A083087 -id:A083087 - cp's OEIS frontend

A083088 First column of table A083087.

1, 2, 4, 6, 7, 9, 11, 12, 14, 16, 18, 19, 21, 23, 24, 26, 28, 30, 31, 33, 35, 36, 38, 40, 41, 43, 45, 47, 48, 50, 52, 53, 55, 57, 59, 60, 62, 64, 65, 67, 69, 70, 72, 74, 76, 77, 79, 81, 82, 84, 86, 88, 89, 91, 93, 94, 96, 98, 100, 101, 103, 105, 106, 108, 110, 111, 113, 115

Offset: 0

Paul D. Hanna, Apr 21 2003

It appears that A188937 gives the positions of 0 in the zero-one sequence A188037; complement of A080754. - Clark Kimberling, Mar 19 2011

Is this (apart from the prefixed a(0)) the same as A080755? - R. J. Mathar, Jul 31 2025

Cf. A083087, A083044, A083047, A083050.

Cf. A080755.

z:=70; x:=1+Sqrt(2); [ Floor(n*x/(x-1))+1: n in [0..z] ]; // Klaus Brockhaus, Jan 04 2011

f[n_] := Floor[n/Sqrt@2 + n + 1]; Array[f, 68, 0]

a(n) = floor(n*x/(x-1)) + 1, n>=0, where x=1+sqrt(2).

a(n) = floor(n/sqrt(2)) + n + 1 = 1+n+A049472(n).

This entry formerly contained an erroneous comment, which was deleted by N. J. A. Sloane, Jan 30 2008

A083090 Main diagonal of table A083087.

Original entry on oeis.org

1, 5, 25, 90, 247, 766, 2258, 5860, 16526, 45639, 124043, 313327, 837221, 2216256, 5545541, 14524810, 37810204, 97907017, 242993558, 625252309, 1602714963, 3962518559, 10109705767, 25718726896, 63402237401, 160711911464

Offset: 0

Paul D. Hanna, Apr 22 2003

nonn

Cf. A083087, A083044, A083047, A083050.

a(n) = T(n, n), where T(n, 0) = floor(n*x/(x-1)) + 1, T(n, k+1) = ceiling(x*T(n, k)), for n >= 0, k >= 0, where x = 1+sqrt(2).

A083091 Antidiagonal sums of table A083087.

Original entry on oeis.org

1, 5, 17, 49, 128, 321, 790, 1924, 4664, 11282, 27262, 65843, 158988, 383863, 926761, 2237435, 5401685, 13040863, 31483472, 76007871, 183499282, 443006506, 1069512368, 2582031320, 6233575089, 15049181582, 36331938341, 87713058356

Offset: 0

Paul D. Hanna, Apr 22 2003

nonn

Cf. A083087, A083044, A083047, A083050.

a(n) = Sum_{k=0..n} T(k, n-k), where T(n, 0) = floor(n*x/(x-1)) + 1, T(n, k+1) = ceiling(x*T(n, k)), for n>=0, k>=0, with x = 1 + sqrt(2).

A113782 a(n) = k such that A083087(k-1) = n.

Original entry on oeis.org

1, 3, 2, 6, 5, 10, 15, 4, 21, 9, 28, 36, 8, 45, 14, 55, 20, 66, 78, 7, 91, 27, 105, 120, 13, 136, 35, 153, 44, 171, 190, 12, 210, 54, 231, 253, 19, 276, 65, 300, 325, 26, 351, 77, 378, 90, 406, 435, 11, 465, 104, 496, 528, 34, 561, 119, 595, 135, 630, 666, 18, 703

Offset: 1

Klaus Brockhaus, Jan 19 2006

nonn

Sequence is a permutation of the positive integers; inverse permutation to sequence A083087 taken with offset 1.

			A083087(9) = 6, hence a(6) = 10.

Cf. A083087.

A048739 Expansion of 1/((1 - x)(1 - 2x - x^2)).

Original entry on oeis.org

1, 3, 8, 20, 49, 119, 288, 696, 1681, 4059, 9800, 23660, 57121, 137903, 332928, 803760, 1940449, 4684659, 11309768, 27304196, 65918161, 159140519, 384199200, 927538920, 2239277041, 5406093003, 13051463048, 31509019100, 76069501249

Offset: 0

Barry E. Williams

Partial sums of Pell numbers A000129.
W(n){1,3;2,-1,1} = Sum_{i=1..n} W(i){1,2;2,-1,0}, where W(n){a,b; p,q,r} implies x(n) = p*x(n-1) - q*x(n-2) + r; x(0)=a, x(1)=b.
Number of 2 X (n+1) binary arrays with path of adjacent 1's from upper left to lower right corner. - R. H. Hardin, Mar 16 2002
Binomial transform of A029744. - Paul Barry, Apr 23 2004
Number of (s(0), s(1), ..., s(n+2)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+2, s(0) = 1, s(n+2) = 3. - Herbert Kociemba, Jun 16 2004
Equals row sums of triangle A153346. - Gary W. Adamson, Dec 24 2008
Equals the sum of the terms of the antidiagonals of A142978. - J. M. Bergot, Nov 13 2012
a(p-2) == 0 mod p where p is an odd prime, see A270342. - Altug Alkan, Mar 15 2016
Also, the lexicographically earliest sequence of positive integers such that for n > 3, {sqrt(2)*a(n)} is located strictly between {sqrt(2)*a(n-1)} and {sqrt(2)*a(n-2)} where {} denotes the fractional part. - Ivan Neretin, May 02 2017
a(n+1) is the number of weak orderings on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1 < ... < n. - J. Devillet, Oct 06 2017

Allombert, Bill, Nicolas Brisebarre, and Alain Lasjaunias. "On a two-valued sequence and related continued fractions in power series fields." The Ramanujan Journal 45.3 (2018): 859-871. See Theorem 3, d_{4n+3}.

T. D. Noe, Table of n, a(n) for n = 0..200
M. Bicknell, A Primer on the Pell Sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
M. Bicknell-Johnson and G. E. Bergum, The Generalized Fibonacci Numbers {C(n)}, C(n)=C(n-1)+C(n-2)+K, Applications of Fibonacci Numbers, 1986, pp. 193-205.
B. Bradie, Extensions and Refinements of some properties of sums involving Pell Numbers, Miss. J. Math. Sci 22 (1) (2010) 37-43
M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA], 2017.
Jimmy Devillet, On the single-peakedness property, International summer school "Preferences, decisions and games" (Sorbonne Université, Paris, 2019).
I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1065
Yun-Tak Oh, Hosho Katsura, Hyun-Yong Lee, Jung Hoon Han, Proposal of a spin-one chain model with competing dimer and trimer interactions, arXiv:1709.01344 [cond-mat.str-el], 2017.
Ahmet Öteleş, On the sum of Pell and Jacobsthal numbers by the determinants of Hessenberg matrices, AIP Conference Proceedings 1863, 310003 (2017).
Wipawee Tangjai, A Non-standard Ternary Representation of Integers, Thai J. Math (2020) Special Issue: Annual Meeting in Mathematics 2019, 269-283.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

First row of table A083087.
With a different offset, a(4n)=A008843(n), a(4n-2)=8*A001110(n), a(2n-1)=A001652(n).
Cf. A001333, A048654, A048655, A083044, A083047, A083050, A153346.

a:=n->sum(fibonacci(i,2), i=0..n): seq(a(n), n=1..29); # Zerinvary Lajos, Mar 20 2008

Join[{a=1,b=3},Table[c=2*b+a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[1/(1-3x+x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,-1,-1},{1,3,8},30] (* Harvey P. Dale, Jun 13 2011 *)

a(n)=local(w=quadgen(8));-1/2+(3/4+1/2*w)*(1+w)^n+(3/4-1/2*w)*(1-w)^n

vector(100, n, n--; floor((1+sqrt(2))^(n+2)/4)) \\ Altug Alkan, Oct 07 2015

Vec(1/((1-x)*(1-2*x-x^2)) + O(x^40)) \\ Michel Marcus, May 06 2017

a(n) = 2*a(n-1) + a(n-2) + 1 with n > 1, a(0)=1, a(1)=3.
a(n) = ((2 + (3*sqrt(2))/2)*(1 + sqrt(2))^n - (2 - (3*sqrt(2))/2)*(1 - sqrt(2))^n )/(2*sqrt(2)) - 1/2.
a(0)=1, a(n+1) = ceiling(x*a(n)) for n > 0, where x = 1+sqrt(2). - Paul D. Hanna, Apr 22 2003
a(n) = 3*a(n-1) - a(n-2) - a(n-3). With two leading zeros, e.g.f. is exp(x)(cosh(sqrt(2)x)-1)/2. a(n) = Sum_{k=0..floor((n+2)/2)} binomial(n+2, 2k+2)2^k. - Paul Barry, Aug 16 2003
-a(-3-n) = A077921(n). - N. J. A. Sloane, Sep 13 2003
E.g.f.: exp(x)(cosh(x/sqrt(2)) + sqrt(2)sinh(x/sqrt(2)))^2. - N. J. A. Sloane, Sep 13 2003
a(n) = floor((1+sqrt(2))^(n+2)/4). - Bruno Berselli, Feb 06 2013
a(n) = (((1-sqrt(2))^(n+2) + (1+sqrt(2))^(n+2) - 2) / 4). - Altug Alkan, Mar 16 2016
2*a(n) = A001333(n+2)-1. - R. J. Mathar, Oct 11 2017
a(n) = Sum_{k=0..n} binomial(n+1,k+1)*2^floor(k/2). - Tony Foster III, Oct 12 2017

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 11 2002

A192386 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

1, 0, 8, 8, 96, 224, 1408, 4608, 22784, 86016, 386048, 1548288, 6676480, 27467776, 116490240, 484409344, 2040135680, 8521777152, 35786063872, 149761818624, 628140015616, 2630784909312, 11028578435072, 46205266558976, 193656954814464

Offset: 1

Clark Kimberling, Jun 30 2011

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+5). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0, x) = 1 -> 1
  p(1, x) =     2*x -> 2*x
  p(2, x) = 3 +   x +  3*x^2 -> 8 + 4*x
  p(3, x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 32*x
  p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 96 + 96*x.
From these, read A192386 = (1, 0, 8, 8, 96, ...) and A192387 = (0, 2, 4, 32, 96, ...).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,12,-8,-16).

Cf. A083087, A192232, A192387, A192388.

R:=PowerSeriesRing(Integers(), 41);
Coefficients(R!( x*(1-2*x-4*x^2)/(1-2*x-12*x^2+8*x^3+16*x^4) )); // G. C. Greubel, Jul 10 2023

q[x_]:= x+1; d= Sqrt[x+5];
p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2*d); (* suggested by A162517 *)
Table[Expand[p[n, x]], {n,6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[ FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}];
Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192386 *)
Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192387 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n,30}] (* A192388 *)
LinearRecurrence[{2,12,-8,-16}, {1,0,8,8}, 40] (* G. C. Greubel, Jul 10 2023 *)

@CachedFunction
def a(n): # a = A192386
    if (n<5): return (0,1,0,8,8)[n]
    else: return 2*a(n-1) +12*a(n-2) -8*a(n-3) -16*a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023

From Colin Barker, May 11 2014: (Start)
a(n) = 2*a(n-1) + 12*a(n-2) - 8*a(n-3) - 16*a(n-4).
G.f.: x*(1-2*x-4*x^2)/(1-2*x-12*x^2+8*x^3+16*x^4). (End)
From G. C. Greubel, Jul 10 2023: (Start)
T(n, k) = [x^k] ((x+sqrt(x+5))^n - (x-sqrt(x+5))^n)/(2*sqrt(x+5)).
a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1). (End)

A191440 Dispersion of ([n*sqrt(2)+n+3/2]), where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 8, 6, 4, 20, 15, 11, 5, 49, 37, 28, 13, 7, 119, 90, 69, 32, 18, 9, 288, 218, 168, 78, 44, 23, 10, 696, 527, 407, 189, 107, 57, 25, 12, 1681, 1273, 984, 457, 259, 139, 61, 30, 14, 4059, 3074, 2377, 1104, 626, 337, 148, 73, 35, 16, 9800, 7422, 5740

Offset: 1

Clark Kimberling, Jun 04 2011

Background discussion:  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)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1....3....8....20...49
  2....6....15...37...90
  4....11...28...69...168
  5....13...32...78...189
  7....18...44...107..259

Cf. A114537, A035513, A035506, A191438, A191439, A083087.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[2];
f[n_] := Floor[n*x+n+3/2] (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191440 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191440 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

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A083088 First column of table A083087.

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A083090 Main diagonal of table A083087.

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A083091 Antidiagonal sums of table A083087.

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A113782 a(n) = k such that A083087(k-1) = n.

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A048739 Expansion of 1/((1 - x)*(1 - 2*x - x^2)).

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A192386 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

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A191440 Dispersion of ([n*sqrt(2)+n+3/2]), where [ ]=floor, by antidiagonals.

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A048739 Expansion of 1/((1 - x)(1 - 2x - x^2)).