A136496 -id:A136496 - cp's OEIS frontend

A202342 Numbers occurring exactly twice in Hofstadter H-sequence A005374.

1, 4, 5, 7, 10, 13, 14, 17, 18, 20, 23, 24, 26, 29, 32, 33, 35, 38, 41, 42, 45, 46, 48, 51, 54, 55, 58, 59, 61, 64, 65, 67, 70, 73, 74, 77, 78, 80, 83, 84, 86, 89, 92, 93, 95, 98, 101, 102, 105, 106, 108, 111, 112, 114, 117, 120, 121, 123, 126, 129, 130, 133

Offset: 1

Reinhard Zumkeller, Dec 17 2011

nonn

Position of the n-th occurrence of the digit 1 in A105083(n-1) for n>=1. - Jeffrey Shallit, Mar 08 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.
Eric Weisstein's World of Mathematics, Hofstadter H-Sequence.
Wikipedia, Hofstadter sequence
Index entries for Hofstadter-type sequences

Cf. A005374, A105083, A202340, A136495, A136496, A202341 (complement).

import Data.List (elemIndices)
a202342 n = a202342_list !! (n-1)
a202342_list = elemIndices 2 a202340_list

A202340(a(n)) = 2.
a(n) = A005374(A136496(n)). - Alan Michael Gómez Calderón, Dec 22 2024
a(n) = A136495(A136495(n)). - Alan Michael Gómez Calderón, Jan 06 2025

A020942 First column of 3rd-order Zeckendorf array A136189.

Original entry on oeis.org

1, 5, 7, 10, 14, 18, 20, 24, 26, 29, 33, 35, 38, 42, 46, 48, 51, 55, 59, 61, 65, 67, 70, 74, 78, 80, 84, 86, 89, 93, 95, 98, 102, 106, 108, 112, 114, 117, 121, 123, 126, 130, 134, 136, 139, 143, 147, 149, 153, 155, 158, 162, 164, 167, 171, 175, 177, 180, 184

Offset: 1

Clark Kimberling

I would like to get similar sequences where the least term in the representation is 2 [gives 2 8 11 15 21 27 30..., which is now A064105], 3, 4, 6, etc. They are the 2nd, 3rd, etc. columns of the 3rd-order Zeckendorf array. [See cross-references. - N. J. A. Sloane, Apr 29 2024]
These have now been entered in the OEIS as
column 1: A020942.
column 2: A064105.
column 3: A064106.
column 4: A372749.
column 5: A372750.
column 6: A372752.
column 7: A372756.
column 8: A372757.

			1=1; 5=4+1; 7=6+1; 10=9+1; etc.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, February 2012. - _N. J. A. Sloane_, Jun 10 2012
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.

Cf. A000930, A064105, A064106, A136189, A136496, A202342, A372749, A372750, A372752, A372756, A372757.

Any number n has unique representation as a sum of terms from {1, 2, 3, 4, 6, 9, 13, 19, ...} (cf. A000930) such that no two terms are adjacent or pen-adjacent; e.g., 7=6+1. Sequence gives all n where that representation involves 1.
Conjecture: a(n) = A202342(n) + n. - Sean A. Irvine, May 05 2019 [proved in corrected form in Shallit (2025); it should read a(n) = A202342(n) + n-1]
a(n) = A136496(n) - 1. - Jeffrey Shallit, Mar 08 2025

More terms from Naohiro Nomoto, Sep 17 2001

A136495 Solution of the complementary equation b(n)=a(a(n))+n.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 12, 13, 14, 16, 17, 18, 20, 22, 23, 24, 26, 28, 29, 31, 32, 33, 35, 37, 38, 40, 41, 42, 44, 45, 46, 48, 50, 51, 53, 54, 55, 57, 58, 59, 61, 63, 64, 65, 67, 69, 70, 72, 73, 74, 76, 77, 78, 80, 82, 83, 84, 86, 88, 89, 91, 92, 93, 95, 97, 98, 100, 101, 102

Offset: 1

Clark Kimberling, Jan 01 2008

nonn

b = 1 + (column 1 of Z) = 1 + A020942. The pair (a,b) also satisfy the following complementary equations: b(n)=a(a(a(n)))+1; a(b(n))=a(n)+b(n); b(a(n))=a(n)+b(n)-1; (and others).
Let Z = (3rd order Zeckendorf array) = A136189. Then a = ordered union of columns 1,3,4,6,7,9,10,12,13,... of Z, b = ordered union of columns 2,5,8,11,14,... of Z.
Position of the n-th occurrence of either 1 or 3 in A105083(n-1) for n>=1. - Jeffrey Shallit, Mar 08 2025

			b(1) = a(a(1))+1 = a(1)+1 = 1+1 = 2;
b(2) = a(a(2))+2 = a(3)+2 = 4+2 = 6;
b(3) = a(a(3))+3 = a(4)+3 = 5+3 = 8;
b(4) = a(a(4))+4 = a(5)+4 = 7+4 = 11.

Clark Kimberling and Peter J. C. Moses, Complementary equations and Zeckendorf arrays, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Thirteenth International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 201 (2010) 161-178.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.
Eric Weisstein's World of Mathematics, Hofstadter H-Sequence.

Cf. A005374, A020942, A035513, A105083, A136189, A136496.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a136495 n = (fromJust $ n `elemIndex` tail a005374_list) + 1
-- Reinhard Zumkeller, Dec 17 2011

A005374(a(n)) = n. - Reinhard Zumkeller, Dec 17 2011

a(n) = A005374(A005374(n-1)) + n. - Alan Michael Gómez Calderón, Jul 16 2025

A372749 4th column of the 3-Zeckendorf array (A136189).

Original entry on oeis.org

4, 17, 23, 32, 45, 58, 64, 77, 83, 92, 105, 111, 120, 133, 146, 152, 161, 174, 187, 193, 206, 212, 221, 234, 247, 253, 266, 272, 281, 294, 300, 309, 322, 335, 341, 354, 360, 369, 382, 388, 397, 410, 423, 429, 438, 451, 464, 470, 483, 489, 498, 511, 517, 526, 539

Offset: 1

A.H.M. Smeets, May 12 2024

nonn

The 3-Zeckendorf array (A136189) is based on the Narayana (Narayana's cow sequence A000930) weighted representation of n (see A350215).

Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.

Cf. A000930, A136189, A350215.
The k-th row: A000930(n+2) (k=1)
The k-th column: A020942 (k=1), A064105 (k=2), A064106 (k=3), this sequence (k=4),  A372750 (k=5).
The k-th prepended column: A005374 (k=1), A202342 (k=4)

a(n) = A202342(n) + A136496(n) + A381841(n) - 2. - Jeffrey Shallit, Mar 08 2025

A372750 5th column of the 3-Zeckendorf array (A136189).

Original entry on oeis.org

6, 25, 34, 47, 66, 85, 94, 113, 122, 135, 154, 163, 176, 195, 214, 223, 236, 255, 274, 283, 302, 311, 324, 343, 362, 371, 390, 399, 412, 431, 440, 453, 472, 491, 500, 519, 528, 541, 560, 569, 582, 601, 620, 629, 642, 661, 680, 689, 708, 717, 730, 749, 758, 771

Offset: 1

A.H.M. Smeets, May 12 2024

nonn

The 3-Zeckendorf array (A136189) is based on the Narayana (Narayana's cow sequence A000930) weighted representation of n (see A350215).

Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.

Cf. A000930, A136189, A350215.
The k-th row: A000930(n+2) (k=1)
The k-th column: A020942 (k=1), A064105 (k=2), A064106 (k=3), A372749 (k=4), this sequence (k=5).
The k-th prepended column: A005374 (k=1), A202342 (k=4)

a(n) = A202342(n) + A136496(n) + 2*A381841(n) - 3. - Jeffrey Shallit, Mar 08 2025

A372752 6th column of the 3-Zeckendorf array (A136189).

Original entry on oeis.org

9, 37, 50, 69, 97, 125, 138, 166, 179, 198, 226, 239, 258, 286, 314, 327, 346, 374, 402, 415, 443, 456, 475, 503, 531, 544, 572, 585, 604, 632, 645, 664, 692, 720, 733, 761, 774, 793, 821, 834, 853, 881, 909, 922, 941, 969, 997, 1010, 1038, 1051, 1070, 1098

Offset: 1

A.H.M. Smeets, May 12 2024

nonn

The 3-Zeckendorf array (A136189) is based on the Narayana (Narayana's cow sequence A000930) weighted representation of n (see A350215).

Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.

Cf. A000930, A136189, A350215.
The k-th row: A000930(n+2) (k=1)
The k-th column: A020942 (k=1), A064105 (k=2), A064106 (k=3), A372749 (k=4), A372750 (k=5), this sequence (k=6).
The k-th prepended column: A005374 (k=1), A202342 (k=4)

a(n) = 2*A202342(n) + A136496(n) + 3*A381841(n) - 4. - Jeffrey Shallit, Mar 08 2025

A381841 Position of the n-th occurrence of the digit 3 in A105083(n-1) for n>=1.

Original entry on oeis.org

3, 9, 12, 16, 22, 28, 31, 37, 40, 44, 50, 53, 57, 63, 69, 72, 76, 82, 88, 91, 97, 100, 104, 110, 116, 119, 125, 128, 132, 138, 141, 145, 151, 157, 160, 166, 169, 173, 179, 182, 186, 192, 198, 201, 205, 211, 217, 220, 226, 229, 233, 239, 242, 246, 252, 258

Offset: 1

Jeffrey Shallit, Mar 08 2025

nonn

Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.

Cf. A064105, A105083, A136495, A136496, A202342.

a(n) = A064105(n) + 1.

a(n) = A136495(n) + A136496(n).

A372756 7th column of the 3-Zeckendorf array (A136189).

Original entry on oeis.org

13, 54, 73, 101, 142, 183, 202, 243, 262, 290, 331, 350, 378, 419, 460, 479, 507, 548, 589, 608, 649, 668, 696, 737, 778, 797, 838, 857, 885, 926, 945, 973, 1014, 1055, 1074, 1115, 1134, 1162, 1203, 1222, 1250, 1291, 1332, 1351, 1379, 1420, 1461, 1480, 1521

Offset: 1

A.H.M. Smeets, May 12 2024

nonn

The 3-Zeckendorf array (A136189) is based on the Narayana (Narayana's cow sequence A000930) weighted representation of n (see A350215).

Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.

Cf. A000930, A136189, A350215.
The k-th row: A000930(n+2) (k=1), A372760 (k=2).
The k-th column: A020942 (k=1), A064105 (k=2), A064106 (k=3), A372749 (k=4), A372750 (k=5), A372752 (k=6), this sequence (k=7), A372757 (k=8).
The k-th prepended column: A005374 (k=1), A136495 (k=2), A023443 (k=3), A202342 (k=4), A372758 (k=5), A372759 (k=6).

a(n) = 3*A202342(n) + 2*A136496(n) + 4*A381841(n) - 6. - Jeffrey Shallit, Mar 08 2025

A372757 8th column of the 3-Zeckendorf array (A136189).

Original entry on oeis.org

19, 79, 107, 148, 208, 268, 296, 356, 384, 425, 485, 513, 554, 614, 674, 702, 743, 803, 863, 891, 951, 979, 1020, 1080, 1140, 1168, 1228, 1256, 1297, 1357, 1385, 1426, 1486, 1546, 1574, 1634, 1662, 1703, 1763, 1791, 1832, 1892, 1952, 1980, 2021, 2081, 2141

Offset: 1

A.H.M. Smeets, May 12 2024

nonn

The 3-Zeckendorf array (A136189) is based on the Narayana (Narayana's cow sequence A000930) weighted representation of n (see A350215).

Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.

Cf. A000930, A136189, A350215.
The k-th row: A000930(n+2) (k=1), A372760 (k=2).
The k-th column: A020942 (k=1), A064105 (k=2), A064106 (k=3), A372749 (k=4), A372750 (k=5), A372752 (k=6), A372756 (k=7), this sequence (k=8).
The k-th prepended column: A005374 (k=1), A136495 (k=2), A023443 (k=3), A202342 (k=4), A372758 (k=5), A372759 (k=6).

a(n) = 4*A202342(n) + 3*A136496(n) + 6*A381841(n) - 9. - Jeffrey Shallit, Mar 08 2025

A138253 Beatty discrepancy of the complementary equation b(n) = a(a(n)) + n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Mar 09 2008

nonn

Suppose that (a(n)) and (b(n)) are complementary sequences that satisfy a complementary equation b(n) = f(a(n), n) and that the limits r = lim_{n->inf} a(n)/n and s = lim_{n->inf} b(n)/n exist and are both in the open interval (0,1). Let c(n) = floor(a(n)) and d(n) = floor(b(n)), so that (c(n)) and (d(n)) are a pair of Beatty sequences. Define e(n) = d(n) - f(c(n), n). The sequence (e(n)) is here introduced as the Beatty discrepancy of the complementary equation b(n) = f(a(n), n). In the case at hand, (e(n)) measures the closeness of the pair (A136495, A136496) to the Beatty pair (A138251, A138252).

			d(1) - c(c(1)) - 1 =  3 - 1 - 1 = 1;
d(2) - c(c(2)) - 2 =  6 - 2 - 2 = 2;
d(3) - c(c(3)) - 3 =  9 - 5 - 3 = 1;
d(4) - c(c(4)) - 4 = 12 - 7 - 4 = 1.

Cf. A136495, A136496, A138251, A138252.

A138253(n) = d(n) - c(c(n)) - n, where c(n) = A138251(n), d(n) = A138252(n).

cp's OEIS Frontend

A202342 Numbers occurring exactly twice in Hofstadter H-sequence A005374.

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A020942 First column of 3rd-order Zeckendorf array A136189.

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A136495 Solution of the complementary equation b(n)=a(a(n))+n.

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A372749 4th column of the 3-Zeckendorf array (A136189).

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A372750 5th column of the 3-Zeckendorf array (A136189).

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A372752 6th column of the 3-Zeckendorf array (A136189).

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A381841 Position of the n-th occurrence of the digit 3 in A105083(n-1) for n>=1.

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A372756 7th column of the 3-Zeckendorf array (A136189).

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A372757 8th column of the 3-Zeckendorf array (A136189).

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