A278619 -id:A278619 - cp's OEIS frontend

A278180 Square spiral in which each new term is the sum of its two largest neighbors.

1, 1, 2, 3, 4, 7, 8, 15, 16, 17, 33, 35, 37, 72, 76, 80, 84, 164, 172, 180, 188, 368, 384, 401, 418, 435, 853, 888, 925, 962, 999, 1961, 2037, 2117, 2201, 2285, 2369, 4654, 4826, 5006, 5194, 5382, 5570, 10952, 11336, 11737, 12155, 12590, 13025, 13460, 26485, 27373, 28298, 29260, 30259, 31258, 32257, 63515

Offset: 1

Omar E. Pol, Nov 14 2016

nonn

To evaluate a(n) consider the neighbors of a(n) that are present in the spiral when a(n) should be a new term in the spiral.

For the same idea but for a hexagonal spiral see A278619; and for a right triangle see A278645. It appears that the same idea for an isosceles triangle and also for a square array gives A030237. - Omar E. Pol, Dec 04 2016

			Illustration of initial terms as a square spiral:
.
.          84----80----76-----72----37
.           |                        |
.          164    4-----3-----2     35
.           |     |           |      |
.          172    7     1-----1     33
.           |     |                  |
.          180    8-----15----16----17
.           |
.          188---368---384---401---418
.
a(21) = 188 because the sum of its two largest neighbors is 180 + 8 = 188.
a(22) = 368 because the sum of its two largest neighbors is 180 + 188 = 368.
a(23) = 384 because the sum of its two largest neighbors is 368 + 16 = 384.
a(24) = 401 because the sum of its two largest neighbors is 384 + 17 = 401.
a(25) = 418 because the sum of its two largest neighbors is 401 + 17 = 418.
a(26) = 435 because the sum of its two largest neighbors is 418 + 17 = 435.

Peter Kagey, Table of n, a(n) for n = 1..10000
Peter Kagey, Bitmap illustrating the parity of the first one million terms. (Even and odd numbers are represented with black and white pixels respeectively.)

Cf. A030237, A078510, A141481, A274917, A278181, A278619, A278645.

A278645 Triangle read by rows in which each new term is the sum of its two largest neighbors in the structure.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 8, 15, 22, 29, 23, 45, 74, 103, 132, 68, 142, 245, 377, 509, 641, 210, 455, 832, 1341, 1982, 2623, 3264, 665, 1497, 2838, 4820, 7443, 10707, 13971, 17235, 2162, 5000, 9820, 17263, 27970, 41941, 59176, 76411, 93646, 7162, 16982, 34245, 62215, 104156, 163332, 239743, 333389, 427035, 520681

Offset: 1

Omar E. Pol, Nov 24 2016

To evaluate T(n,k) consider only the two largest neighbors of T(n,k) that are present in the triangle when T(n,k) should be a new term in the triangle.
For the same idea but for a square spiral see A278180; and for a hexagonal spiral see A278619.
It appears that the same idea for an isosceles triangle and also for a square array gives A030237.

			Triangle begins:
1;
1,    2;
3,    5,     7;
8,    15,    22,    29;
23,   45,    74,    103,   132;
68,   142,   245,   377,   509,    641;
210,  455,   832,   1341,  1982,   2623,   3264;
665,  1497,  2838,  4820,  7443,   10707,  13971,  17235;
2162, 5000,  9820,  17263, 27970,  41941,  59176,  76411,  93646;
7162, 16982, 34245, 62215, 104156, 163332, 239743, 333389, 427035, 520681;
...

Cf. A030237, A278180, A278317, A278619.

A365960 Sum of the 6 nearest neighbors of n in a hexagonal spiral with positive integers.

Original entry on oeis.org

27, 38, 40, 48, 56, 64, 54, 78, 86, 72, 100, 84, 114, 96, 128, 108, 142, 120, 126, 162, 138, 176, 150, 156, 196, 168, 174, 216, 186, 192, 236, 204, 210, 256, 222, 228, 234, 282, 246, 252, 302, 264, 270, 276, 328, 288, 294, 300, 354, 312, 318, 324, 380, 336, 342, 348, 406, 360, 366, 372, 378, 438, 390, 396, 402

Offset: 1

Eric Angelini and Giorgos Kalogeropoulos, Sep 23 2023

nonn

			Spiral begins:
                  49--48--47--46--45
                  /                 \
                50  28--27--26--25  44
                /   /             \   \
              51  29  13--12--11  24  43
              /   /   /         \   \   \
            52  30  14   4---3  10  23  42
            /   /   /   /     \   \   \   \
          53  31  15   5   1---2   9  22  41
            \   \   \   \         /   /   /
            54  32  16   6---7---8  21  40
              \   \   \             /   /
               55  33  17--18--19--20  39
                \   \                 /
                56  34--35--36--37--38
                  \
                  57--58--59--60--61
.
The 6 nearest neighbors of 2 are 1,3,7,8,9,10. Their sum is a(2)=38.

Eric Angelini, An hexagonal seq?, Personal blog "Cinquante signes", Sept 2023.
Eric Angelini, An hexagonal seq?, Personal blog "Cinquante signes", Sept 2023 [Cached copy]

Cf. A214176, A278181, A278619.

step=9; ta[x_]:=Table[12,x];f=Flatten[Table[Table[{ta[If[m==2,k-1,k]],16+2m+12k},{m,6}],{k,0,step}]][[3;;]];Join[{27,38},f+6Range[3,Length@f+2]]

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A278180 Square spiral in which each new term is the sum of its two largest neighbors.

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A278645 Triangle read by rows in which each new term is the sum of its two largest neighbors in the structure.

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A365960 Sum of the 6 nearest neighbors of n in a hexagonal spiral with positive integers.

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Mathematica