A014166 -id:A014166 - cp's OEIS frontend

A257838 Main diagonal of iterated partial sums array of Fibonacci numbers (starting with the first partial sums).

0, 1, 4, 16, 63, 247, 967, 3785, 14820, 58060, 227612, 892926, 3505386, 13770404, 54129602, 212904952, 837885495, 3299264407, 12997784803, 51230474669, 202014314769, 796928589755, 3145066003589, 12416625685891, 49037912997003, 193734379979677, 765632076098287, 3026670770970925, 11968378998073935

Offset: 0

Luciano Ancora, May 10 2015

The array used here starts in row n=0 with the first partial sums of A000045. The array which starts with the Fibonacci numbers in row k=0 is shown in A136431. The diagonal of that array is given in A176085. - Wolfdieter Lang, Jun 03 2015

			This sequence is the main diagonal of the following array (see the comment and Example field of A136431):
0, 1, 2,  4,  7,  12, ...  A000071
0, 1, 3,  7, 14,  26, ...  A001924
0, 1, 4, 11, 25,  51, ...  A014162
0, 1, 5, 16, 41,  92, ...  A014166
0, 1, 6, 22, 63, 155, ...  A053739
0, 1, 7, 29, 92, 247, ...  A053295

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000045, A000071, A001924, A014162, A014166, A136431, A176085.

Table[DifferenceRoot[Function[{a, n},{(2*n + 4*n^2)*a[n] + (2 + 7*n + 15*n^2)*a[1 + n] + (8 - 6*n - 8*n^2)*a[2 + n] + (-2 + n + n^2)*a[3 + n] == 0, a[1] == 0, a[2] == 1, a[3] == 4, a[4] == 16}]][n], {n, 30}]

a(n):=sum(binomial(2*n-k,n-k)*fib(k),k,0,n); /* Vladimir Kruchinin, Oct 09 2016 */

x='x+O('x^50); concat([0], Vec(-(4*x+sqrt(1-4*x)-1)/(8*x^2+sqrt(1-4*x)*(8*x-2)-2*x))) \\ G. C. Greubel, Apr 08 2017

a(n) = F^{n+1}(n), n >= 0, with the k-th iterated partial sum F^{k} of the Fibonacci number A000045. - Wolfdieter Lang, Jun 03 2015
Conjecture: n*(n-3)*a(n) +2*(-4*n^2+13*n-6)*a(n-1) +(15*n^2-53*n+48)*a(n-2) +2*(2*n-3)*(n-2)*a(n-3)=0. - R. J. Mathar, Dec 10 2015
G.f.: -(4*x+sqrt(1-4*x)-1)/(8*x^2+sqrt(1-4*x)*(8*x-2)-2*x). - Vladimir Kruchinin, Oct 09 2016
a(n) = Sum_{k=0..n} binomial(2*n-k,n-k)*F(k), where F(k) = A000045(k). - Vladimir Kruchinin, Oct 09 2016
a(n) ~ 2^(2*n+1)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 09 2016

Name edited by Wolfdieter Lang, Jun 03 2015

A136338 Primes in the array A136431 that are not Fibonacci numbers.

Original entry on oeis.org

7, 11, 29, 37, 41, 67, 79, 97, 137, 191, 211, 277, 379, 631, 709, 821, 947, 967, 991, 1129, 1327, 1597, 1831, 2017, 2081, 2267, 2347, 2557, 2683, 2851, 2927, 3571, 3917, 4561, 4657, 4951, 5051, 5779, 6217, 6329, 6763, 8273, 8647, 8779, 9181, 9871, 10093

Offset: 1

Jonathan Vos Post, Apr 12 2008

A generalization of prime Fibonacci numbers (A005478) are the prime hyperfibonacci numbers (primes in A136431). Referring to the array A(k,n) = Apply partial sum operator k times to Fibonacci numbers, we see that every prime occurs in the n=2 column (as it contains every positive integer).
So excluding n=2 and k=0 (A005478) we have the nontrivially prime hyperfibonacci numbers which are not Fibonacci numbers.
Note that this sequence does not indicate multiplicity (e.g., 7 occurs twice in the valid part of the table).
Continuing the table of primes in the examples, from a computation by Joshua Zucker, we have:
k=1: {7, ...} no more through n = 1000.
k=2: {7, 79, 514201, 14930317, 956722025983, 5527939700884681 4660046610375530219, ...}
k=3: {11, 97, 17519, next value has 60 digits, ...}
k=4: {41, 10093, 16703, 3520457, 591286703533, 6557470285501, 19740274219868101499, ...}
k=5: {709, 8273, 14323, 466004661037329684,1 298611126818977061133263, ...}
k=6: {29, 2683, 23945893, 1835540197, 4052735290427, 27777884012083, ...}
k=7: {37, 967, 2267, 127921, 226007, 62048869, 1131463777, 7540113804271826929, ...}
k=8: {27777538280521, 1409869790947669143312035590804646728957, ...}
k=9: {1033628323428189498226451492123369099, next value has 60 digits, ...}
k=10: {67, 5972304273877744135569337875802249660927, ...}
k=11: {79, 4478413, 19008291293, 61305228407581679, ...}
k=12: {6763, 1982269, 37886753582095837, 2791715456569622316696636389, ...}.

			k=1: primes in A000071 = {A000071(4) = 7}, no more through n = 1000.
k=2: primes in A001924 = {A001924(3) = 7, A001924(7) = 79, A001924(25) = 514201, ...}
k=3: primes in A014162 = {A014162(3) = 11, A014162(6) = 97, A014162(16) = 17519}, no more through n = 30.
k=4: primes in A014166 = {A014166(4) = 41, A014166(13) = 10093, A014166(14) = 16703}
k=5: primes in A053739 = {A053739(7) = 709, A053739(10) = 8273, A053739(11) = 14323}, no more through n = 27.
k=6: primes in A053295 = {A053295(3) = 29, A053295(8) = 2683, 23945893(24) = 23945893}, no more through n = 27.
k=7: primes in A053296 = {A053296(3) = 37, A053296(6) = 967, A053296(7) = 2267, A053296(12) = 127921, A053296(13) = 226007}, no more through n = 27.

Cf. A000040, A005478, A136431, A137176.

Cf. A136431.

A136431 := proc(k,n) local x ; coeftayl(x/(1-x-x^2)/(1-x)^k,x=0,n) ; end: A136338 := proc(amax) local a,k,n,a136431; a := [] ; for k from 1 do if A136431(k,3) > amax then break ; fi ; for n from 3 do a136431 := A136431(k,n) ; if a136431 > amax then break ; fi ; if isprime(a136431) and not a136431 in a then a := [op(a),a136431] ; fi ; od: od: sort(a) ; end: A136338(20000) ; # R. J. Mathar, Apr 21 2008

partsumfib(N,s=[],P=[])={ for( n=1+#s,N, s=concat(s,n+1); forstep( i=n,1,-1, isprime( s[i]+= if( i>1, s[i-1], fibonacci(n+2) ) ) & P=setunion(P,[s[i]]) ); print(s); );vecsort(eval(P))} \\ M. F. Hasler

Primes in the hyperfibonacci number array of A136431, excluding the n=2 column (which contains every positive integer).

Revised definition from N. J. A. Sloane, May 09 2008

More terms from R. J. Mathar, Apr 21 2008

A220888 a(n) = F(n+7) - (1/2)(n^3+2n^2+13*n+26) where F(i) is a Fibonacci number (A000045).

Original entry on oeis.org

0, 0, 0, 0, 2, 11, 37, 98, 225, 470, 919, 1713, 3082, 5400, 9274, 15688, 26236, 43499, 71655, 117466, 191875, 312590, 508265, 825265, 1338612, 2169696, 3514932, 5692128, 9215510, 14917115, 24143209, 39072098, 63228357, 102314870, 165559099, 267891393, 433469566, 701381784, 1134874030

Offset: 0

N. J. A. Sloane, Dec 29 2012

nonn

J. Freixas and S. Kurz, The golden number and Fibonacci sequences in the design of voting structures, 2012.
J. Freixas and S. Kurz, The golden number and Fibonacci sequences in the design of voting structures, arXiv:1401.8180 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

Cf. A000045, A014166.

LinearRecurrence[{5, -9, 6, 1, -3, 1}, {0, 0, 0, 0, 2, 11}, 39] (* Jean-François Alcover, Feb 12 2019 *)

G.f.: -x^4*(2+x) / ( (x^2+x-1)*(x-1)^4 ). - R. J. Mathar, Jan 11 2013

a(n) = A014166(n-4)+2*A014166(n-3). - R. J. Mathar, Mar 24 2013

A351898 Decimal expansion of metallic ratio for N = 14.

Original entry on oeis.org

1, 4, 0, 7, 1, 0, 6, 7, 8, 1, 1, 8, 6, 5, 4, 7, 5, 2, 4, 4, 0, 0, 8, 4, 4, 3, 6, 2, 1, 0, 4, 8, 4, 9, 0, 3, 9, 2, 8, 4, 8, 3, 5, 9, 3, 7, 6, 8, 8, 4, 7, 4, 0, 3, 6, 5, 8, 8, 3, 3, 9, 8, 6, 8, 9, 9, 5, 3, 6, 6, 2, 3, 9, 2, 3, 1, 0, 5, 3, 5, 1, 9, 4, 2, 5, 1, 9

Offset: 2

A.H.M. Smeets, Feb 24 2022

Decimal expansion of continued fraction [14; 14, 14, 14, ...].
Also largest solution of x^2 - 14 x - 1 = 0.
Essentially the same digit sequence as A010503, A157214, A174968 and A268683.
The metallic ratio's for N = A077444(n) are equal to powers of the silver ratio, i.e., A014166^(2n-1); this constant represents the special case for N = A077444(2).

			14.0710678118654752440084436210484903928483593...

Michael Penn, 7+5√2 is a perfect cube??, YouTube video, 2022.
Wikipedia, Metallic mean.

Cf. A010503, A077444, A157214, A174968, A268683.

Metallic ratios: A001622 (N=1), A014176 (N=2), A098316 (N=3), A098317 (N=4), A098318 (N=5), A176398 (N=6), A176439 (N=7), A176458 (N=8), A176522 (N=9), A176537 (N=10), A244593 (N=11).

RealDigits[7 + 5*Sqrt[2], 10, 100][[1]] (* Amiram Eldar, Feb 24 2022 *)

PARI
```
(1+sqrt(2))^3
```

Equals 2 + 5*A014176.
Equals A014176^3.
Equals exp(arcsinh(7)). - Amiram Eldar, Jul 04 2023

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A257838 Main diagonal of iterated partial sums array of Fibonacci numbers (starting with the first partial sums).

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A136338 Primes in the array A136431 that are not Fibonacci numbers.

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A220888 a(n) = F(n+7) - (1/2)*(n^3+2*n^2+13*n+26) where F(i) is a Fibonacci number (A000045).

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A351898 Decimal expansion of metallic ratio for N = 14.

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A220888 a(n) = F(n+7) - (1/2)(n^3+2n^2+13*n+26) where F(i) is a Fibonacci number (A000045).