A055096 -id:A055096 - cp's OEIS frontend

A055630 Table T(k,m) = k^2 + m read by antidiagonals.

0, 1, 1, 4, 2, 2, 9, 5, 3, 3, 16, 10, 6, 4, 4, 25, 17, 11, 7, 5, 5, 36, 26, 18, 12, 8, 6, 6, 49, 37, 27, 19, 13, 9, 7, 7, 64, 50, 38, 28, 20, 14, 10, 8, 8, 81, 65, 51, 39, 29, 21, 15, 11, 9, 9, 100, 82, 66, 52, 40, 30, 22, 16, 12, 10, 10, 121, 101, 83, 67, 53, 41, 31, 23, 17, 13

Offset: 0

Henry Bottomley, Jun 05 2000

			Table begins:
..0...1...4...9..16..25..36..49..64..81.100.121.144...
..1...2...5..10..17..26..37..50..65..82.101.122.145...
..2...3...6..11..18..27..38..51..66..83.102.123.146...
..3...4...7..12..19..28..39..52..67..84.103.124.147...
..4...5...8..13..20..29..40..53..68..85.104.125.148...
..5...6...9..14..21..30..41..54..69..86.105.126.149...
..6...7..10..15..22..31..42..55..70..87.106.127.150...
..7...8..11..16..23..32..43..56..71..88.107.128.151...
..8...9..12..17..24..33..44..57..72..89.108.129.152...
..9..10..13..18..25..34..45..58..73..90.109.130.153...
.10..11..14..19..26..35..46..59..74..91.110.131.154...
... - _Philippe Deléham_, Mar 31 2013

First column is A001477, second column is A000027, first row is A000290, second row is A002522, third row (apart from first term) is A010000, main diagonal is A002378, other diagonals include A028387, A028552, A014209, A002061, A014206, A027688-A027694, each row of A055096 (as upper right triangle) is right hand part of some row of this table

Cf. A185910

A256560 Triangle read by rows, sums of 2 distinct nonzero squares plus sums of 2 distinct nonzero cubes: T(n,k) = n^2 + k^2 + n^3 + k^3, 1 <= k <= n-1.

Original entry on oeis.org

14, 38, 48, 82, 92, 116, 152, 162, 186, 230, 254, 264, 288, 332, 402, 394, 404, 428, 472, 542, 644, 578, 588, 612, 656, 726, 828, 968, 812, 822, 846, 890, 960, 1062, 1202, 1386, 1102, 1112, 1136, 1180, 1250, 1352, 1492, 1676, 1910

Offset: 2

Bob Selcoe, Apr 02 2015

All terms are even.
T(n,1) = A011379(n) + 2.
When n=k+1, T(n,k+1) = A011379(n-1) + A011379(n) = 2n^3 - n^2 + n.

			Triangle starts T(2,1):
n\k   1    2    3    4    5    6    7     8    9   10
2:   14
3:   38   48
4:   82   92   116
5:   152  162  186  230
6:   254  264  288  332  402
7:   394  404  428  472  542  644
8:   578  588  612  656  726  828  968
9:   812  822  846  890  960  1062 1202 1386
10:  1102 1112 1136 1180 1250 1352 1492 1676 1910
11:  1454 1464 1488 1532 1602 1704 1844 2028 2262 2552
...
The successive terms are: (2^2 + 1^2 + 2^3 + 1^3), (3^2 + 1^2 + 3^3 + 1^3), (3^2 + 2^2 + 3^3 + 2^3), (4^2 + 1^2 + 4^3 + 1^3), (4^2 + 2^2 + 4^3 + 2^3), (4^2 + 3^2 + 4^3 + 3^3), ...
T(7,4) = 472 because 7^2 + 7^3 + 4^2 + 4^3 = 472.

Cf. A055096 (sums of 2 distinct nonzero squares), A256497 (sums of 2 distinct nonzero cubes), A011379, A024670, A004431, A049450.

a(n) = A055096(n) + A256497(n-1).
T(n,k) = T055096(n,k) + T256547(n-1,k).
T(n,k) = T(n-1,k) + A049450(n).
T(n,k) = T(n,k-1) + A049450(k).
T(n,k) = A011379(n) + A011379(k).

A272063 a(n) = largest k such that A004431(n) +/- k are both positive squares.

Original entry on oeis.org

4, 6, 12, 8, 16, 24, 10, 20, 30, 12, 24, 40, 36, 14, 48, 28, 42, 60, 56, 32, 48, 70, 64, 18, 84, 80, 54, 72, 96, 20, 40, 90, 60, 112, 80, 108, 22, 100, 126, 120, 88, 144, 110, 48, 140, 72, 132, 96, 160, 120, 154, 52, 78, 144, 180

Offset: 1

Bob Selcoe, Apr 19 2016

nonn

There can be more than one value of k such that A004431(n) +/- k are both positive squares; i.e., when there are multiple ways to express A004431(n) as the sum of positive squares. These are the terms which appear more than once in A055096. For example A004431(19) = 65 = {(1^2 + 8^2), (4^2 + 7^2)}: 65 +/- 16 = {7^2, 9^2} and 65 +/- 56 = {3^2, 11^2}. So a(19) = 56 rather than 16.
Similar to A270835; differences occur for n<56 at n = {19,25,38,39,42,51}; i.e., terms A004431(n) which appear more than once in A055096.
Sequence contains every even number >=4 and no odd numbers.

			a(11)=24 because A004431(11) = 40; 40+24 = 8^2 and 40-24 = 4^2.

Cf. A001844, A004431, A055096, A270835.

a(n) = A004431(n)-1 when A004431(n) = k^2 + (k+1)^2 == A001844(k), k>=1.

A332850 Numbers k = a^2 + b^2 such that reversal(k) = a^2 - b^2 for a > b > 0, where reversal is A004086.

Original entry on oeis.org

699796, 4854634, 6752626, 84036010, 931910661, 21584860960, 52554850525, 467170024564, 637843128736, 638730439636, 638734039636, 638943127636, 727830438745, 727834038745, 746710459825, 746754019825, 748943127625, 9894192267061, 401309596403104, 844181015028970

Offset: 1

Metin Sariyar, Feb 26 2020

When b=0, the palindromic numbers m = a^2 + b^2 such that reversal(m) = a^2 - b^2, are A002779 (palindromic squares).

a(19) > 3*10^14, if it exists. - Giovanni Resta, Feb 27 2020

			699796 = 836^2 + 30^2 and 697996 = 836^2 - 30^2.

Cf. A002779, A004086, A035519, A055096, A202386.

Do[If[IntegerReverse[a^2+b^2]==a^2-b^2,Print[{a^2+b^2,a,b}]],{a,1,50000},{b,1,a-1}]

isok(k) = {my(r = fromdigits(Vecrev(digits(k))), s = r+k, d = k-r); d && !(s % 2) && issquare(s/2) && !(d % 2) && issquare(d/2); } \\ Michel Marcus, Feb 27 2020

a(6)-a(18) from Giovanni Resta, Feb 27 2020

a(19)-a(20) from Jinyuan Wang, Apr 10 2025

cp's OEIS Frontend

A055630 Table T(k,m) = k^2 + m read by antidiagonals.

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A256560 Triangle read by rows, sums of 2 distinct nonzero squares plus sums of 2 distinct nonzero cubes: T(n,k) = n^2 + k^2 + n^3 + k^3, 1 <= k <= n-1.

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A272063 a(n) = largest k such that A004431(n) +/- k are both positive squares.

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A332850 Numbers k = a^2 + b^2 such that reversal(k) = a^2 - b^2 for a > b > 0, where reversal is A004086.

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