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User: Iskender Ozturk

Iskender Ozturk has authored 2 sequences.

A373629 a(n) = sum of all numbers whose binary expansion is n bits long, starts and ends with a 1 bit, and contains no 00 bit pairs.

1, 3, 12, 39, 131, 426, 1389, 4503, 14596, 47259, 152991, 495162, 1602521, 5186067, 16782828, 54310911, 175754731, 568755690, 1840534485, 5956098495, 19274345876, 62373103443, 201843619047, 653179698234, 2113733947681, 6840186809691, 22135309606524, 71631366769623

Offset: 1

Iskender Ozturk, Melike Caliskan, Betül Küçükgök, Ecem Yanik, Irem Türker, Rüya Kılıçarslan, Jun 11 2024

The numbers that are summed are the terms t of A247648 in the range 2^(n-1) <= t < 2^n.

There are Fibonacci(n) of these numbers (per Grimaldi's exercise, in which closed walks on the u-v graph there are a 1 bit at a visit to u and a 0 bit at a visit to v), and this allows recurrences etc. for a(n).

			For n=5, the terms of A247648 that are in the interval [16, 31] are 21, 23, 27, 29, and 31, so a(5) = 21+23+27+29+31 = 131.

R. Grimaldi, (2012). Fibonacci and Catalan Numbers: An Introduction, page 80, Example 12.1.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Iskender Ozturk, Walking paths.
Index entries for linear recurrences with constant coefficients, signature (3,3,-6,-4).

Cf. A000045 (Fibonacci numbers), A247648.

LinearRecurrence[{3, 3, -6, -4}, {1, 3, 12, 39}, 30] (* Paolo Xausa, Jun 19 2024 *)

Vec(x/((1 - x - x^2)*(1 - 2*x - 4*x^2)) + O(x^40)) \\ Michel Marcus, Jun 16 2024

a(n) = Sum_{i=F(n+1)..F(n+2)-1} A247648(i) where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) = a(n-1) + a(n-2) + F(n)*2^(n-1).
a(n) = 3*a(n-1) + 3*a(n-2) - 6*a(n-3) - 4*a(n-4).
a(n) = F(n)*(2^n-1) - Sum_{i=1..n-1} F(i)*F(n-i-1)*2^(n-i-1).
G.f.: x/((1 - x - x^2)*(1 - 2*x - 4*x^2)).
E.g.f.: 2*(exp(x)*(sqrt(5)*cosh(sqrt(5)*x) + 7*sinh(sqrt(5)*x)) - exp(x/2)*(sqrt(5)*cosh(sqrt(5)*x/2) + 4*sinh(sqrt(5)*x/2)))/(11*sqrt(5)). - Stefano Spezia, Jun 19 2024

A367636 a(n) is the number of distinct combinations that can be created by painting the sections on a shape with n divisions that rotates around its center and consists of 4 identical arms at 90-degree intervals.

Original entry on oeis.org

2, 12, 140, 2088, 32912, 524832, 8390720, 134226048, 2147516672, 34359869952, 549756339200, 8796095121408, 140737496748032, 2251799847247872, 36028797153198080, 576460752840327168, 9223372039002324992, 147573952598266478592, 2361183241469182607360, 37778931863094601187328

Offset: 1

Kadir E. Celik, Alp Giray Datlar, Iskender Ozturk, and Gulnur Ozbek, Nov 25 2023

A shape/object consists of n divisions (cells) that rotates around its center and consists of 4 identical arms at 90-degree intervals.
Each division (cell) can be unpainted (white) or painted (black).
(4n-3) is the number of divisions (cells) on the object/shape which consists of 4 identical arms at 90-degree intervals.

			In the figures below, "[ ]" represents an unpainted cell; "[o]" represents a painted cell.
For n = 1, there are a(1) = 2 combinations:
.
  [ ]  [o]
.
For n = 2, there are a(2) = 12 combinations:
.
    [ ]         [ ]         [ ]        [ ]
 [ ][ ][ ]   [ ][ ][o]   [ ][o][ ]  [ ][o][o]
    [ ]         [ ]         [ ]        [ ]
.
    [ ]         [o]         [o]        [o]
 [o][ ][o]   [ ][ ][o]   [o][ ][ ]  [ ][o][ ]
    [ ]         [ ]         [o]        [o]
.
    [ ]         [o]         [o]        [o]
 [o][o][ ]   [o][o][ ]   [o][ ][o]  [o][o][o]
    [o]         [o]         [o]        [o]
.
For n = 3, there are a(3) = 140 combinations:
.
       [ ]               [ ]               [ ]              [ ]
       [ ]               [ ]               [ ]              [ ]
 [ ][ ][ ][ ][ ]   [ ][ ][ ][o][ ]   [ ][ ][ ][ ][o]  [ ][ ][o][ ][ ]
       [ ]               [ ]               [ ]              [ ]
       [ ]               [ ]               [ ]              [ ]
.
       [ ]               [ ]               [ ]              [ ]
       [ ]               [ ]               [ ]              [ ]
 [ ][ ][o][o][ ]   [ ][ ][o][ ][o]   [ ][ ][ ][o][o]  [ ][o][ ][o][ ]
       [ ]               [ ]               [ ]              [ ]
       [ ]               [ ]               [ ]              [ ]
...

A. Nesin, Matematik ve sonsuz [Math and infinity], Nesin Yayıncılık, 2019, pages 137-143.

Gulnur Ozbek, Illustrations for initial terms (with painted cells black, unpainted cells white).
Index entries for linear recurrences with constant coefficients, signature (22,-104,128).

CoefficientList[Series[2*(1 - 16*x + 42*x^2)/((1 - 2*x)*(1 - 4*x)*(1 - 16*x)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Dec 10 2023 *)

a(n) = 2^(4n-5) + 2^(2n-3) + 2^(n-1).
a(n) is the sum of the terms in the n-th row of the following triangle, where k is the number of divisions (cells) which are colored/painted black.
.
n\k| 0 1  2   3   4    5    6    7    8    9   10   11   12   13  14 ...  4n-3
---+--------------------------------------------------------------------------
 1 | 1 1
 2 | 1 2  3   3   2    1
 3 | 1 3 10  22  34   34   22   10    3    1
 4 | 1 4 21  73 184  327  434  434  327  184   73   21    4   1
 5 | 1 5 36 172 604 1556 3108 4876 6098 6098 4876 3108 1556 604 172 36 5 1
...|
 n | 1 n ...
The term at the intersection of any row and column is
    C((4n-3),k)/4 + C([(4n-3)/2],[k/2])/4
                  + C([(4n-3)/4],[k/4])/2  for k == 0 or 1 (mod 4),
    C((4n-3),k)/4 + C([(4n-3)/2],[k/2])/4  for k == 2 or 3 (mod 4)
  where [] is the floor function.
G.f.: 2*x*(1 - 16*x + 42*x^2)/((1 - 2*x)*(1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Dec 03 2023

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User: Iskender Ozturk

A373629 a(n) = sum of all numbers whose binary expansion is n bits long, starts and ends with a 1 bit, and contains no 00 bit pairs.

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