cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A208588 Row square-sums of triangle A185384.

Original entry on oeis.org

1, 5, 65, 979, 15345, 247535, 4069155, 67773805, 1139789185, 19311870095, 329149434263, 5637030686105, 96925730626035, 1672193347218577, 28932082285914005, 501821453320612915, 8722842168045249345, 151912536408383664095, 2650102280875677625415
Offset: 0

Views

Author

Emanuele Munarini, Feb 29 2012

Keywords

Links

Crossrefs

Cf. A185384.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]^2Fibonacci[2n-k+1]^2,{k,0,n}],{n,0,20}]
  • Maxima
    makelist(sum(binomial(n,k)^2*fib(2*n-k+1)^2,k,0,n),n,0,20);

Formula

a(n) = sum(binomial(n,k)^2*F(2n-k+1)^2, k=0..n), where F(n) are the Fibonacci numbers (A000045).
Recurrence: (n-2)*(n-1)*n*(1305*n^4 - 14094*n^3 + 57321*n^2 - 104304*n + 71944)*a(n) = 4*(n-2)*(n-1)*(6525*n^5 - 74385*n^4 + 327843*n^3 - 685683*n^2 + 653864*n - 201144)*a(n-1) - 6*(n-2)*(9135*n^6 - 125193*n^5 + 695046*n^4 - 1997365*n^3 + 3132821*n^2 - 2544304*n + 838848)*a(n-2) + 4*(32625*n^7 - 514170*n^6 + 3412449*n^5 - 12326760*n^4 + 26068504*n^3 - 32079108*n^2 + 21061664*n - 5595312)*a(n-3) - 3*(n-3)*(66555*n^6 - 918894*n^5 + 5135607*n^4 - 14853260*n^3 + 23457868*n^2 - 19205728*n + 6392000)*a(n-4) + 16*(n-4)*(n-3)*(6525*n^5 - 73080*n^4 + 309312*n^3 - 634788*n^2 + 644096*n - 264112)*a(n-5) - 4*(n-5)*(n-4)*(n-3)*(1305*n^4 - 8874*n^3 + 22869*n^2 - 26724*n + 12172)*a(n-6). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ sqrt(58+26*sqrt(5)) * (9+4*sqrt(5))^n/(20*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 13 2013
Equivalently, a(n) ~ phi^(6*n + 7/2) / (10*sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021

A208473 Central coefficients of triangle A185384.

Original entry on oeis.org

1, 6, 78, 1100, 16310, 248724, 3863244, 60784152, 965571750, 15451970820, 248731275364, 4022998982184, 65326505787452, 1064336612493000, 17390322931354200, 284844148339840560, 4675649304522319110, 76895312195988615780
Offset: 0

Views

Author

Emanuele Munarini, Feb 29 2012

Keywords

Programs

  • Mathematica
    Table[Binomial[2n,n]Fibonacci[3n+1],{n,0,100}]
  • Maxima
    makelist(binomial(2*n,n)*fib(3*n+1),n,0,17);

Formula

a(n) = binomial(2n,n)*F(3n+1), where F(n) are the Fibonacci numbers (A000045).

A208481 Diagonal sums of triangle A185384.

Original entry on oeis.org

1, 2, 6, 19, 60, 188, 589, 1846, 5786, 18135, 56840, 178152, 558377, 1750106, 5485310, 17192459, 53885860, 168892996, 529356757, 1659148590, 5200224626, 16298923631, 51085276240, 160115201936, 501844754065, 1572918462066, 4929955864374, 15451827549123
Offset: 0

Views

Author

Emanuele Munarini, Feb 29 2012

Keywords

Links

Crossrefs

Cf. A185384.

Programs

  • Mathematica
    Table[Sum[Binomial[n-k,k]Fibonacci[2n-3k+1],{k,0,Floor[n/2]}],{n,0,100}]
  • Maxima
    makelist(sum(binomial(n-k,k)*fib(2*n-3*k+1),k,0,floor(n/2)),n,0,27);

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*F(2*n-3*k+1), where F(n) are the Fibonacci numbers (A000045).
G.f.: (1 - x)/(1 - 3*x - x^3 - x^4).
a(n) = 3*a(n-1) + a(n-3) + a(n-4).

A033887 a(n) = Fibonacci(3*n + 1).

Original entry on oeis.org

1, 3, 13, 55, 233, 987, 4181, 17711, 75025, 317811, 1346269, 5702887, 24157817, 102334155, 433494437, 1836311903, 7778742049, 32951280099, 139583862445, 591286729879, 2504730781961, 10610209857723, 44945570212853, 190392490709135, 806515533049393, 3416454622906707
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Binomial transform of A063727, and second binomial transform of (1,1,5,5,25,25,...), which is A074872 with offset 0. - Paul Barry, Jul 16 2003
Equals INVERT transform of A104934: (1, 2, 8, 28, 100, 356, ...) and INVERTi transform of A005054: (1, 4, 20, 100, 500, ...). - Gary W. Adamson, Jul 22 2010
a(n) is the number of compositions of n when there are 3 types of 1 and 4 types of other natural numbers. - Milan Janjic, Aug 13 2010
F(3*n+1) = 3^n*a(n;2/3), where a(n;d), n = 0, 1, ..., d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also the papers by Witula et al.). - Roman Witula, Jul 12 2012
We note that the remark above by Paul Barry can be easily obtained from the following scaling identity for delta-Fibonacci numbers y^n a(n;x/y) = Sum_{k=0..n} binomial(n,k) (y-1)^(n-k) a(k;x) and the fact that a(n;2)=5^floor(n/2). Indeed, for x=y=2 we get 2^n a(n;1) = Sum_{k=0..n} binomial(n,k) a(k;2) and, by A000045: Sum_{k=0..n} binomial(n,k) 2^k a(k;1) = Sum_{k=0..n} binomial(n,k) F(k+1) 2^k = 3^n a(n;2/3) = F(3n+1). - Roman Witula, Jul 12 2012
Except for the first term, this sequence can be generated by Corollary 1 (iv) of Azarian's paper in the references for this sequence. - Mohammad K. Azarian, Jul 02 2015
Number of 1’s in the substitution system {0 -> 110, 1 -> 11100} at step n from initial string "1" (1 -> 11100 -> 111001110011100110110 -> ...). - Ilya Gutkovskiy, Apr 10 2017
The o.g.f. of {F(m*n + 1)}A000045%20and%20L%20=%20A000032.%20-%20_Wolfdieter%20Lang">{n>=0}, for m = 1, 2, ..., is G(m,x) = (1 - F(m-1)*x) / (1 - L(m)*x + (-1)^m*x^2), with F = A000045 and L = A000032. - _Wolfdieter Lang, Feb 06 2023

Examples

			a(5) = Fibonacci(3*5 + 1) = Fibonacci(16) = 987. - _Indranil Ghosh_, Feb 04 2017

Links

Crossrefs

Cf. A000032, A000045, A104934, A005054, A063727 (inverse binomial transform), A082761 (binomial transform), A001076, A001077.
Cf. A016777, A049310, A049651, A074872, A122070, A167808, A185384.

Programs

Formula

a(n) = A001076(n) + A001077(n) = A001076(n+1) - A001076(n).
a(n) = 2*A049651(n) + 1.
a(n) = 4*a(n-1) + a(n-2) for n>1, a(0)=1, a(1)=3;
G.f.: (1 - x)/(1 - 4*x - x^2).
a(n) = ((1 + sqrt(5))*(2 + sqrt(5))^n - (1 - sqrt(5))*(2 - sqrt(5))^n)/(2*sqrt(5)).
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n,j)*C(n-j,k)*F(n-j+1). - Paul Barry, May 19 2006
First differences of A001076. - Al Hakanson (hawkuu(AT)gmail.com), May 02 2009
a(n) = A167808(3*n+1). - Reinhard Zumkeller, Nov 12 2009
a(n) = Sum_{k=0..n} C(n,k)*F(n+k+1). - Paul Barry, Apr 19 2010
Let p[1]=3, p[i]=4, (i>1), and A be a Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1] (i <= j), A[i,j]=-1 (i = j+1), and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = det A. - Milan Janjic, Apr 29 2010
a(n) = Sum_{i=0..n} C(n,n-i)*A063727(i). - Seiichi Kirikami, Mar 06 2012
a(n) = Sum_{k=0..n} A122070(n,k) = Sum_{k=0..n} A185384(n,k). - Philippe Deléham, Mar 13 2012
a(n) = A000045(A016777(n)). - Michel Marcus, Dec 10 2015
a(n) = F(2*n)*L(n+1) + F(n-1)*(-1)^n for n > 0. - J. M. Bergot, Feb 09 2016
a(n) = Sum_{k=0..n} binomial(n,k)*5^floor(k/2)*2^(n-k). - Tony Foster III, Sep 03 2017
2*a(n) = Fibonacci(3*n) + Lucas(3*n). - Bruno Berselli, Oct 13 2017
a(n)^2 is the denominator of continued fraction [4,...,4, 2, 4,...,4], which has n 4's before, and n 4's after, the middle 2. - Greg Dresden and Hexuan Wang, Aug 30 2021
a(n) = i^n*(S(n, -4*i) + i*S(n-1, -4*i)), with i = sqrt(-1), and the Chebyshev S-polynomials (see A049310) with S(n, -1) = 0. From the simplified trisection formula. See the first entry above with A001076. - Gary Detlefs and Wolfdieter Lang, Mar 06 2023
E.g.f.: exp(2*x)*(5*cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x))/5. - Stefano Spezia, May 24 2024

A181154 Number of connected 8-regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 13, 1
Offset: 0

Views

Author

Jason Kimberley, week to Jan 31 2011

Keywords

Comments

a(20) and a(21) were computed by the author, using GENREG, over 79 processor hours and 294 processor days, respectively, during Dec 2009.

Examples

			The a( 0)=1 null graph is vacuously 8-regular and connected; since it is acyclic then it has infinite girth.
The a(16)=1 graph is the complete bipartite graph K_{8,8}.
The a(21)=1 graph has girth 4, automorphism group of order 829440, and the following adjacency lists:
01 : 02 03 04 05 06 07 08 09
02 : 01 10 11 12 13 14 15 16
03 : 01 10 11 12 13 14 15 16
04 : 01 10 11 12 13 14 15 16
05 : 01 10 11 12 13 14 15 16
06 : 01 10 11 12 17 18 19 20
07 : 01 10 11 13 17 18 19 20
08 : 01 10 12 13 17 18 19 20
09 : 01 11 12 13 17 18 19 20
10 : 02 03 04 05 06 07 08 21
11 : 02 03 04 05 06 07 09 21
12 : 02 03 04 05 06 08 09 21
13 : 02 03 04 05 07 08 09 21
14 : 02 03 04 05 17 18 19 20
15 : 02 03 04 05 17 18 19 20
16 : 02 03 04 05 17 18 19 20
17 : 06 07 08 09 14 15 16 21
18 : 06 07 08 09 14 15 16 21
19 : 06 07 08 09 14 15 16 21
20 : 06 07 08 09 14 15 16 21
21 : 10 11 12 13 17 18 19 20

References

  • M. Meringer, Fast Generation of Regular Graphs and Construction of Cages. Journal of Graph Theory, 30 (1999), 137-146.

Links

Crossrefs

8-regular simple graphs with girth at least 4: this sequence (connected), A185284 (disconnected), A185384 (not necessarily connected).
Connected k-regular simple graphs with girth at least 4: A186724 (any k), A186714 (triangle); specified degree k: A185114 (k=2), A014371 (k=3), A033886 (k=4), A058275 (k=5), A058276 (k=6), A181153 (k=7), this sequence (k=8), A181170 (k=9).
Connected 8-regular simple graphs with girth at least g: A184981 (triangle); chosen g: A014378 (g=3), this sequence (g=4).
Connected 8-regular simple graphs with girth exactly g: A184980 (triangle); chosen g: A184983 (g=3).

A122070 Triangle given by T(n,k) = Fibonacci(n+k+1)*binomial(n,k) for 0<=k<=n.

Original entry on oeis.org

1, 1, 2, 2, 6, 5, 3, 15, 24, 13, 5, 32, 78, 84, 34, 8, 65, 210, 340, 275, 89, 13, 126, 510, 1100, 1335, 864, 233, 21, 238, 1155, 3115, 5040, 4893, 2639, 610, 34, 440, 2492, 8064, 16310, 21112, 17080, 7896, 1597, 55, 801, 5184, 19572, 47502, 76860, 82908, 57492, 23256, 4181
Offset: 0

Views

Author

Philippe Deléham, Oct 15 2006, Mar 13 2012

Keywords

Comments

Subtriangle of (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Mirror image of the triangle in A185384.

Examples

			Triangle begins:
   1;
   1,   2;
   2,   6,   5;
   3,  15,  24,   13;
   5,  32,  78,   84,   34;
   8,  65, 210,  340,  275,  89;
  13, 126, 510, 1100, 1335, 864, 233;
(0, 1, 1, -1, 0, 0, ...) DELTA (1, 1, 1, 0, 0, ...) begins :
  1;
  0,  1;
  0,  1,   2;
  0,  2,   6,   5;
  0,  3,  15,  24,   13;
  0,  5,  32,  78,   84,   34;
  0,  8,  65, 210,  340,  275,  89;
  0, 13, 126, 510, 1100, 1335, 864, 233;

Links

Crossrefs

Cf. A000045, A001519, A033887, A033889, A185384.

Programs

  • GAP
    Flat(List([0..10], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n+ k+1) ))); # G. C. Greubel, Oct 02 2019
  • Magma
    [Binomial(n,k)*Fibonacci(n+k+1): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 02 2019
  • Maple
    with(combinat): seq(seq(binomial(n,k)*fibonacci(n+k+1), k=0..n), n=0..10); # G. C. Greubel, Oct 02 2019
  • Mathematica
    Table[Fibonacci[n+k+1]*Binomial[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 02 2019 *)
  • PARI
    T(n,k) = binomial(n,k)*fibonacci(n+k+1);
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 02 2019
  • Sage
    [[binomial(n,k)*fibonacci(n+k+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Oct 02 2019

Formula

T(n,k) = A000045(n+k+1)*A007318(n,k) .
T(n,n) = Fibonacci(2*n+1) = A001519(n+1) .
Sum_{k=0..n} T(n,k) = Fibonacci(3*n+1) = A033887(n) .
Sum_{k=0..n}(-1)^k*T(n,k) = (-1)^n = A033999(n) .
Sum_{k=0..floor(n/2)} T(n-k,k) = (Fibonacci(n+1))^2 = A007598(n+1).
Sum_{k=0..n} T(n,k)*2^k = Fibonacci(4*n+1) = A033889(n).
Sum_{k=0..n} T(n,k)^2 = A208588(n).
G.f.: (1-y*x)/(1-(1+3y)*x-(1+y-y^2)*x^2).
T(n,k) = T(n-1,k) + 3*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n.
T(n,k) = A185384(n,n-k).
T(2n,n) = binomial(2n,n)*Fibonacci(3*n+1) = A208473(n).

Extensions

Corrected and extended by Philippe Deléham, Mar 13 2012
Term a(50) corrected by G. C. Greubel, Oct 02 2019
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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