A078510 -id:A078510 - cp's OEIS frontend

A265404 a(n) = number of Spironacci numbers (A078510) needed to sum to n using the greedy algorithm.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2

Offset: 0

Antti Karttunen, Dec 16 2015

nonn

a(0) = 0, because no numbers are needed to form an empty sum, which is zero.

First 2 occurs as a(17), first 3 at a(234), first 4 at a(3266).

			For n=17, the largest Spironacci number <= 17 is 16 (= A078510(22)). 17 - 16 = 1, which is A078510(1), thus 17 = A078510(22) + A078510(1), requiring only two such numbers for its sum, thus a(17) = 2.
For n=234, the largest Spironacci number <= 234 is 217 (= A078510(45)). 234-217 = 17 (whose decomposition is shown above), so 234 = A078510(45) + A078510(22) + A078510(1), thus a(234) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..10132

Cf. A078510 (from its term a(7) onward gives also the positions of ones here).

Cf. also A007895, A053610, A265743, A265744, A265745.

A265370 a(0) = 0; for n >= 1, a(n) = A070939(A078510(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14

Offset: 0

Antti Karttunen, Dec 16 2015

nonn

Number of significant bits in the binary representation of n-th Spironacci number, A078510(n).

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A070939, A078510.

Cf. also A020909.

a(0) = 0; for n >= 1, a(n) = A070939(A078510(n)).

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

Original entry on oeis.org

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.

A265408 Prime factorization representation of Spironacci polynomials: a(0) = 1, a(1) = 2, and for n > 1, a(n) = A003961(a(n-1)) * a(A265409(n)).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 38, 138, 870, 9765, 213675, 4309305, 201226025, 9679967297, 810726926009, 40855897091009, 4259653632223561, 380804291082185737, 44319264099050115071, 4644246052673250585913

Offset: 0

Antti Karttunen, Dec 13 2015

nonn

The polynomials encoded by these numbers could also be called "Fernandez spiral polynomials" after Neil Fernandez, who discovered sequence A078510, which is obtained when they are evaluated at X=1.

The polynomial recurrence uses the same composition rules as the Fibonacci polynomials (A206296), but with the neighborhood rules of A078510, where the other polynomial is taken from the nearest inner neighbor (A265409) when the polynomials are arranged as a spiral into a square grid. See A265409 for the illustration.

			n    a(n)   prime factorization    Spironacci polynomial
------------------------------------------------------------
0       1   (empty)                S_0(x) = 0
1       2   p_1                    S_1(x) = 1
2       3   p_2                    S_2(x) = x
3       5   p_3                    S_3(x) = x^2
4       7   p_4                    S_4(x) = x^3
5      11   p_5                    S_5(x) = x^4
6      13   p_6                    S_6(x) = x^5
7      17   p_7                    S_7(x) = x^6
8      38   p_8 * p_1              S_8(x) = x^7 + 1
9     138   p_9 * p_2 * p_1        S_9(x) = x^8 + x + 1

Cf. A003961, A265407, A265409.

Cf. also A078510, A206296 and A265752, A265753.

a(0) = 1, a(1) = 2, and for n >= 2, a(n) = A003961(a(n-1)) * a(A265409(n)).
Other identities. For all n >= 0:
A078510(n) = A001222(a(n)). [when each polynomial is evaluated at x=1]
A265407(n) = A248663(a(n)). [at x=2 over the field GF(2)]

A265409 a(n) = index to the nearest inner neighbor in Ulam-style square-spirals using zero-based indexing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 9, 9, 9, 10, 11, 12, 12, 12, 13, 14, 15, 16, 16, 16, 17, 18, 19, 20, 20, 20, 21, 22, 23, 24, 25, 25, 25, 26, 27, 28, 29, 30, 30, 30, 31, 32, 33, 34, 35, 36, 36, 36, 37, 38, 39, 40, 41, 42, 42, 42, 43, 44, 45, 46, 47, 48, 49, 49, 49, 50

Offset: 1

Antti Karttunen, Dec 13 2015

nonn

Each n occurs A265411(n+1) times.

Useful when defining recurrences like A078510 and A265408.

			We arrange natural numbers as a counterclockwise spiral into the square grid in the following manner (here A stands for 10, B for 11). The first square corresponds with n (where the initial term 0 is at the center), and the second square with the value of a(n). This sequence doesn't specify a(0), thus it is shown as an asterisk (*):
                    44322
            432B    40002B
            501A    50*01A
            6789    600119
                    667899
-
For each n > 0, we look for the nearest horizontally or vertically adjacent neighbor of n towards the center that is not n-1, which will then be value of a(n) [e.g., it is 0 for 3, 5 and 7, while it is 1 for 8, 9 and A (10) and 2 for B (11)] unless n is in the corner (one of the terms of A002620), in which case the value is the nearest diagonally adjacent neighbor towards the center, e.g. 0 for 2, 4 and 6, while it is 1 for 9).
See also the illustration at A078510.

Antti Karttunen, Table of n, a(n) for n = 1..10000

One less than A265410(n+1).
Cf. A002620, A240025, A265359.
Cf. also A033951, A063826, A078510, A265408.

If n <= 7, a(n) = 0 for n >= 8: if either A240025(n) or A240025(n-1) is not zero [when n or n-1 is in A002620], then a(n) = a(n-1), otherwise, a(n) = 1 + a(n-1).
If n <= 7, a(n) = 0, for n >= 8, a(n) = a(n-1) + (1-A240025(n))*(1-A240025(n-1)). [The same formula in a more compact form.]
a(n) = A265410(n+1) - 1.
Other identities. For all n >= 0:
a(n) = n - A265359(n).

A265407 Spironacci-style recurrence: a(0)=0, a(1)=1, a(n) = 2*a(n) XOR a(A265409(n)).

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 64, 129, 259, 519, 1036, 2074, 4150, 8296, 16600, 33208, 66424, 132832, 265696, 531424, 1062880, 2125696, 4251521, 8502785, 17005825, 34011905, 68023301, 136047622, 272093206, 544188470, 1088378998, 2176753882, 4353515996, 8707015520, 17414063992, 34828160840, 69656354600, 139312643368

Offset: 0

Antti Karttunen, Dec 13 2015

nonn

Spironacci-polynomials evaluated at X=2 over the field GF(2).

This is otherwise computed like A078510, which starts with a(0)=0 placed in the center of spiral (in square grid), followed by a(1) = 1, after which each term is a sum of two previous terms that are nearest when terms are arranged in a spiral, that is terms a(n-1) and a(A265409(n)), except here we first multiply the term a(n-1) by 2, and use carryless XOR (A003987) instead of normal addition.

Antti Karttunen, Table of n, a(n) for n = 0..256

Cf. A003987, A248663, A265408, A265409.

Cf. also A078510, A264977.

a(0)=0, a(1)=1; after which, a(n) = 2*a(n) XOR a(A265409(n)).

a(n) = A248663(A265408(n)).

A079421 Spiro-Fibonacci differences, a(n) = difference of two previous terms that are nearest when terms arranged in a spiral.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 0

Neil Fernandez, Jan 07 2003

nonn

			Terms are written in square boxes radiating spirally (cf. Ulam prime spiral). a(0)=0 and a(1)=1, so write 0 and then 1 to its right. a(2) goes in the box below a(1). The nearest two filled boxes contain a(0) and a(1), so a(2)=abs(a(0)-a(1))=abs(0-1)=1. a(3) goes in the box to the left of a(2). The nearest two filled boxes contain a(0) and a(2), so a(3)=abs(a(0)-a(2))=abs(0-1)=1.

Sean A. Irvine, Table of n, a(n) for n = 0..10000
N. Fernandez, Graphical representations of some Spiro-Fibonacci sequences

Cf. A063826, A078510.

A283474 a(0) = 0, a(1) = 1, for n > 1, a(n) = a(n-1) + a(n-A002487(n)).

Original entry on oeis.org

0, 1, 2, 3, 6, 8, 14, 20, 40, 48, 68, 82, 150, 190, 272, 354, 708, 858, 1130, 1280, 2138, 2328, 3186, 3894, 7080, 8210, 10538, 11818, 20028, 23214, 33752, 44290, 88580, 100398, 123612, 134150, 222730, 233268, 277558, 300772, 534040, 567792, 691404, 725156, 1025928, 1126326, 1427098, 1704656, 3131754, 3665794

Offset: 0

Antti Karttunen, Mar 23 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A000045, A002487, A078510, A284007, A284013.

Cf. A283479 (first differences).

a(0) = 0, a(1) = 1, for n > 1, a(n) = a(n-1) + a(A284013(n)).

A094926 A hexagonal spiral Fibonacci sequence.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 14, 23, 38, 63, 102, 168, 272, 445, 720, 1173, 1898, 3084, 5004, 8102, 13143, 21268, 34472, 55841, 90376, 146382, 237028, 383578, 621046, 1005341, 1626832, 2633338, 4262063, 6896574, 11161708, 18063264, 29233060, 47301328, 76547494, 123870067

Offset: 0

Yasutoshi Kohmoto

Consider the following spiral:
..........a(5)..a(6)..a(7)
.......a(4)..a(0)..a(1)..a(8)
....a(13).a(3)..a(2)..a(9)
.......a(12).a(11).a(10)
Then a(0)=0, a(1)=1, a(n)=a(n-1)+Sum{a(i), a(i) is adjacent to a(n-1)}; 6 terms around a(m) touch a(m).
From Manfred Scheucher, Jun 03 2015: (Start)
Since a(n-1)+a(n-2) <= a(n) <= a(n-1)+a(n-2)+a(n-k)+a(n-k-1) holds for some k where k=Theta(sqrt(n)), and also 2^n >= a(n) >= F(n) holds, I believe that a(n) = (a(n-1)+a(n-2))/(1-c*d^(-sqrt(n))) can be proofen properly. This would lead to a similar asymptotic behavior as F(n), i.e., a(n) ~ c*phi^n where phi=1.61803... denotes the golden ratio and c=0.54172... is a constant.
Actually, the terms in the b-file seem to confirm this conjecture because exp(log(a(n))/n) seem to converge to phi. In particular, g(100)=1.60..., g(1000)=1.616..., g(10000)=1.6178..., g(30602)=1.61800..., where g(n):=exp(log(a(n))/n).
(End)

			Spiral with 2 rings:
... ..5 ... ..8 ... .14 ...
..3 ... ..0 ... ..1 ... .23
... ..2 ... ..1 ... .38 ...
... ... 102 ... .63 ... ...
Spiral with 3 rings:
...... ...... ..1173 ...... ..1898 ...... ..3084 ...... ..5004 ...... ......
...... ...720 ...... .....5 ...... .....8 ...... ....14 ...... ..8102 ......
...445 ...... .....3 ...... .....0 ...... .....1 ...... ....23 ...... .13143
...... ...272 ...... .....2 ...... .....1 ...... ....38 ...... .21268 ......
...... ...... ...168 ...... ...102 ...... ....63 ...... .34472 ...... ......
...... ...... ...... 146382 ...... .90376 ...... .55841 ...... ...... ......

Manfred Scheucher, Table of n, a(n) for n = 0..1322
N. Fernandez, Spiro-Fibonacci numbers
Vaclav Kotesovec, Graph - the asymptotic ratio
Manfred Scheucher, Sage Script

Cf. A094925, A078510, A079421, A079422.

a(n) ~ c*phi^n with phi=1.61803... being the golden ratio and c = A258639 = 0.54172002195814443386932... (conjectured). - Manfred Scheucher, Jun 03 2015

Offset changed and more terms from Manfred Scheucher, Jun 03 2015

A079422 a(n) = number of 1's in the first n^2 Spiro-Fibonacci differences.

Original entry on oeis.org

0, 3, 7, 11, 15, 19, 24, 26, 32, 41, 49, 55, 61, 67, 75, 78, 83, 111, 119, 135, 139, 155, 160, 168, 182, 217, 229, 249, 259, 279, 293, 303, 312, 342, 399, 454, 493, 530, 566, 603, 642, 681, 722, 765, 817, 875, 909, 958, 992, 1044, 1107, 1160, 1215, 1267, 1315

Offset: 1

Neil Fernandez, Jan 07 2003

nonn

			The Spiro-Fibonacci differences are 0,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,...(A079421). Terms are written in square boxes radiating spirally. a(n) = the sum of the first n^2 terms in A079421, i.e. the number of 1's in a spiral of height n and width n.

N. Fernandez, Graphical representations of some Spiro-Fibonacci sequences

Cf. A063826, A078510, A079421.

More terms from Sean A. Irvine, Aug 13 2025

cp's OEIS Frontend

A265404 a(n) = number of Spironacci numbers (A078510) needed to sum to n using the greedy algorithm.

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A265370 a(0) = 0; for n >= 1, a(n) = A070939(A078510(n)).

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

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A265408 Prime factorization representation of Spironacci polynomials: a(0) = 1, a(1) = 2, and for n > 1, a(n) = A003961(a(n-1)) * a(A265409(n)).

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A265409 a(n) = index to the nearest inner neighbor in Ulam-style square-spirals using zero-based indexing.

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A265407 Spironacci-style recurrence: a(0)=0, a(1)=1, a(n) = 2*a(n) XOR a(A265409(n)).

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A079421 Spiro-Fibonacci differences, a(n) = difference of two previous terms that are nearest when terms arranged in a spiral.

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A283474 a(0) = 0, a(1) = 1, for n > 1, a(n) = a(n-1) + a(n-A002487(n)).

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A094926 A hexagonal spiral Fibonacci sequence.

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A079422 a(n) = number of 1's in the first n^2 Spiro-Fibonacci differences.

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