A022096 -id:A022096 - cp's OEIS frontend

A192956 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

1, 0, 4, 9, 20, 38, 69, 120, 204, 341, 564, 926, 1513, 2464, 4004, 6497, 10532, 17062, 27629, 44728, 72396, 117165, 189604, 306814, 496465, 803328, 1299844, 2103225, 3403124, 5506406, 8909589, 14416056, 23325708, 37741829, 61067604, 98809502

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) +- 1 + n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

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

Cf. A000045, A192232, A192744, A192951, A192957.

F:=Fibonacci;; List([0..40], n-> F(n+3)+4*F(n+1)-(2*n+5)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [F(n+3)+4*F(n+1)-(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n^2 - 1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192956 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192957 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[n+3]+4*F[n+1]-(2*n+5), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

vector(40, n, n--; f=fibonacci; f(n+3)+4*f(n+1)-(2*n+5)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [f(n+3)+4*f(n+1)-(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From R. J. Mathar, May 09 2014: (Start)
G.f.: (1 -3*x +6*x^2 -2*x^3)/((1-x-x^2)*(1-x)^2).
a(n) -2*a(n+1) +a(n+2) = A022096(n-3). (End)
a(n) = Fibonacci(n+3) + 4*Fibonacci(n+1) - (2*n+5). - G. C. Greubel, Jul 12 2019

A210209 GCD of all sums of n consecutive Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 4, 1, 3, 2, 11, 1, 8, 1, 29, 2, 21, 1, 76, 1, 55, 2, 199, 1, 144, 1, 521, 2, 377, 1, 1364, 1, 987, 2, 3571, 1, 2584, 1, 9349, 2, 6765, 1, 24476, 1, 17711, 2, 64079, 1, 46368, 1, 167761, 2, 121393, 1, 439204, 1, 317811, 2, 1149851, 1, 832040

Offset: 0

Alonso del Arte, Mar 18 2012

Early on in the Posamentier & Lehmann (2007) book, the fact that the sum of any ten consecutive Fibonacci numbers is a multiple of 11 is presented as an interesting property of the Fibonacci numbers. Much later in the book a proof of this fact is given, using arithmetic modulo 11. An alternative proof could demonstrate that 11*F(n + 6) = Sum_{i=n..n+9} F(i).

			a(3) = 2 because all sums of three consecutive Fibonacci numbers are divisible by 2 (F(n) + F(n-1) + F(n-2) = 2F(n)), but since the GCD of 3 + 5 + 8 = 16 and 5 + 8 + 13 = 26 is 2, no number larger than 2 divides all sums of three consecutive Fibonacci numbers.
a(4) = 1 because the GCD of 1 + 1 + 2 + 3 = 7 and 1 + 2 + 3 + 5 = 11 is 1, so the sums of four consecutive Fibonacci numbers have no factors in common.

Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Prometheus Books, New York (2007) p. 33.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Dan Guyer and aBa Mbirika, GCD of sums of k consecutive Fibonacci, Lucas, and generalized Fibonacci numbers, Journal of Integer Sequences, 24 No.9, Article 21.9.8 (2021), 25pp; arXiv preprint, arXiv:2104.12262 [math.NT], 2021.
I. D. Ruggles, Elementary problem B-1, Fibonacci Quarterly, Vol. 1, No. 1, February 1963, p. 73; Solution by Marjorie R. Bicknell, published in Vol. 1, No. 3, October 1963, pp. 76-77.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,1,0,-1,0,-3,0,0,0,1).

Cf. A000045, A000071, sum of the first n Fibonacci numbers, A001175 (Pisano periods). Cf. also A229339.
Bisections give: A005013 (even part), A131534 (odd part).
Sums of m consecutive Fibonacci numbers: A055389 (m = 3, ignoring the initial 1); A000032 (m = 4, these are the Lucas numbers); A013655 (m = 5); A022087 (m = 6); A022096 (m = 7); A022379 (m = 8).

a:= n-> (Matrix(7, (i, j)-> `if`(i=j-1, 1, `if`(i=7, [1, 0, -3, -1, 1, 3, 0][j], 0)))^iquo(n, 2, 'r'). `if`(r=0, <<0, 1, 1, 4, 3, 11, 8>>, <<1, 2, 1, 1, 2, 1, 1>>))[1, 1]: seq(a(n), n=0..80);  # Alois P. Heinz, Mar 18 2012

Table[GCD[Fibonacci[n + 1] - 1, Fibonacci[n]], {n, 1, 50}] (* Horst H. Manninger, Dec 19 2021 *)

a(n)=([0,1,0,0,0,0,0,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0,0,0,0,0,0; 0,0,0,0,1,0,0,0,0,0,0,0,0,0; 0,0,0,0,0,1,0,0,0,0,0,0,0,0; 0,0,0,0,0,0,1,0,0,0,0,0,0,0; 0,0,0,0,0,0,0,1,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,1,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,1,0,0,0,0; 0,0,0,0,0,0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,1; 1,0,0,0,-3,0,-1,0,1,0,3,0,0,0]^n*[0;1;1;2;1;1;4;1;3;2;11;1;8;1])[1,1] \\ Charles R Greathouse IV, Jun 20 2017

G.f.: -x*(x^12-x^11+2*x^10-x^9-2*x^8-x^7-6*x^6+x^5-2*x^4+x^3+2*x^2+x+1) / (x^14-3*x^10-x^8+x^6+3*x^4-1) = -1/(x^4+x^2-1) + (x^2+1)/(x^4-x^2-1) + (x+2)/(6*(x^2+x+1)) + (x-2)/(6*(x^2-x+1)) - 2/(3*(x+1)) - 2/(3*(x-1)). - Alois P. Heinz, Mar 18 2012
a(n) = gcd(Fibonacci(n+1)-1, Fibonacci(n)). - Horst H. Manninger, Dec 19 2021
From Aba Mbirika, Jan 21 2022: (Start)
a(n) = gcd(F(n+1)-1, F(n+2)-1).
a(n) = Lcm_{A001175(m) divides n} m.
Proofs of these formulas are given in Theorems 15 and 25 of the Guyer-Mbirika paper. (End)

More terms from Alois P. Heinz, Mar 18 2012

A353595 Array read by ascending antidiagonals. Generalized Fibonacci numbers F(n, k) = (psi^(k - 1)(phi + n) - phi^(k - 1)(psi + n)) / (psi - phi) where phi = (1+sqrt(5))/2 and psi = (1-sqrt(5))/2. F(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 3, 1, 2, 2, 4, 1, 3, 3, 3, 5, 1, 4, 4, 5, 5, 6, 1, 5, 5, 7, 8, 8, 7, 1, 6, 6, 9, 11, 13, 13, 8, 1, 7, 7, 11, 14, 18, 21, 21, 9, 1, 8, 8, 13, 17, 23, 29, 34, 34, 10, 1, 9, 9, 15, 20, 28, 37, 47, 55, 55, 11, 1, 10, 10, 17, 23, 33, 45, 60, 76, 89, 89

Offset: 0

Peter Luschny, May 09 2022

The definition declares the Fibonacci numbers for all integers n and k. It gives the classical Fibonacci numbers as F(0, n) = A000045(n). A different enumeration is given in A352744.

			Array starts:
n\k 0, 1,  2,  3,  4,  5,  6,   7,   8,   9, ...
--------------------------------------------------------
[0]  0, 1,  1,  2,  3,  5,  8, 13,  21,  34, ... A000045
[1]  1, 1,  2,  3,  5,  8, 13, 21,  34,  55, ... A000045 (shifted once)
[2]  2, 1,  3,  4,  7, 11, 18, 29,  47,  76, ... A000032
[3]  3, 1,  4,  5,  9, 14, 23, 37,  60,  97, ... A104449
[4]  4, 1,  5,  6, 11, 17, 28, 45,  73, 118, ... [4] + A022095
[5]  5, 1,  6,  7, 13, 20, 33, 53,  86, 139, ... [5] + A022096
[6]  6, 1,  7,  8, 15, 23, 38, 61,  99, 160, ... [6] + A022097
[7]  7, 1,  8,  9, 17, 26, 43, 69, 112, 181, ... [7] + A022098
[8]  8, 1,  9, 10, 19, 29, 48, 77, 125, 202, ... [8] + A022099
[9]  9, 1, 10, 11, 21, 32, 53, 85, 138, 223, ... [9] + A022100

Peter Luschny, The Fibonacci Function.
Wikipedia, Cassini and Catalan identities.

Cf. A000045, A000032, A104449, A094588 (main diagonal).

Cf. A352744, A354265 (generalized Lucas numbers).

function fibrec(n::Int)
    n == 0 && return (BigInt(0), BigInt(1))
    a, b = fibrec(div(n, 2))
    c = a * (b * 2 - a)
    d = a * a + b * b
    iseven(n) ? (c, d) : (d, c + d)
end
function Fibonacci(n::Int, k::Int)
    k == 0 && return BigInt(n)
    k == 1 && return BigInt(1)
    k  < 0 && return (-1)^(k-1)*Fibonacci(-n - 1, 2 - k)
    a, b = fibrec(k - 1)
    a*n + b
end
for n in -6:6
    println([n], [Fibonacci(n, k) for k in -6:6])
end

f := n -> combinat:-fibonacci(n): F := (n, k) -> n*f(k - 1) + f(k):
seq(seq(F(n - k, k), k = 0..n), n = 0..11);
# The next implementation is for illustration only but is not recommended
# as it relies on floating point arithmetic. Illustrates the case n,k < 0.
phi := (1 + sqrt(5))/2: psi := (1 - sqrt(5))/2:
F := (n, k) -> (psi^(k-1)*(psi + n) - phi^(k-1)*(phi + n)) / (psi - phi):
for n from -6 to 6 do lprint(seq(simplify(F(n, k)), k = -6..6)) od;

(* Works also for n < 0 and k < 0. Uses a remark from Bill Gosper. *)
c := I*ArcSinh[1/2] - Pi/2;
F[n_, k_] := (n Sin[(k - 1) c] - I Sin[k c]) / (I^k Sqrt[5/4]);
Table[Simplify[F[n, k]], {n, 0, 6}, {k, 0, 6}] // TableForm

Functional equation extends Cassini's theorem:
F(n, k) = (-1)^(k - 1)*F(-n - 1, 2 - k).
F(n, k) = ((1 - phi)^(k - 1)*(1 - phi + n) - phi^(k - 1)*(phi + n))/(1 - 2*phi).
F(n, k) = n*fib(k - 1) + fib(k), where fib(n) are the classical Fibonacci numbers A000045 extended in the usual way for negative n.
F(n, k) - F(n-1, k) = fib(k-1).
F(n, k) = F(n, k-1) + F(n, k-2).
F(n, k) = (n*sin((k - 1)*c) - i*sin(k*c))/(i^k*sqrt(5/4)) where c = i*arcsinh(1/2) - Pi/2, for all n, k in Z. Based on a remark of Bill Gosper.

A199535 Clark Kimberling's even first column Stolarsky array read by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 5, 11, 9, 10, 8, 18, 15, 17, 12, 13, 29, 24, 27, 19, 14, 21, 47, 39, 44, 31, 23, 16, 34, 76, 63, 71, 50, 37, 25, 20, 55, 123, 102, 115, 81, 60, 41, 33, 22, 89, 199, 165, 186, 131, 97, 66, 53, 35, 26, 144, 322, 267, 301, 212, 157, 107, 86, 57, 43, 28

Offset: 1

Casey Mongoven, Nov 07 2011

The rows of the array can be seen to have the form A(n, k) = p(n)*Fibonacci(k) + q(n)*Fibonacci(k+1) where p(n) is the sequence {0, 1, 3, 3, 3, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, ...}{n >= 1} and q(n) is the sequence {1, 3, 3, 7, 2, 9, 9, 13, 13, 17, 17, 19, 19, 23, 23, 25, ...}{n >= 1}. - G. C. Greubel, Jun 23 2022

			The even first column stolarsky array (EFC array), northwest corner:
  1......2.....3.....5.....8....13....21....34....55....89...144 ... A000045;
  4......7....11....18....29....47....76...123...199...322...521 ... A000032;
  6......9....15....24....39....63...102...165...267...432...699 ... A022086;
  10....17....27....44....71...115...186...301...487...788..1275 ... A022120;
  12....19....31....50....81...131...212...343...555...898..1453 ... A013655;
  14....23....37....60....97...157...254...411...665..1076..1741 ... A000285;
  16....25....41....66...107...173...280...453...733..1186..1919 ... A022113;
  20....33....53....86...139...225...364...589...953..1542..2495 ... A022096;
  22....35....57....92...149...241...390...631..1021..1652..2673 ... A022130;
Antidiagonal rows (T(n, k)):
   1;
   2,   4;
   3,   7,   6;
   5,  11,   9,  10;
   8,  18,  15,  17, 12;
  13,  29,  24,  27, 19, 14;
  21,  47,  39,  44, 31, 23, 16;
  34,  76,  63,  71, 50, 37, 25, 20;
  55, 123, 102, 115, 81, 60, 41, 33, 22;

Clark Kimberling, The first column of an interspersion, Fibonacci Quarterly 32 (1994), pp. 301-314.

Cf. A000032, A000045, A000285, A013655, A022086, A022088, A022095.
Cf. A022096, A022097, A022113, A022120, A022121, A022122, A022130.
Cf. A022388, A022390, A035506, A035513, A097135, A199536, A199537.

From G. C. Greubel, Jun 23 2022: (Start)
T(n, 1) = A000045(n+1).
T(n, 2) = A000032(n+1), n >= 2.
T(n, 3) = A022086(n) = A097135(n), n >= 3.
T(n, 4) = A022120(n-2), n >= 4.
T(n, 5) = A013655(n-1), n >= 5.
T(n, 6) = A000285(n-2), n >= 6.
T(n, 7) = A022113(n-4), n >= 7.
T(n, 8) = A022096(n-4), n >= 8.
T(n, 9) = A022130(n-6), n >= 9.
T(n, 10) = A022098(n-5), n >= 10.
T(n, 11) = A022095(n-7), n >= 11.
T(n, 12) = A022121(n-8), n >= 12.
T(n, 13) = A022388(n-10), n >= 13.
T(n, 14) = A022122(n-10), n >= 14.
T(n, 15) = A022097(n-10), n >= 15.
T(n, 16) = A022088(n-10), n >= 16.
T(n, 17) = A022390(n-14), n >= 17.
T(n, n) = A199536(n).
T(n, n-1) = A199537(n-1), n >= 2. (End)

More terms added by G. C. Greubel, Jun 23 2022

A017125 Table read by antidiagonals of Fibonacci-type sequences.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 3, 3, 3, 1, 4, 5, 5, 4, 4, 1, 5, 8, 8, 7, 5, 5, 1, 6, 13, 13, 11, 9, 6, 6, 1, 7, 21, 21, 18, 14, 11, 7, 7, 1, 8, 34, 34, 29, 23, 17, 13, 8, 8, 1, 9, 55, 55, 47, 37, 28, 20, 15, 9, 9, 1, 10, 89, 89, 76, 60, 45, 33, 23, 17, 10, 10, 1, 11, 144, 144, 123, 97, 73

Offset: 0

Henry Bottomley, Jul 31 2000

Rows are (essentially) A000045, A000045, A000032, A000285, A022095, A022096, A022097, etc. Columns are (essentially) A001477, A000012, A000027, A005408, A016789, A016885, etc. One of the diagonals is A007502.

Antidiagonal sums are in A019274.

T(n, k) = T(n, k-1)+T(n, k-2) [with T(n, 0) = n and T(n, 1) = 1] = 2*T(n-1, k)-T(n-2, k) = Fib(k)+n*Fib(k-1) = (s^k*(1+2n/s)-t^k*(1+2n/t))/(2^k*sqrt(5)) where s = (1+sqrt(5))/2 and t = (1-sqrt(5))/2 = 1-s.

G.f. for n-th row: (n+x-nx)/(1-x-x^2).

A022310 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=5.

Original entry on oeis.org

0, 5, 6, 12, 19, 32, 52, 85, 138, 224, 363, 588, 952, 1541, 2494, 4036, 6531, 10568, 17100, 27669, 44770, 72440, 117211, 189652, 306864, 496517, 803382, 1299900, 2103283, 3403184, 5506468, 8909653, 14416122, 23325776, 37741899, 61067676, 98809576, 159877253

Offset: 0

N. J. A. Sloane

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

Cf. A000045, A022096.

LinearRecurrence[{2, 0, -1}, {0, 5, 6}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)

concat(0, Vec(x*(5-4*x) / ((1-x)*(1-x-x^2)) + O(x^50))) \\ Colin Barker, Feb 20 2017

From R. J. Mathar, Apr 07 2011: (Start)
G.f.: -x*(-5 + 4*x)/((x - 1)*(x^2 + x - 1)).
a(n) = A022096(n) - 1. (End)
a(n) = 6*F(n) + F(n-1) - 1, where F = A000045. - Bruno Berselli, Feb 20 2017
From Colin Barker, Feb 20 2017: (Start)
a(n) = -1 + (2^(-1-n)*((1-t)^n*(-11+t) + (1+t)^n*(11+t))) / t where t=sqrt(5).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)

A161611 Primes that are a sum of 7 consecutive Fibonacci numbers.

Original entry on oeis.org

53, 139, 953, 44771, 189653, 1494692464747, 26821168009247, 49900729438853608883, 5173807429197933544505507, 235602509965217050702320581591948981

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 14 2009

nonn

			a(1) = 53 = 1+2+3+5+8+13+21 is prime;
a(2) = 139 = 3+5+8+13+21+34+55 is prime.

Select[Total/@Partition[Fibonacci[Range[200]],7,1],PrimeQ] (* Harvey P. Dale, Oct 14 2022 *)

{prime(j): prime(j) = A022096(k) for some k>4}. [R. J. Mathar, Jun 18 2009]

A122012 G.f.: x^2(3+3x-2x^2)/ ( (x^2-x-1) (x^2+x-1)).

Original entry on oeis.org

0, 3, 3, 7, 9, 18, 24, 47, 63, 123, 165, 322, 432, 843, 1131, 2207, 2961, 5778, 7752, 15127, 20295, 39603, 53133, 103682, 139104, 271443, 364179, 710647, 953433, 1860498, 2496120, 4870847, 6534927, 12752043, 17108661, 33385282, 44791056

Offset: 1

Roger L. Bagula, Sep 11 2006

Index entries for linear recurrences with constant coefficients, signature (0, 3, 0, -1).

M = {{0, 1, 1, 0}, {1, 0, 0, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}} v[1] = {0, 1, 2, 3} v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]

a(n)= 3*a(n-2) -a(n-4).
a(n)= A022096(n-1)/2 + (-1)^n*A000045(n-2)/2, n >1.
a(2n+1)= A099256(2n-2), n>=1. [Mar 27 2010]

Definition replaced with generating function by the Assoc. Eds. of the OEIS, Mar 27 2010

cp's OEIS Frontend

A192956 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A210209 GCD of all sums of n consecutive Fibonacci numbers.

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A353595 Array read by ascending antidiagonals. Generalized Fibonacci numbers F(n, k) = (psi^(k - 1)*(phi + n) - phi^(k - 1)*(psi + n)) / (psi - phi) where phi = (1+sqrt(5))/2 and psi = (1-sqrt(5))/2. F(n, k) for n >= 0 and k >= 0.

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Julia

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A199535 Clark Kimberling's even first column Stolarsky array read by antidiagonals.

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A017125 Table read by antidiagonals of Fibonacci-type sequences.

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A022310 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=5.

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A161611 Primes that are a sum of 7 consecutive Fibonacci numbers.

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Mathematica

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A122012 G.f.: x^2*(3+3*x-2*x^2)/ ( (x^2-x-1) * (x^2+x-1)).

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A353595 Array read by ascending antidiagonals. Generalized Fibonacci numbers F(n, k) = (psi^(k - 1)(phi + n) - phi^(k - 1)(psi + n)) / (psi - phi) where phi = (1+sqrt(5))/2 and psi = (1-sqrt(5))/2. F(n, k) for n >= 0 and k >= 0.

A122012 G.f.: x^2(3+3x-2x^2)/ ( (x^2-x-1) (x^2+x-1)).