A095768 -id:A095768 - cp's OEIS frontend

A254105 Dispersion of A055938; starting from its complementary sequence A005187 as the first column of square array A(row,col), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

1, 2, 3, 5, 6, 4, 12, 13, 9, 7, 27, 28, 20, 14, 8, 58, 59, 43, 29, 17, 10, 121, 122, 90, 60, 36, 21, 11, 248, 249, 185, 123, 75, 44, 24, 15, 503, 504, 376, 250, 154, 91, 51, 30, 16, 1014, 1015, 759, 505, 313, 186, 106, 61, 33, 18, 2037, 2038, 1526, 1016, 632, 377, 217, 124, 68, 37, 19, 4084, 4085, 3061, 2039, 1271, 760, 440, 251, 139, 76, 40, 22

Offset: 1

Antti Karttunen, Jan 26 2015

This sequence is one instance of Clark Kimberling's generic dispersion arrays. Paraphrasing his explanation in A191450, mutatis mutandis, we have the following definition:
Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n) = {index of the row of D that contains n} is a fractal sequence. In this case s(n) = A055938(n), t(n) = A005187(n) [from term A005187(1) onward] and u(n) = A254112(n).
For other examples of such sequences, see the Crossrefs section. For a general introduction, please follow the Kimberling references.
The main diagonal: 1, 6, 20, 60, 154, 377, 887, 2040, 4598, 10229, 22515, 49139, ...

			The top left corner of the array:
   1,  2,  5,  12,  27,  58,  121,  248,  503,  1014,  2037,  4084
   3,  6, 13,  28,  59, 122,  249,  504, 1015,  2038,  4085,  8180
   4,  9, 20,  43,  90, 185,  376,  759, 1526,  3061,  6132, 12275
   7, 14, 29,  60, 123, 250,  505, 1016, 2039,  4086,  8181, 16372
   8, 17, 36,  75, 154, 313,  632, 1271, 2550,  5109, 10228, 20467
  10, 21, 44,  91, 186, 377,  760, 1527, 3062,  6133, 12276, 24563
  11, 24, 51, 106, 217, 440,  887, 1782, 3573,  7156, 14323, 28658
  15, 30, 61, 124, 251, 506, 1017, 2040, 4087,  8182, 16373, 32756
  16, 33, 68, 139, 282, 569, 1144, 2295, 4598,  9205, 18420, 36851
  18, 37, 76, 155, 314, 633, 1272, 2551, 5110, 10229, 20468, 40947
etc.

Antti Karttunen, Table of n, a(n) for n = 1..120; the first 15 antidiagonals of array
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and Dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Index entries for sequences that are permutations of the natural numbers

Inverse: A254106.
Transpose: A254107.
Column 1: A005187.
Cf. also A000325, A095768, A123720 (Seem to be rows 1 - 3, the last one from its second term onward.)
Columnd index of n: A254111, Row index: A254112.
Cf. A002260, A004736, A055938, A233275-A233278.
Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191450, A191426-A191455.

(define (A254105 n) (A254105bi (A002260 n) (A004736 n)))
(define (A254105bi row col) (if (= 1 col) (A005187 row) (A055938 (A254105bi row (- col 1)))))

If col = 1, then A(row,col) = A005187(row), otherwise A(row,col) = A055938(A(row,col-1)).

A132045 Row sums of triangle A132044.

Original entry on oeis.org

1, 2, 3, 6, 13, 28, 59, 122, 249, 504, 1015, 2038, 4085, 8180, 16371, 32754, 65521, 131056, 262127, 524270, 1048557, 2097132, 4194283, 8388586, 16777193, 33554408, 67108839, 134217702, 268435429, 536870884, 1073741795, 2147483618, 4294967265, 8589934560

Offset: 0

Gary W. Adamson, Aug 08 2007

Apart from initial terms, and with a change of offset, same as A095768. - Jon E. Schoenfield, Jun 15 2017

			a(4) = 13 = sum of row 4 terms of triangle A132044: (1 + 3 + 5 + 3 + 1).
a(4) = 13 = (1, 4, 6, 4, 1) dot (1, 1, 0, 2, 0) = (1 + 4 + 0 + 8 + 0).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000325, A095768, A132044.

[1] cat [2^n -n +1: n in [1..35]]; // G. C. Greubel, Feb 12 2021

Table[2^n -(n-1) -Boole[n==0], {n, 0, 35}] (* G. C. Greubel, Feb 12 2021 *)

Vec((1-2*x+2*x^3)/((1-x)^2*(1-2*x)) + O(x^100)) \\ Colin Barker, Mar 14 2014

[1]+[2^n -n +1 for n in (1..35)] # G. C. Greubel, Feb 12 2021

Binomial transform of (1, 1, 0, 2, 0, 2, 0, 2, 0, 2, ...).
For n>=1, a(n) = 2^n - n + 1 = A000325(n) + 1. - Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 17 2009. (Corrected by Franklin T. Adams-Watters, Jan 17 2009)
E.g.f.: U(0) - 1, where U(k) = 1 - x/(2^k + 2^k/(x - 1 - x^2*2^(k+1)/(x*2^(k+1) + (k+1)/U(k+1)))). - Sergei N. Gladkovskii, Dec 01 2012
From Colin Barker, Mar 14 2014: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n>3.
G.f.: (1-2*x+2*x^3) / ((1-x)^2*(1-2*x)). (End)

A084172 a(n) = 3a(n-1) - a(n-2) - 3a(n-3) + 2*a(n-4).

Original entry on oeis.org

1, 2, 4, 9, 19, 40, 82, 167, 337, 678, 1360, 2725, 5455, 10916, 21838, 43683, 87373, 174754, 349516, 699041, 1398091, 2796192, 5592394, 11184799, 22369609, 44739230, 89478472, 178956957, 357913927, 715827868, 1431655750, 2863311515

Offset: 0

Paul Barry, May 18 2003

Original name was: Generalized Jacobsthal numbers.

Sums of rows of the triangle in A109225. - Reinhard Zumkeller, Jun 23 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

[2^(n+2)/3-(-1)^n/12-(2*n+1)/4: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

LinearRecurrence[{3,-1,-3,2},{1,2,4,9},40] (* Harvey P. Dale, Nov 13 2013 *)

a(n) = 2^(n+2)/3 - (-1)^n/12 - (2*n+1)/4.
G.f: (2*x^3 - x^2 - x + 1)/( (x+1)*(1-2*x)*(1-x)^2).
a(n+2) = a(n+1) + 2*a(n) + n, a(0)=0, a(1)=2.
a(n) = A001045(n+1) + A083579(n).
a(n+1) = 2*a(n) + floor(n/2). Franklin T. Adams-Watters, Oct 17 2013
a(n)+a(n+1) = A095768(n+1). - R. J. Mathar, Apr 15 2024

A135087 Triangle T(n, k) = 2*A134058(n, k) - 1, read by rows.

Original entry on oeis.org

1, 3, 3, 3, 7, 3, 3, 11, 11, 3, 3, 15, 23, 15, 3, 3, 19, 39, 39, 19, 3, 3, 23, 59, 79, 59, 23, 3, 3, 27, 83, 139, 139, 83, 27, 3, 3, 31, 111, 223, 279, 223, 111, 31, 3, 3, 35, 143, 335, 503, 503, 335, 143, 35, 3

Offset: 0

Gary W. Adamson, Nov 18 2007

			First few rows of the triangle are:
  1;
  3,  3;
  3,  7,  3;
  3, 11, 11,  3;
  3, 15, 23, 15,  3;
  3, 19, 39, 39, 19,  3;
  3, 23, 59, 79, 59, 23, 3;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A095768, A134058.

[1] cat [4*Binomial(n,k) -1: k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021

Table[4*Binomial[n, k] -2*Boole[n==0] -1, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 03 2021 *)

def A135087(n,k): return 4*binomial(n,k) -2*bool(n==0) -1
flatten([[A135087(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

T(n, k) = 2*A134058(n, k) - 1.
From G. C. Greubel, May 03 2021: (Start)
T(n, k) = 4*binomial(n, k) - 2*[n=0] - 1.
Sum_{k=0..n} T(n, k) = 2^(n+2) - (n + 1 + 2*[n=0]) = A095768(n) - 2*[n=0]. (End)

A095767 a(n) = valuation(A004001(n),2).

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 2, 2, 0, 1, 0, 0, 3, 3, 3, 3, 0, 1, 0, 2, 2, 0, 1, 1, 0, 0, 0, 4, 4, 4, 4, 4, 0, 1, 0, 2, 0, 0, 1, 0, 3, 3, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 1, 1, 0, 0, 0, 0, 5, 5, 5, 5, 5, 5, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 1, 1, 0, 2, 0, 0, 1, 0, 0, 4, 4, 4, 0, 1, 0, 0, 2, 0, 0, 1, 1, 1, 0, 3, 3, 0, 0, 0, 1, 1, 1

Offset: 1

Benoit Cloitre, Jun 05 2004

nonn

Cf. A095768.

a[1] = a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; f[n_] := Length[ NestWhileList[ #/2 &, n, IntegerQ[ # ] &]] - 2; Table[ f[ a[n]], {n, 105}] (* Robert G. Wilson v, Jun 11 2004 *)

Partial formula: a(2^(n+1) - n + i) = n for 0<=i<=n.

cp's OEIS Frontend

A254105 Dispersion of A055938; starting from its complementary sequence A005187 as the first column of square array A(row,col), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A132045 Row sums of triangle A132044.

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A084172 a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).

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A135087 Triangle T(n, k) = 2*A134058(n, k) - 1, read by rows.

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Magma

Mathematica

Sage

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A095767 a(n) = valuation(A004001(n),2).

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Mathematica

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A084172 a(n) = 3a(n-1) - a(n-2) - 3a(n-3) + 2*a(n-4).