A116699 -id:A116699 - cp's OEIS frontend

A114697 Expansion of (1+x+x^2)/((1-x^2)(1-2x-x^2)); a Pellian-related sequence.

1, 3, 9, 22, 55, 133, 323, 780, 1885, 4551, 10989, 26530, 64051, 154633, 373319, 901272, 2175865, 5253003, 12681873, 30616750, 73915375, 178447501, 430810379, 1040068260, 2510946901, 6061962063, 14634871029, 35331704122, 85298279275, 205928262673

Offset: 0

Creighton Dement, Feb 18 2006

Generating floretion: (- .5'j + .5'k - .5j' + .5k' + 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki')*('i + 'j + i').

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

Cf. A000129, A002203, A005409, A100828, A111954, A113224.

Cf. A114647, A114688, A114689, A114695, A116699.

Table[(3*LucasL[n, 2] +10*Fibonacci[n, 2] -3 +(-1)^n)/4, {n,0,30}] (* G. C. Greubel, May 24 2021 *)

Vec((1+x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^40)) \\ Colin Barker, Jun 24 2015

[(4*lucas_number1(n+2,2,-1) -2*lucas_number1(n+1,2,-1) -3 +(-1)^n)/4 for n in (0..30)] # G. C. Greubel, May 24 2021

a(n+2) - 2*a(n+1) + a(n) = A111955(n+2).
G.f.: (1+x+x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
From Raphie Frank, Oct 01 2012: (Start)
a(2*n) = A216134(2*n+1).
a(2*n+1) = A006452(2*n+3)-1.
Lim_{n->infinity} a(n+1)/a(n) = A014176. (End)
a(n) = (2*A078343(n+2) - A010694(n))/4. - R. J. Mathar, Oct 02 2012
From Colin Barker, May 26 2016: (Start)
a(n) = ( 2*(-3 +(-1)^n) + (6-5*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(6+5*sqrt(2)) )/8.
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n>3. (End)
a(n) = (3*A002203(n) + 10*A000129(n) - 3 + (-1)^n)/4. - G. C. Greubel, May 24 2021

A115008 a(n) = a(n-1) + a(n-3) + a(n-4).

Original entry on oeis.org

1, 0, 2, 4, 5, 7, 13, 22, 34, 54, 89, 145, 233, 376, 610, 988, 1597, 2583, 4181, 6766, 10946, 17710, 28657, 46369, 75025, 121392, 196418, 317812, 514229, 832039, 1346269, 2178310, 3524578, 5702886, 9227465, 14930353, 24157817, 39088168

Offset: 0

Creighton Dement, Feb 23 2006

a(n+2) - a(n+1) - a(n) gives match to A000034, apart from signs.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Cf. A000034, A000045, A001519, A001906, A006498, A116697, A116698, A116699.

A115008:= func< n | Fibonacci(n+1) - (n mod 2) + 2*0^((n+1) mod 4) >;
[A115008(n): n in [0..50]]; // G. C. Greubel, Aug 24 2025

Table[Fibonacci[n+1] -I^(n-1)*Mod[n,2], {n,0,50}] (* G. C. Greubel, Aug 24 2025 *)

def A115008(n): return fibonacci(n+1) -i**(n-1)*(n%2)
print([A115008(n) for n in range(51)]) # G. C. Greubel, Aug 24 2025

a(2*n) = A000045(2*n+1) = A001519(n).
G.f.: (1-x+2*x^2+x^3)/((1+x^2)*(1-x-x^2)).
a(2*n+1) = (-1)^(n+1) + A001906(n+1) (compare with a similar property for A116697) - Creighton Dement, Mar 31 2006
From G. C. Greubel, Aug 24 2025: (Start)
a(n) = A000045(n+1) - i^(n-1)*(n mod 2).
E.g.f.: exp(x/2)*(cosh(p*x) + (1/(2*p))*sinh(p*x)) - sin(x), where 2*p = sqrt(5). (End)

A116697 a(n) = -a(n-1) - a(n-3) + a(n-4).

Original entry on oeis.org

1, 1, -2, 2, -2, 5, -9, 13, -20, 34, -56, 89, -143, 233, -378, 610, -986, 1597, -2585, 4181, -6764, 10946, -17712, 28657, -46367, 75025, -121394, 196418, -317810, 514229, -832041, 1346269, -2178308, 3524578, -5702888

Offset: 0

Creighton Dement, Feb 23 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,0,-1,1).

Cf. A000045, A001519, A006498, A056594, A115008, A116698, A116699, A128535.

Cf. A186679 (first differences).

a116697 n = a116697_list !! n
a116697_list = [1,1,-2,2]
               ++ (zipWith (-) a116697_list
                               $ zipWith (+) (tail a116697_list)
                                             (drop 3 a116697_list))
a128535_list = 0 : (map negate $ map a116697 [0,2..])
a001519_list = 1 : map a116697 [1,3..]
a186679_list = zipWith (-) (tail a116697_list) a116697_list
a128533_list = map a186679 [0,2..]
a081714_list = 0 : (map negate $ map a186679 [1,3..])
a075193_list = 1 : -3 : (zipWith (+) a186679_list $ drop 2 a186679_list)
-- Reinhard Zumkeller, Feb 25 2011

A116697:= func< n | (-1)^Floor((n+1)/2)*(1+(-1)^n)/2 -(-1)^n*Fibonacci(n) >;
[A116697(n): n in [0..50]]; // G. C. Greubel, Jun 08 2025

LinearRecurrence[{-1,0,-1,1},{1,1,-2,2},40] (* Harvey P. Dale, Nov 04 2011 *)

def A116697(n): return (-1)^(n//2)*((n+1)%2) - (-1)^n*fibonacci(n)
print([A116697(n) for n in range(51)]) # G. C. Greubel, Jun 08 2025

G.f.: (1 + 2*x - x^2 + x^3)/((1 + x^2)*(1 + x - x^2)).
a(2*n+1) = A000045(2*n+1) = A001519(n).
a(2*n) = - A128535(n+1). - Reinhard Zumkeller, Feb 25 2011
a(n) = A056594(n) - (-1)^n*A000045(n). - Bruno Berselli, Feb 26 2011
E.g.f.: cos(x) + (2/sqrt(5))*exp(-x/2)*sinh(sqrt(5)*x/2). - G. C. Greubel, Jun 08 2025

A116698 Expansion of (1-x+3x^2+x^3) / ((1-x-x^2)(1+2*x^2)).

Original entry on oeis.org

1, 0, 2, 5, 5, 4, 13, 29, 34, 39, 89, 176, 233, 313, 610, 1115, 1597, 2328, 4181, 7277, 10946, 16687, 28657, 48416, 75025, 117297, 196418, 326003, 514229, 815656, 1346269, 2211077, 3524578, 5637351, 9227465

Offset: 0

Creighton Dement, Feb 23 2006

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

Cf. A000045, A001519, A006498, A115008, A116697, A116699.

A116698:= func< n | Fibonacci(n+1) -((n mod 2) -2*0^((n+1) mod 4))*2^Floor(n/2) >;
[A116898(n): n in [0..50]]; // G. C. Greubel, Aug 24 2025

CoefficientList[Series[(1-x+3x^2+x^3)/((1-x-x^2)(1+2x^2)),{x,0,100}],x] (* or *) LinearRecurrence[{1,-1,2,2},{1,0,2,5},100] (* Harvey P. Dale, May 14 2022 *)
Table[Fibonacci[n+1] -I^(n-1)*Mod[n,2]*2^Floor[n/2], {n,0,50}] (* G. C. Greubel, Aug 24 2025 *)

Vec((1-x +3*x^2 +x^3)/((1-x-x^2)*(1+2*x^2)) + O(x^40)) \\ Colin Barker, May 18 2019

def A116898(n): return fibonacci(n+1) - (-1)**((n-1)//2)*(n%2)*2**(n//2)
print([A116898(n) for n in range(51)]) # G. C. Greubel, Aug 24 2025

a(2*n) = A000045(2*n+1) = A001519(n).
a(n) = a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) for n > 3. - Colin Barker, May 18 2019
From G. C. Greubel, Aug 24 2025: (Start)
a(n) = A000045(n+1) - (-1)^floor((n-1)/2) * (n mod 2) * 2^floor(n/2).
E.g.f.: exp(x/2)*(cosh(sqrt(5)*x/2) + (1/sqrt(5))*sinh(sqrt(5)*x/2))  - sin(sqrt(2)*x)/sqrt(2). (End)

A248126 a(n) = n^2 with each digit repeated.

Original entry on oeis.org

11, 44, 99, 1166, 2255, 3366, 4499, 6644, 8811, 110000, 112211, 114444, 116699, 119966, 222255, 225566, 228899, 332244, 336611, 440000, 444411, 448844, 552299, 557766, 662255, 667766, 772299, 778844, 884411, 990000, 996611, 11002244, 11008899, 11115566

Offset: 1

Jon Perry, Nov 01 2014

Inspired by A116699.

			13^2 = 169, so a(13) = 116699.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A020338, A044836.

Cf. A116699, A338754.

for (i=1;i<40;i++) {
s=(i*i).toString();
for (j=0;j

a248126[n_Integer] :=
Module[{m}, m := IntegerDigits[n^2];
  FromDigits[Flatten[Transpose[List[m, m]]]]]; a248126 /@ Range[34] (* Michael De Vlieger, Nov 06 2014 *)
Table[FromDigits[Riffle[id=IntegerDigits[n^2],id]],{n,40}] (* Harvey P. Dale, Dec 19 2015 *)

a(n)= my(d = 11*digits(n^2)); fromdigits(d, 100) \\ David A. Corneth, Dec 02 2023

def a(n): return int("".join(d*2 for d in str(n**2)))
print([a(n) for n in range(1, 35)]) # Michael S. Branicky, Dec 02 2023

A179257 Number of permutations of length n which avoid the patterns 321 and 1324.

Original entry on oeis.org

1, 1, 2, 5, 13, 32, 72, 148, 281, 499, 838, 1343, 2069, 3082, 4460, 6294, 8689, 11765, 15658, 20521, 26525, 33860, 42736, 53384, 66057, 81031, 98606, 119107, 142885, 170318, 201812, 237802, 278753, 325161, 377554, 436493, 502573, 576424, 658712, 750140, 851449

Offset: 0

Vincent Vatter, Jul 05 2010

			There are 13 permutations of length 4 which avoid these two patterns, so a(4)=13.

M. D. Atkinson, Restricted permutations, Discrete Math., 195 (1999), 27-38.
Christian Bean, Bjarki Gudmundsson, Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.
J. West, Generating trees and forbidden subsequences, Discrete Math., 157 (1996), 363-374.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A116699, A116701, A116702, A088921, A005183, A116703, A001519.

LinearRecurrence[{6,-15,20,-15,6,-1},{1,1,2,5,13,32},50] (* Harvey P. Dale, May 19 2024 *)

a(n) = 1+binomial(n,2)+binomial(n+2,5).
G.f.: 1-x*(x^5-4*x^4+7*x^3-8*x^2+4*x-1)/(x-1)^6. - Colin Barker, Aug 02 2012
a(n) = 1+A027658(n-2). - R. J. Mathar, Aug 19 2022

a(0)=1 prepended by Alois P. Heinz, Jul 05 2018

cp's OEIS Frontend

A114697 Expansion of (1+x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

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A115008 a(n) = a(n-1) + a(n-3) + a(n-4).

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A116697 a(n) = -a(n-1) - a(n-3) + a(n-4).

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A116698 Expansion of (1-x+3*x^2+x^3) / ((1-x-x^2)*(1+2*x^2)).

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A248126 a(n) = n^2 with each digit repeated.

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A179257 Number of permutations of length n which avoid the patterns 321 and 1324.

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A114697 Expansion of (1+x+x^2)/((1-x^2)(1-2x-x^2)); a Pellian-related sequence.

A116698 Expansion of (1-x+3x^2+x^3) / ((1-x-x^2)(1+2*x^2)).