A101450 Areas of (not necessarily primitive) Pythagorean triangles which are palindromes.

6, 63336, 474474, 666666, 4053504, 4244424, 4383834, 42066024, 42666624, 43177134, 610262016, 6390000936, 48576067584, 460962269064, 654513315456, 4872873782784, 60471399317406, 60490233209406, 643869171968346

Offset

1,1

Comments

Compare areas of primitive Pythagorean triangles which are palindromes, A101439. Are these lists full? What are the next cases?

Other parts of the n-th triangle are: for a(1): a=3, b=4, c=5 & area = 6 = a*b/2; for a(2): a=377, b=336, c=505; a(3): 6083, 156, 6085; a(4): 693, 1924, 2045;

a(5): 2688, 3016, 4040; a(6): 408, 20806, 20810; a(7): 1443, 6076, 6245;

a(8): 8008, 10506, 13210; a(9): 15392, 5544, 16360; a(10): 24843, 3476, 25085;

a(11): 12432, 98176, 98960; a(12): 155448, 82214, 175850;

a(13): 23168, 4193376, 4193440; a(14): 81073, 11371536, 11371825;

a(15): 304928, 4292904, 4303720; a(16): 3420732, 2849024, 4451780;

a(17): 2724403, 44392404, 44475925; a(18): 5390853, 22441804, 23080205;

a(19): 17453637, 73780516, 75816845; a(20): 1454034783, 9227944, 1454064065;

a(21): 53643247, 1666695504, 1667558545;

a(22): 1019664547, 1194231396, 1570319845;

a(23): 1804499368, 5363316426, 5658743770, ....

The list seems incomplete: why is 46464 = a*b/2 for (a=264, b=352) with a^2 + b^2 = 440^2 not in the list? I get (6, 46464, 48384, 63336, 65856, 66066, 474474, ...) - M. F. Hasler, Jun 06 2024

Links

Table of n, a(n) for n=1..19.

Formula

Intersection of A002113 and A009112. - M. F. Hasler, Jun 06 2024

Example

666666 is a member, as it is a palindromic number and is area of
Pythagorean triangle with legs a=693, b=1924 and hypotenuse c=2045.

Mathematica

lst = {}; Do[ a = IntegerDigits[m*n^3 - n*m^3]; If[ Reverse[a] == a, lst = Sort[ AppendTo[ lst, FromDigits[a]]]; Print[{n^2 - m^2, 2m*n, n^2 + m^2, m*n^3 - n*m^3}]], {n, 50000}, {m, n - 1}] (* Robert G. Wilson v, Jan 29 2005 *)

Prog

(PARI) for(n=1, oo, is_A009112(a=A002113(n)) && print1(a", ")) \\ or better:
A101450_row(n) = select(is_A009112, A002113_row(n)) \\ row n = the n-digit terms - M. F. Hasler, Jun 06 2024

Crossrefs

Cf. A101439.

Cf. A002113, A009112.

Sequence in context: A323725 A259162 A076913 * A101439 A099112 A172750

Adjacent sequences: A101447 A101448 A101449 * A101451 A101452 A101453

Keyword

base,nonn

Author

Zak Seidov, Jan 19 2005

Extensions

a(16), a(17), a(19)-a(23) from Robert G. Wilson v, Jan 31 2005

Status

approved