/2 )))))

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

PARI

Sage

Formula

Extensions

A193769 Numbers of the form (3^a+5^b)/2 ; a,b >= 0 (with repetition).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI