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### -- Authors: Tobias Boege
### -- File: t/data/Fibonacci-Search-all.txt
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### -- Chunk 11 of 15
21104 a(n) = Fibonacci(n - 1) modulo the n-th prime number.
%H A121104 Vincenzo Librandi, <a href="/A121104/b121104.txt">Table of n, a(n) for n = 2..5000</a>
%F A121104 a(n) = Fibonacci(n - 1) modulo Prime(n).
%e A121104 a(10)=5 because the 9th Fibonacci=34, the 10th Prime=29, and 34 mod 29=5.
%t A121104 With[{nn=70},Mod[First[#],Last[#]]&/@Thread[{Fibonacci[Range[nn-1]], Prime[ Range[2,nn]]}]] (* _Harvey P. Dale_, Feb 27 2013 *)
%t A121104 Table[Mod[Fibonacci[n - 1], Prime[n]], {n, 2, 70}] (* _Vincenzo Librandi_, Jun 19 2017 *)
%o A121104 (PARI) a(n) = fibonacci(n-1) % prime(n); \\ _Michel Marcus_, Jun 18 2017
%o A121104 (PARI) fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
%o A121104 a(n)=lift(fibmod(n-1,prime(n))) \\ _Charles R Greathouse IV_, Jun 19 2017
%o A121104 (MAGMA) [Fibonacci(n-1) mod NthPrime(n): n in [2..70]]; // _Vincenzo Librandi_, Jun 19 2017
%Y A121104 Cf. A000040, A000045, A072123.
%K A121104 nonn
%O A121104 2,3
%A A121104 _Gil Broussard_, Aug 12 2006

%I A147316
%S A147316 -6765,4181,-2584,1597,-987,610,-377,233,-144,89,-55,34,-21,13,-8,5,
%T A147316 -3,2,-1,1,0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,
%U A147316 4181,6765,10946,17711,28657,46368,75025,121393,196418,317811,514229,832040
%N A147316 Fibonacci numbers (A000045) starting at offset -20.
%C A147316 The recurrence relation a(n+1) = a(n) + a(n-1) defines the Fibonacci sequence for all (positive and negative) integer indices, given any two values with indices of opposite parity, e.g., a(0) and a(1), or a(-1) and a(42). Any other Fibonacci-type sequence {b(n)} satisfying this recurrence relation can be written as b(n) = b(1)*A000045(n) + b(0)*A000045(n-1). This can be seen from the fact that the set of all sequences satisfying a given linear recurrence relation of order 2 with constant coefficients forms a vector space of dimension two. So each element (sequence) of this space is a linear combination of any two elements which are not proportional to each other and thus form a base. The most natural choice of such a base could be the two sequences having (b(0), b(1)) = (0, 1) resp (1, 0). These are A000045 and n -> A000045(n-1) = A212804 (extended to negative indices, if needed). - _M. F. Hasler_, May 10 2017
%H A147316 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%H A147316 Philipp Fahr and Claus Michael Ringel, <a href="http://www.mathematik.uni-bielefeld.de/~ringel/opus/fr-zwei.pdf">Categorification of the Fibonacci Numbers Using Representations of Quivers</a>
%F A147316 a(n) = a(n-1) + a(n-2). - _R. J. Mathar_, Nov 30 2008
%t A147316 Array[Fibonacci, 51, -20] (* _Michael De Vlieger_, May 10 2017 *)
%o A147316 (PARI) a(n)=fibonacci(n) \\ _M. F. Hasler_, May 10 2017
%Y A147316 Cf. A000285, A022113, A001060.
%Y A147316 Of course A000045 is the main entry for the Fibonacci numbers.
%Y A147316 See also A039834 for A000045 extended to negative indices.
%K A147316 sign,easy,less
%O A147316 -20,1
%A A147316 _Roger L. Bagula_, Nov 05 2008
%E A147316 Extended to n = -20 .. 30 by _M. F. Hasler_, May 10 2017

%I A177372
%S A177372 1,1,2,3,5,8,13,21,34,55,89,144,233,610,2584,4181,10946,46368,121393,
%T A177372 196418,514229,832040,1346269,14930352,39088169,63245986,102334155,
%U A177372 165580141,1836311903,12586269025,32951280099,139583862445,365435296162
%N A177372 Fibonacci numbers whose decimal expansion does not contain any digit 7
%C A177372 Probability that Fib(n) contains no digit 7 goes to zero as n grows to infinity. I suppose that the maximum number is Fib(224) having 47 digits.
%e A177372 a(14)=610 since it is the 14th Fibonacci containing no 7's.
%t A177372 Select[Fibonacci[Range[100]],DigitCount[#,10,7]==0&] (* _Harvey P. Dale_, Dec 13 2014 *)
%Y A177372 Cf. A000045, A177194, A177195, A177231, A177245, A177246, A176253, A177247
%K A177372 nonn,base
%O A177372 1,3
%A A177372 _Carmine Suriano_, May 07 2010

%I A206139
%S A206139 1,1,1,2,3,5,8,13,21,34,55,88,141,224,356,563,890,1401,2202,3448,5386,
%T A206139 8386,13025,20175,31180,48077,73976,113588,174057,266174,406224,
%U A206139 618729,940552,1427038,2161122,3266956,4930052,7427314,11171332,16776169,25154204
%N A206139 G.f.: A(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1-x^k)^(n-k+1).
%H A206139 Vaclav Kotesovec, <a href="/A206139/b206139.txt">Table of n, a(n) for n = 0..400</a>
%e A206139 G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 13*x^7 +...
%e A206139 where
%e A206139 A(x) = 1 + x/(1-x) + x^3/((1-x)^2*(1-x^2)) + x^6/((1-x)^3*(1-x^2)^2*(1-x^3)) + x^10/((1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)) +...
%o A206139 (PARI) {a(n)=polcoeff(sum(m=0,n,x^(m*(m+1)/2)/prod(k=1,m,(1-x^k +x*O(x^n))^(m-k+1))),n)}
%o A206139 for(n=0,61,print1(a(n),", "))
%Y A206139 Cf. A206119.
%K A206139 nonn
%O A206139 0,4
%A A206139 _Paul D. Hanna_, Feb 04 2012

%I A217737
%S A217737 1,1,2,3,5,8,13,21,34,55,89,144,51,167,130,171,67,190,1,45,320,1,505,
%T A217737 168,275,649,614,319,59,620,125,837,376,407,485,1296,1331,419,466,
%U A217737 1435,1231,1420,1289,1653,830,2069,2161,1344,1849,1975,746,1167,1589,872,2645,2205
%N A217737 a(n) = Fibonacci(n) mod n*(n+1).
%H A217737 Alois P. Heinz, <a href="/A217737/b217737.txt">Table of n, a(n) for n = 1..10000</a>
%F A217737 A000045(n) modulo A002378(n).
%p A217737 a:= proc(n) local r, M, p, m; r, M, p, m:=
%p A217737       <<1|0>, <0|1>>, <<0|1>, <1|1>>, n, n*(n+1);
%p A217737       do if irem(p, 2, 'p')=1 then r:= r.M mod m fi;
%p A217737          if p=0 then break fi; M:= M.M mod m
%p A217737       od; r[1, 2]
%p A217737     end:
%p A217737 seq(a(n), n=1..100);  # _Alois P. Heinz_, Nov 26 2016
%t A217737 Table[Mod[Fibonacci[n],n(n+1)],{n,60}] (* _Harvey P. Dale_, Oct 02 2017 *)
%o A217737 (Python)
%o A217737 prpr, prev = 0, 1
%o A217737 for i in range(1, 333):
%o A217737     cur = prpr + prev
%o A217737     print str(prev % (i*(i+1))) + ',',
%o A217737     prpr, prev = prev, cur
%o A217737 (PARI) a(n)=fibonacci(n)%(n*(n+1)) \\ _Charles R Greathouse IV_, Jun 23 2017
%Y A217737 Cf. A000045, A002708, A121343, A132634, A132636.
%K A217737 nonn,easy
%O A217737 1,3
%A A217737 _Alex Ratushnyak_, Mar 22 2013

%I A273715
%S A273715 0,1,1,1,2,3,5,8,13,21,1,34,57,6,90,158,27,241,445,107,1,652,1269,396,
%T A273715 10,1780,3655,1404,66,4899,10611,4838,356,1,13581,31002,16344,1700,15,
%U A273715 37893,91048,54429,7482,135,106340,268536,179332,31070,940,1
%N A273715 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k peaks of width 1 (i.e., UHD configurations, where U=(0,1), H(1,0), D=(0,-1)), (n>=2, k>=0).
%C A273715 Sum of entries in row n = A082582(n).
%C A273715 Sum(k*T(n,k),k>=1) = A273716(n).
%H A273715 Alois P. Heinz, <a href="/A273715/b273715.txt">Rows n = 2..250, flattened</a>
%H A273715 M. Bousquet-Mélou and A. Rechnitzer <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a> Adv. Appl. Math., 31, 2003, 86-112.
%H A273715 Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088, 2016
%F A273715 G.f.: G(t,z) satisfies z*G^2 - (1-2*z-z^2-z^3+t*z^3)G + z^2*(t+z-t*z) = 0.
%e A273715 Row 4 is 2,3 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] which, clearly, have 0,1,1,0,1 peaks of width 1.
%e A273715 Triangle T(n,k) begins:
%e A273715 :    0,    1;
%e A273715 :    1,    1;
%e A273715 :    2,    3;
%e A273715 :    5,    8;
%e A273715 :   13,   21,   1;
%e A273715 :   34,   57,   6;
%e A273715 :   90,  158,  27;
%e A273715 :  241,  445, 107,  1;
%e A273715 :  652, 1269, 396, 10;
%p A273715 eq := z*G^2-(1-2*z-z^2-z^3+t*z^3)*G+z^2*(t+z-t*z) = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): for n from 2 to 20 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 20 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%p A273715 # second Maple program:
%p A273715 b:= proc(n, y, t, h) option remember; expand(
%p A273715       `if`(n=0, (1-t)*z^h, `if`(t<0, 0, b(n-1, y+1, 1, 0))+
%p A273715       `if`(t>0 or y<2, 0, b(n, y-1, -1, 0)*z^h)+
%p A273715       `if`(y<1, 0, b(n-1, y, 0, `if`(t>0, 1, 0)))))
%p A273715     end:
%p A273715 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
%p A273715 seq(T(n), n=2..20);  # _Alois P. Heinz_, Jun 06 2016
%t A273715 b[n_, y_, t_, h_] := b[n, y, t, h] = Expand[ If[n == 0, (1 - t)*z^h, If[t < 0, 0, b[n - 1, y + 1, 1, 0]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]*z^h] + If[y < 1, 0, b[n - 1, y, 0, If[t > 0, 1, 0]]]]] ; T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0, 0]]; Table[T[n], {n, 2, 20}] // Flatten (* _Jean-François Alcover_, Nov 29 2016 after _Alois P. Heinz_ *)
%Y A273715 Cf. A082582, A273716.
%K A273715 nonn,tabf
%O A273715 2,5
%A A273715 _Emeric Deutsch_, May 28 2016

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# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 51-60 of 92

%I A302019
%S A302019 1,1,2,3,5,8,13,21,34,56,91,149,243,397,648,1058,1727,2819,4602,7512,
%T A302019 12263,20018,32678,53344,87080,142151,232050,378803,618366,1009433,
%U A302019 1647819,2689933,4391101,7168122,11701387,19101580,31181804,50901806,83093134,135642908,221426218,361460624
%N A302019 Expansion of 1/(1 - x*Sum_{k>=0} x^(k^3)).
%H A302019 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A302019 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A302019 G.f.: 1/(1 - x*Sum_{k>=0} x^(k^3)).
%F A302019 a(0) = 1; a(n) = Sum_{k=1..n} A010057(k-1)*a(n-k).
%t A302019 nmax = 41; CoefficientList[Series[1/(1 - x Sum[x^k^3, {k, 0, nmax}]), {x, 0, nmax}], x]
%Y A302019 Antidiagonal sums of A290054.
%Y A302019 Cf. A010057, A302018.
%K A302019 nonn
%O A302019 0,3
%A A302019 _Ilya Gutkovskiy_, Mar 30 2018

%I A304790
%S A304790 1,1,2,3,5,8,13,21,36,55,95,149,281,430,781,1211,2245,3456,6728,10092,
%T A304790 18061,31529,51378,85659,167089,252748,431819,817991,1292697
%N A304790 The maximal number of different domino tilings allowed by the Ferrers-Young diagram of a single partition of 2n.
%H A304790 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram</a>
%H A304790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_(mathematics)">Domino</a>
%H A304790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_tiling">Domino tiling</a>
%H A304790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ferrers_diagram">Ferrers diagram</a>
%H A304790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mutilated_chessboard_problem">Mutilated chessboard problem</a>
%H A304790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A304790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau#Diagrams">Young tableau, Diagrams</a>
%F A304790 a(n) = max { k : A304789(n,k) > 0 }.
%F A304790 a(A001105(n)) = A004003(n).
%F A304790 a(n) = A000045(n+1) for n < 8.
%e A304790 a(11) = 149 different domino tilings are possible for 444442 and 6655.
%e A304790 a(18) = 6728 different domino tilings are possible for 666666.
%p A304790 h:= proc(l, f) option remember; local k; if min(l[])>0 then
%p A304790      `if`(nops(f)=0, 1, h(map(u-> u-1, l[1..f[1]]), subsop(1=[][], f)))
%p A304790     else for k from nops(l) while l[k]>0 by -1 do od;
%p A304790         `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
%p A304790         `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
%p A304790       fi
%p A304790     end:
%p A304790 g:= l-> `if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
%p A304790         `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
%p A304790 b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), max(b(n, i-1, l),
%p A304790                    b(n-i, min(n-i, i), [l[], i]))):
%p A304790 a:= n-> b(2*n$2, []):
%p A304790 seq(a(n), n=0..15);
%Y A304790 Cf. A000045, A001105, A004003, A304789.
%K A304790 nonn
%O A304790 0,3
%A A304790 _Alois P. Heinz_, May 18 2018

%I A306486
%S A306486 0,1,1,2,3,5,8,13,21,36,58,96,159,262,431,712,1172,1934,3189,5256,
%T A306486 8667,14289,23559,38841,64039,105583,174076,287003,473188,780155,
%U A306486 1286258,2120681,3496412,5764609,9504233,15669832,25835185,42595018,70227313,115785266
%N A306486 Number of squares in the interval [e^(n-1), e^n).
%C A306486 The lower endpoint e^(n-1) is included; the upper endpoint is not included. The terms a(0) to a(8) coincide with the Fibonacci numbers.
%H A306486 Alois P. Heinz, <a href="/A306486/b306486.txt">Table of n, a(n) for n = 0..4607</a> (first 501 terms from Alexei Kourbatov)
%F A306486 a(n) = ceiling(sqrt(exp(n))) - ceiling(sqrt(exp(n-1))).
%F A306486 From _Alois P. Heinz_, Feb 19 2019: (Start)
%F A306486 Lim_{n->oo} a(n+1)/a(n) = sqrt(e) = 1.64872127... = A019774.
%F A306486 a(n) = A005181(n+1) - A005181(n). (End)
%F A306486 a(n) = (1-1/sqrt(e))*e^(n/2)+O(1) ~ 0.39346934...*e^(n/2) ~ A290506*e^(n/2). - _Alexei Kourbatov_, Feb 20 2019
%e A306486 Between exp(2) and exp(3) there are two squares, namely, 9 and 16; therefore, a(3)=2.
%p A306486 a:= n-> (f-> f(n)-f(n-1))(i-> ceil(exp(i/2))):
%p A306486 seq(a(n), n=0..44);  # _Alois P. Heinz_, Feb 18 2019
%o A306486 (PARI) a(n)=ceil(sqrt(exp(n)))-ceil(sqrt(exp(n-1)));
%o A306486 for(n=0,50,print1(a(n)", "))
%Y A306486 Cf. A000045, A000290, A001113, A005181, A019774, A290506, A306604.
%K A306486 nonn
%O A306486 0,4
%A A306486 _Alexei Kourbatov_, Feb 18 2019

%I A324969
%S A324969 1,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,
%T A324969 10946,17711,28657,46368
%N A324969 Number of unlabeled rooted identity trees with n vertices whose non-leaf terminal subtrees are all different.
%C A324969 A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. This sequence counts rooted identity trees satisfying the additional condition that all non-leaf terminal subtrees are different.
%C A324969 Appears to be essentially the same as the Fibonacci sequence A000045. - _R. J. Mathar_, Mar 28 2019
