A048492 -id:A048492 - cp's OEIS frontend

A046698 a(0) = 0, a(1) = 1, a(n) = a(a(n-1)) + a(a(n-2)) if n > 1.

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

Offset: 0

N. J. A. Sloane, R. K. Guy

Partial sums are A004275. Binomial transform is A048492, starting with 0. - Paul Barry, Feb 28 2003
From Elmo R. Oliveira, Jul 25 2024: (Start)
Continued fraction expansion of 2 - sqrt(2) = A101465.
Decimal expansion of 101/9000. (End)

Sequence proposed by Reg Allenby.

Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, On the Generalized Fibonacci-circulant-Hurwitz numbers, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 1, 179-190.
O. Deveci, Y. Akuzum, E. Karaduman, and O. Erdag, The Cyclic Groups via Bezout Matrices, Journal of Mathematics Research, Vol. 7, No. 2, 2015, pp. 34-41.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A004275, A048492, A101465 (decimal expansion of 2 - sqrt(2)).

Cf. A033324, A114955, A343461.

CoefficientList[Series[x (1 + x^2)/(1 - x), {x, 0, 104}], x] (* or *)
Nest[Append[#, #[[#[[-1]] + 1]] + #[[#[[-2]] + 1 ]]] &, {0, 1}, 105] (* Michael De Vlieger, Jul 31 2020 *)

PARI
```
a(n)=(n>0)+(n>2)
```

G.f.: x*(1+x^2)/(1-x). - Paul Barry, Feb 28 2003
From Elmo R. Oliveira, Jul 25 2024: (Start)
E.g.f.: 2*exp(x) - x - 1.
a(n) = 2 for n > 2.
a(n) = 2 - A033324(n+2) = 4 - A343461(n+4) = A114955(n+6) - 6. (End)

A265901 Square array read by descending antidiagonals: A(n,1) = A188163(n), and for k > 1, A(n,k) = A087686(1+A(n,k-1)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 8, 15, 12, 6, 16, 31, 27, 14, 9, 32, 63, 58, 30, 21, 10, 64, 127, 121, 62, 48, 24, 11, 128, 255, 248, 126, 106, 54, 26, 13, 256, 511, 503, 254, 227, 116, 57, 29, 17, 512, 1023, 1014, 510, 475, 242, 120, 61, 38, 18, 1024, 2047, 2037, 1022, 978, 496, 247, 125, 86, 42, 19, 2048, 4095, 4084, 2046, 1992, 1006, 502, 253, 192, 96, 45, 20

Offset: 1

Antti Karttunen, Dec 18 2015

Square array read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
The topmost row (row 1) of the array is A000079 (powers of 2), and in general each row 2^k contains the sequence (2^n - k), starting from the term (2^(k+1) - k). This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 in PDF).
Moreover, each row 2^k - 1 (for k >= 2) contains the sequence 2^n - n - (k-2), starting from the term (2^(k+1) - (2k-1)). To see why this holds, consider the definitions of sequences A162598 and A265332, the latter which also illustrates how the frequency counts Q_n for A004001 are recursively constructed (in the Kubo & Vakil paper).

			The top left corner of the array:
   1,  2,   4,   8,  16,   32,   64,  128,  256,   512,  1024, ...
   3,  7,  15,  31,  63,  127,  255,  511, 1023,  2047,  4095, ...
   5, 12,  27,  58, 121,  248,  503, 1014, 2037,  4084,  8179, ...
   6, 14,  30,  62, 126,  254,  510, 1022, 2046,  4094,  8190, ...
   9, 21,  48, 106, 227,  475,  978, 1992, 4029,  8113, 16292, ...
  10, 24,  54, 116, 242,  496, 1006, 2028, 4074,  8168, 16358, ...
  11, 26,  57, 120, 247,  502, 1013, 2036, 4083,  8178, 16369, ...
  13, 29,  61, 125, 253,  509, 1021, 2045, 4093,  8189, 16381, ...
  17, 38,  86, 192, 419,  894, 1872, 3864, 7893, 16006, 32298, ...
  18, 42,  96, 212, 454,  950, 1956, 3984, 8058, 16226, 32584, ...
  19, 45, 102, 222, 469,  971, 1984, 4020, 8103, 16281, 32650, ...
  20, 47, 105, 226, 474,  977, 1991, 4028, 8112, 16291, 32661, ...
  22, 51, 112, 237, 490,  999, 2020, 4065, 8158, 16347, 32728, ...
  23, 53, 115, 241, 495, 1005, 2027, 4073, 8167, 16357, 32739, ...
  25, 56, 119, 246, 501, 1012, 2035, 4082, 8177, 16368, 32751, ...
  28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, ...
  ...

Antti Karttunen, Table of n, a(n) for n = 1..210; the first 20 antidiagonals of array
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
Index entries for Hofstadter-type sequences
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A267102.
Transpose: A265903.
Cf. A265900 (main diagonal).
Cf. A162598 (row index of n in array), A265332 (column index of n in array).
Cf. A004001, A051135, A088359, A087686.
Column 1: A188163.
Column 2: A266109.
Row 1: A000079 (2^n).
Row 2: A000225 (2^n - 1, from 3 onward).
Row 3: A000325 (2^n - n, from 5 onward).
Row 4: A000918 (2^n - 2, from 6 onward).
Row 5: A084634 (?, from 9 onward).
Row 6: A132732 (2^n - 2n + 2, from 10 onward).
Row 7: A000295 (2^n - n - 1, from 11 onward).
Row 8: A036563 (2^n - 3).
Row 9: A084635 (?, from 17 onward).
Row 12: A048492 (?, from 20 onward).
Row 13: A249453 (?, from 22 onward).
Row 14: A183155 (2^n - 2n + 1, from 23 onward. Cf. also A244331).
Row 15: A000247 (2^n - n - 2, from 25 onward).
Row 16: A028399 (2^n - 4).
Cf. also permutations A267111, A267112.

(define (A265901 n) (A265901bi (A002260 n) (A004736 n)))
(define (A265901bi row col) (if (= 1 col) (A188163 row) (A087686 (+ 1 (A265901bi row (- col 1))))))

For the first column k=1, A(n,1) = A188163(n), for columns k > 1, A(n,k) = A087686(1+A(n,k-1)).

A145070 Partial sums of A006127, starting at n=1.

Original entry on oeis.org

3, 9, 20, 40, 77, 147, 282, 546, 1067, 2101, 4160, 8268, 16473, 32871, 65654, 131206, 262295, 524457, 1048764, 2097360, 4194533, 8388859, 16777490, 33554730, 67109187, 134218077, 268435832, 536871316, 1073742257, 2147484111

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 30 2008

			a(2) = a(1) + 2^2 + 2 = 3 + 4 + 2 = 9; a(3) = a(2) + 2^3 + 3 = 9 + 8 + 3 = 20.

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A006127 (2^n + n), A000325 (2^n - n), A048492 (partial sums of A000325, starting at n=1).

a:=0; for n:=1 to 30 do a:=a+2**n+n; write(a, ","); end;

A145070:=n->(-4+2^(2+n)+n+n^2)/2: seq(A145070(n), n=1..50); # Wesley Ivan Hurt, Jan 22 2017

lst={};s=0;Do[s+=2^n+n;AppendTo[lst,s],{n,5!}];lst
Accumulate[Table[2^n+n,{n,50}]] (* or *) LinearRecurrence[{5,-9,7,-2},{3,9,20,40},50] (* Harvey P. Dale, Aug 22 2020 *)

Vec(x*(2*x^2-6*x+3) / ((x-1)^3*(2*x-1)) + O(x^100)) \\ Colin Barker, Oct 27 2014

a(1) = 3; a(n) = a(n-1) + 2^n + n for n > 1.
From Colin Barker, Oct 27 2014: (Start)
a(n) = (-4+2^(2+n)+n+n^2)/2.
a(n) = 5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4).
G.f.: x*(2*x^2-6*x+3) / ((x-1)^3*(2*x-1)).
(End)

Edited by Klaus Brockhaus, Oct 14 2008

cp's OEIS Frontend

A046698 a(0) = 0, a(1) = 1, a(n) = a(a(n-1)) + a(a(n-2)) if n > 1.

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A265901 Square array read by descending antidiagonals: A(n,1) = A188163(n), and for k > 1, A(n,k) = A087686(1+A(n,k-1)).

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A145070 Partial sums of A006127, starting at n=1.

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