cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Cino Hilliard, Apr 20 2003

Keywords

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

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

A193770 Table T(m,n) = (5^m + 3^n)/2, m,n = 0,1,2,..., read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 13, 14, 7, 14, 63, 41, 16, 17, 64, 313, 122, 43, 26, 67, 314, 1563, 365, 124, 53, 76, 317, 1564, 7813, 1094, 367, 134, 103, 326, 1567, 7814, 39063, 3281, 1096, 377, 184, 353, 1576, 7817, 39064, 195313, 9842, 3283, 1106, 427, 434, 1603, 7826, 39067, 195314
Offset: 0

Views

Author

M. F. Hasler, Jan 01 2013

Keywords

Comments

Sequence A193769 lists the elements of the array in order of increasing size. Sequence A081458 is the subtable with every other row and column deleted (i.e., m,n=0,2,4,...). (The earlier existence of that table in the OEIS has motivated the definition of the present sequence/table.)
Looking at the example one can notice the periodicity of the final digit(s) of the terms; it is easy to prove these formulas. - M. F. Hasler, Jan 06 2013

Examples

			The upper left part of the infinite square array reads:
[   1    2    5   14   41  122  365 1094  3281 ...]
[   3    4    7   16   43  124  367 1096  3283 ...]
[  13   14   17   26   53  134  377 1106  3293 ...]
[  63   64   67   76  103  184  427 1156  3343 ...]
[ 313  314  317  326  353  434  677 1406  3593 ...]
[1563 1564 1567 1576 1603 1684 1927 2656  4843 ...]
[7813 7814 7817 7826 7853 7934 8177 8906 11093 ...]
[...]

Links

Programs

  • Mathematica
    Flatten@Table[(5^j + 3^(i - j))/2, {i, 0, 8}, {j, 0, i}] (* Ivan Neretin, Sep 07 2017 *)
  • PARI
    for(x=0,10,for(y=0,x, print1((3^(x-y)+5^y)/2 ","))) \\ prints this sequence; to get the table, use matrix(7,9,m,n,3^n/3+5^m/5)/2 \\ M. F. Hasler, Jan 06 2013

Formula

T(m,n+4) = T(m,n) (mod 10),
T(m+1,n) = T(m,n) (mod 10) for m > 0,
T(m+1,n) = T(m,n) + 50 (mod 100) for m > 1, etc. - M. F. Hasler, Jan 06 2013

A226809 Numbers of the form 3^j + 5^k, for j and k >= 0.

Original entry on oeis.org

2, 4, 6, 8, 10, 14, 26, 28, 32, 34, 52, 82, 86, 106, 126, 128, 134, 152, 206, 244, 248, 268, 368, 626, 628, 634, 652, 706, 730, 734, 754, 854, 868, 1354, 2188, 2192, 2212, 2312, 2812, 3126, 3128, 3134, 3152, 3206, 3368, 3854, 5312, 6562, 6566, 6586, 6686, 7186
Offset: 1

Views

Author

T. D. Noe, Jun 19 2013

Keywords

Links

Crossrefs

Cf. A004050 (2^j + 3^k), A226806-A226832 (cases to 8^j + 9^k).
Cf. A193769, A226790 (a(n)/2 with/without repetition).

Programs

  • Mathematica
    a = 3; b = 5; mx = 8000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]
Showing 1-3 of 3 results.

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