A161441 -id:A161441 - cp's OEIS frontend

A001025 Powers of 16: a(n) = 16^n.

1, 16, 256, 4096, 65536, 1048576, 16777216, 268435456, 4294967296, 68719476736, 1099511627776, 17592186044416, 281474976710656, 4503599627370496, 72057594037927936, 1152921504606846976, 18446744073709551616, 295147905179352825856, 4722366482869645213696, 75557863725914323419136, 1208925819614629174706176

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 16), L(1, 16), P(1, 16), T(1, 16). Essentially same as Pisot sequences E(16, 256), L(16, 256), P(16, 256), T(16, 256). See A008776 for definitions of Pisot sequences.
Convolution-square (auto-convolution) of A098430. - R. J. Mathar, May 22 2009
Subsequence of A161441: A160700(a(n)) = 1. - Reinhard Zumkeller, Jun 10 2009
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 16-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Muniru A Asiru, Table of n, a(n) for n = 0..820 (terms n = 0..100 from T. D. Noe)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 280
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for linear recurrences with constant coefficients, signature (16).

Partial sums give A131865.

Cf. A000079, A000302, A001018, A005843, A008586.

List([0..20],n->16^n); # Muniru A Asiru, Nov 07 2018

a001025 = (16 ^)
a001025_list = iterate (* 16) 1  -- Reinhard Zumkeller, Nov 07 2012

A001025:=-1/(-1+16*z); # Simon Plouffe in his 1992 dissertation

Table[4^(2*n), {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Mar 01 2009 *)

A001025(n):=16^n$
makelist(A001025(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=1<<(4*n) \\ Charles R Greathouse IV, Feb 01 2012

print([16**n for n in range(20)]) # Stefano Spezia, Nov 10 2018

[lucas_number1(n,16,0) for n in range(1, 18)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-16*x).
E.g.f.: exp(16*x).
From Muniru A Asiru, Nov 07 2018: (Start)
a(n) = 16^n.
a(0) = 1, a(n) = 16*a(n-1). (End)
a(n) = 4^A005843(n) = 2^A008586(n) = A000302(n)^2 = A000079(n)*A001018(n). - Muniru A Asiru, Nov 10 2018
a(n) = ( Sum_{k = 0..n} (2*k + 1)*binomial(2*n + 1, n - k) ) * ( Sum_{k = 0..n} (-1)^k/(2*k + 1)*binomial(2*n + 1, n - k) ). - Peter Bala, Feb 12 2019
a(n) = Sum_{k = 0..2*n} A000984(k) * A000984(2*n-k). - Peter Bala, Aug 23 2025

A160700 a(n) = if n<16 then n else a(floor(n/16)) XOR (n mod 16).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13, 3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12, 4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11, 5, 4, 7, 6, 1, 0, 3, 2, 13

Offset: 0

Reinhard Zumkeller, Jun 01 2009

A very simple hash function for the nonnegative integers.

a(A000079(n))=A133145(n); a(A000302(n))=A010685(n); a(A001025(n))=A161452(n); a(A161440(n))=0; a(A161441(n))=1; a(A161442(n))=2; a(A161443(n))=3; a(A161444(n))=4; a(A161445(n))=5; a(A161446(n))=6; a(A161447(n))=7; a(A161448(n))=8; a(A161449(n))=9; a(A161450(n))=10; a(A161451(n))=11; a(A161452(n))=12; a(A161453(n))=13; a(A161454(n))=14; a(A161455(n))=15. - Reinhard Zumkeller, Jun 10 2009

R. Zumkeller, Table of n, a(n) for n = 0..10000

import Data.Bits (xor)
a160700 n = a160700_list !! n
a160700_list = [0..15] ++ map f [16..] where
   f x = a160700 x' `xor` m :: Int where (x', m) = divMod x 16
-- Reinhard Zumkeller, Nov 07 2012

read("transforms") ;
A160700 := proc(n)
    if n < 16 then
        n;
    else
        XORnos(procname(floor(n/16)),modp(n,16))
    end if;
end proc: # R. J. Mathar, Jul 12 2016

a[n_] := a[n] = If[n < 16, n, a[Floor[n/16]] ~BitXor~ Mod[n, 16]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 25 2018 *)

load(functs)$
A160700(n):=if n<16 then n else logxor(floor(n/16),mod(n,16))$
makelist(A160700(n),n,0,60); /* Martin Ettl, Nov 05 2012 */

a(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t \\ Charles R Greathouse IV, Jan 25 2018

A161447 Numbers n such that A160700(n) = 7.

Original entry on oeis.org

7, 22, 37, 52, 67, 82, 97, 112, 143, 158, 173, 188, 203, 218, 233, 248, 262, 279, 292, 309, 322, 339, 352, 369, 398, 415, 428, 445, 458, 475, 488, 505, 517, 532, 551, 566, 577, 592, 611, 626, 653, 668, 687, 702, 713, 728, 747, 762, 772, 789, 806, 823, 832

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A161440, A161441, A161442, A161443, A161444, A161445, A161446, A161448, A161449, A161450, A161451, A161452, A161453, A161454, A161455.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==7, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A161440 Numbers m such that A160700(m) = 0.

Original entry on oeis.org

0, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 257, 272, 291, 306, 325, 340, 359, 374, 393, 408, 427, 442, 461, 476, 495, 510, 514, 531, 544, 561, 582, 599, 612, 629, 650, 667, 680, 697, 718, 735, 748, 765, 771, 786, 801, 816, 839

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A160700.

Cf. A161441, A161442, A161443, A161444, A161445, A161446, A161447, A161448, A161449, A161450, A161451, A161452, A161453, A161454, A161455.

b[n_] := b[n] = If[n < 16, n, b[Floor[n/16]]~BitXor~Mod[n, 16]];
Select[Range[0, 1000], b[#] == 0&] (* Jean-François Alcover, Dec 01 2021 *)

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16,16*n-1, if(a(k)==0, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A161442 Numbers n such that A160700(n) = 2.

Original entry on oeis.org

2, 19, 32, 49, 70, 87, 100, 117, 138, 155, 168, 185, 206, 223, 236, 253, 259, 274, 289, 304, 327, 342, 357, 372, 395, 410, 425, 440, 463, 478, 493, 508, 512, 529, 546, 563, 580, 597, 614, 631, 648, 665, 682, 699, 716, 733, 750, 767, 769, 784, 803, 818, 837

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A161440, A161441, A161443, A161444, A161445, A161446, A161447, A161448, A161449, A161450, A161451, A161452, A161453, A161454, A161455.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16,16*n-1, if(a(k)==2, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A161443 Numbers m such that A160700(m) = 3.

Original entry on oeis.org

3, 18, 33, 48, 71, 86, 101, 116, 139, 154, 169, 184, 207, 222, 237, 252, 258, 275, 288, 305, 326, 343, 356, 373, 394, 411, 424, 441, 462, 479, 492, 509, 513, 528, 547, 562, 581, 596, 615, 630, 649, 664, 683, 698, 717, 732, 751, 766, 768, 785, 802, 819, 836

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A161440, A161441, A161442, A161444, A161445, A161446, A161447, A161448, A161449, A161450, A161451, A161452, A161453, A161454, A161455.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==3, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A161444 Numbers n such that A160700(n) = 4.

Original entry on oeis.org

4, 21, 38, 55, 64, 81, 98, 115, 140, 157, 174, 191, 200, 217, 234, 251, 261, 276, 295, 310, 321, 336, 355, 370, 397, 412, 431, 446, 457, 472, 491, 506, 518, 535, 548, 565, 578, 595, 608, 625, 654, 671, 684, 701, 714, 731, 744, 761, 775, 790, 805, 820, 835

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

A161440, A161441, A161442, A161443, A161445, A161446, A161447, A161448, A161449, A161450, A161451, A161452, A161453, A161454, A161455.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==4, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A161445 Numbers n such that A160700(n) = 5.

Original entry on oeis.org

5, 20, 39, 54, 65, 80, 99, 114, 141, 156, 175, 190, 201, 216, 235, 250, 260, 277, 294, 311, 320, 337, 354, 371, 396, 413, 430, 447, 456, 473, 490, 507, 519, 534, 549, 564, 579, 594, 609, 624, 655, 670, 685, 700, 715, 730, 745, 760, 774, 791, 804, 821, 834

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A161440, A161441, A161442, A161443, A161444, A161446, A161447, A161448, A161449, A161450, A161451, A161452, A161453, A161454, A161455.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==5, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A161446 Numbers n such that A160700(n) = 6.

Original entry on oeis.org

6, 23, 36, 53, 66, 83, 96, 113, 142, 159, 172, 189, 202, 219, 232, 249, 263, 278, 293, 308, 323, 338, 353, 368, 399, 414, 429, 444, 459, 474, 489, 504, 516, 533, 550, 567, 576, 593, 610, 627, 652, 669, 686, 703, 712, 729, 746, 763, 773, 788, 807, 822, 833

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A161440, A161441, A161442, A161443, A161444, A161445, A161447, A161448, A161449, A161450, A161451, A161452, A161453, A161454, A161455.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==6, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A161448 Numbers n such that A160700(n) = 8.

Original entry on oeis.org

8, 25, 42, 59, 76, 93, 110, 127, 128, 145, 162, 179, 196, 213, 230, 247, 265, 280, 299, 314, 333, 348, 367, 382, 385, 400, 419, 434, 453, 468, 487, 502, 522, 539, 552, 569, 590, 607, 620, 637, 642, 659, 672, 689, 710, 727, 740, 757, 779, 794, 809, 824, 847

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A161440, A161441, A161442, A161443, A161444, A161445, A161446, A161447, A161449, A161450, A161451, A161452, A161453, A161454, A161455.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==8, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

cp's OEIS Frontend

A001025 Powers of 16: a(n) = 16^n.

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A160700 a(n) = if n<16 then n else a(floor(n/16)) XOR (n mod 16).

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A161447 Numbers n such that A160700(n) = 7.

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A161440 Numbers m such that A160700(m) = 0.

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A161442 Numbers n such that A160700(n) = 2.

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A161443 Numbers m such that A160700(m) = 3.

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A161444 Numbers n such that A160700(n) = 4.

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A161445 Numbers n such that A160700(n) = 5.

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