A015520 -id:A015520 - cp's OEIS frontend

A015541 Expansion of x/(1 - 5x - 7x^2).

0, 1, 5, 32, 195, 1199, 7360, 45193, 277485, 1703776, 10461275, 64232807, 394392960, 2421594449, 14868722965, 91294775968, 560554940595, 3441838134751, 21133075257920, 129758243232857, 796722742969725, 4891921417478624, 30036666288181195

Offset: 0

Olivier Gérard

Pisano period lengths: 1, 3, 8, 6, 8, 24, 6, 6, 24, 24, 5, 24, 12, 6, 8, 12, 16, 24, 120, 24, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,7).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226.

[n le 2 select n-1 else 5*Self(n-1) + 7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

Join[{a=0,b=1},Table[c=5*b+7*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{5, 7}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

x='x+O('x^30); concat([0], Vec(x/(1-5*x-7*x^2))) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,5,-7) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 7*a(n-2).

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 5, 33, 205, 1289, 8085, 50737, 318365, 1997721, 12535525, 78659393, 493581165, 3097180969, 19434554165, 121950218577, 765227526205, 4801739379641, 30130517107845, 189066500576353, 1186376639744525, 7444415203333449, 46713089134623445

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,8).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015441, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A015555 (binomial transform).

[n le 2 select n-1 else 5*Self(n-1) + 8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

a[n_]:=(MatrixPower[{{1,2},{1,-6}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{5, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

A015544(n)=imag((2+quadgen(57))^n) \\ M. F. Hasler, Mar 06 2009

x='x+O('x^30); concat([0], Vec(x/(1 - 5*x - 8*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,5,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 8*a(n-2).

G.f.: x/(1 - 5*x - 8*x^2). - M. F. Hasler, Mar 06 2009

More precise definition by M. F. Hasler, Mar 06 2009

A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 1, 2, 5, 8, 5, 0, 1, 2, 6, 12, 16, 6, 0, 1, 2, 7, 16, 29, 32, 7, 0, 1, 2, 8, 20, 44, 70, 64, 8, 0, 1, 2, 9, 24, 61, 120, 169, 128, 9, 0, 1, 2, 10, 28, 80, 182, 328, 408, 256, 10, 0, 1, 2, 11, 32, 101, 256, 547, 896, 985, 512, 11

Offset: 0

Ralf Stephan, Oct 13 2004

			Square array, A(n, k), begins as:
  0, 1, 2,  3,  4,   5,    6,    7,     8, ... A001477;
  0, 1, 2,  4,  8,  16,   32,   64,   128, ... A000079;
  0, 1, 2,  5, 12,  29,   70,  169,   408, ... A000129;
  0, 1, 2,  6, 16,  44,  120,  328,   896, ... A002605;
  0, 1, 2,  7, 20,  61,  182,  547,  1640, ... A015518;
  0, 1, 2,  8, 24,  80,  256,  832,  2688, ... A063727;
  0, 1, 2,  9, 28, 101,  342, 1189,  4088, ... A002532;
  0, 1, 2, 10, 32, 124,  440, 1624,  5888, ... A083099;
  0, 1, 2, 11, 36, 149,  550, 2143,  8136, ... A015519;
  0, 1, 2, 12, 40, 176,  672, 2752, 10880, ... A003683;
  0, 1, 2, 13, 44, 205,  806, 3457, 14168, ... A002534;
  0, 1, 2, 14, 48, 236,  952, 4264, 18048, ... A083102;
  0, 1, 2, 15, 52, 269, 1110, 5179, 22568, ... A015520;
  0, 1, 2, 16, 56, 304, 1280, 6208, 27776, ... A091914;
Antidiagonal triangle, T(n, k), begins as:
  0;
  0,  1;
  0,  1,  2;
  0,  1,  2,  3;
  0,  1,  2,  4,  4;
  0,  1,  2,  5,  8,  5;
  0,  1,  2,  6, 12, 16,   6;
  0,  1,  2,  7, 16, 29,  32,   7;
  0,  1,  2,  8, 20, 44,  70,  64,   8;
  0,  1,  2,  9, 24, 61, 120, 169, 128,   9;
  0,  1,  2, 10, 28, 80, 182, 328, 408, 256,  10;

G. C. Greubel, Antidiagonals n = 0..50, flattened
Ralf Stephan, Prove or disprove. 100 Conjectures from the OEIS, #16, arXiv:math/0409509 [math.CO], 2004.

Rows m: A001477 (m=0), A000079 (m=1), A000129 (m=2), A002605 (m=3), A015518 (m=4), A063727 (m=5), A002532 (m=6), A083099 (m=7), A015519 (m=8), A003683 (m=9), A002534 (m=10), A083102 (m=11), A015520 (m=12), A091914 (m=13).
Columns q: A000004 (q=0), A000012 (q=1), A009056 (q=2), A008586 (q=3).
Main diagonal gives A357502.

A099173:= func< n,k | (&+[n^j*Binomial(k,2*j+1): j in [0..Floor(k/2)]]) >;
[A099173(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2023

A[k_, n_]:= Which[k==0, n, n==0, 0, True, ((1+Sqrt[k])^n - (1-Sqrt[k])^n)/(2 Sqrt[k])]; Table[A[k-n, n]//Simplify, {k, 0, 12}, {n, 0, k}]//Flatten (* Jean-François Alcover, Jan 21 2019 *)

A(k,n)=sum(i=0,n\2,k^i*binomial(n,2*i+1))

def A099173(n,k): return sum( n^j*binomial(k, 2*j+1) for j in range((k//2)+1) )
flatten([[A099173(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 17 2023

A(n, k) = Sum_{i=0..floor(k/2)} n^i * C(k, 2*i+1) (array).
Recurrence: A(n, k) = 2*A(n, k-1) + (n-1)*A(n, k-2), with A(n, 0) = 0, A(n, 1) = 1.
T(n, k) = A(n-k, k) (antidiagonal triangle).
T(2*n, n) = A357502(n).
A(n, k) = ((1+sqrt(n))^k - (1-sqrt(n))^k)/(2*sqrt(n)). - Jean-François Alcover, Jan 21 2019

A109447 Binomial coefficients C(n,k) with n-k odd, read by rows.

Original entry on oeis.org

1, 2, 1, 3, 4, 4, 1, 10, 5, 6, 20, 6, 1, 21, 35, 7, 8, 56, 56, 8, 1, 36, 126, 84, 9, 10, 120, 252, 120, 10, 1, 55, 330, 462, 165, 11, 12, 220, 792, 792, 220, 12, 1, 78, 715, 1716, 1287, 286, 13, 14, 364, 2002, 3432, 2002, 364, 14, 1, 105, 1365, 5005, 6435, 3003, 455, 15

Offset: 1

Philippe Deléham, Aug 27 2005

The same as A119900 without 0's. A reflected version of A034867 or A202064. - Alois P. Heinz, Feb 07 2014
From Vladimir Shevelev, Feb 07 2014: (Start)
Also table of coefficients of polynomials  P_1(x)=1, P_2(x)=2, for n>=2, P_(n+1)(x) = 2*P_n(x)+(x-1)* P_(n-1)(x).  The polynomials P_n(x)/2^(n-1) are connected with sequences A000045 (x=5), A001045 (x=9), A006130 (x=13), A006131 (x=17), A015440 (x=21), A015441 (x=25), A015442 (x=29), A015443 (x=33), A015445 (x=37), A015446 (x=41), A015447 (x=45), A053404 (x=49); also the polynomials P_n(x) are connected with sequences A000129, A002605, A015518, A063727, A085449, A002532, A083099, A015519, A003683, A002534, A083102, A015520. (End)

			Starred terms in Pascal's triangle (A007318), read by rows:
1;
1*, 1;
1, 2*, 1;
1*, 3, 3*, 1;
1, 4*, 6, 4*, 1;
1*, 5, 10*, 10, 5*, 1;
1, 6*, 15, 20*, 15, 6*, 1;
1*, 7, 21*, 35, 35*, 21, 7*, 1;
1, 8*, 28, 56*, 70, 56*, 28, 8*, 1;
1*, 9, 36*, 84, 126*, 126, 84*, 36, 9*, 1;
Triangle T(n,k) begins:
1;
2;
1,    3;
4,    4;
1,   10,  5;
6,   20,  6;
1,   21,  35,   7;
8,   56,  56,   8;
1,   36, 126,  84,  9;
10, 120, 252, 120, 10;

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A109446.

T:= (n, k)-> binomial(n, 2*k+1-irem(n, 2)):
seq(seq(T(n, k), k=0..ceil((n-2)/2)), n=1..20);  # Alois P. Heinz, Feb 07 2014

Flatten[ Table[ If[ OddQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Aug 30 2005

Corrected offset by Alois P. Heinz, Feb 07 2014

A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2n))^k - (1 - sqrt(2n))^k)/(2sqrt(2n)).

Original entry on oeis.org

5, 7, 101, 11, 13, 269, 17, 19, 509, 23, 709, 821, 29, 31, 46957, 55399, 37, 168846239, 41, 43, 9177868096974864412935432937651459122761, 47, 485329129, 2789, 53, 3229, 3461, 59, 61, 1563353111, 139237612541, 67, 5021, 71, 73, 484639, 6221, 79, 6869, 83, 7549

Offset: 1

XU Pingya, Sep 24 2017

nonn

			For k = {1, 2, 3, 4, 5}, ((1 + sqrt(6))^k - (1 - sqrt(6))^k)/(2*sqrt(6)) = {1, 2, 9, 28, 101}. 101 is odd prime, so a(3) = 101.

Cf. A000129, A002605, A015518, A063727, A002532, A083099, A015519, A003683, A002534, A083102, A015520, A091914, A079773, A161007, A099134.

g[n_, k_] := ((1 + Sqrt[n])^k - (1 - Sqrt[n])^k)/(2Sqrt[n]);
Table[k = 3; While[! PrimeQ[Expand@g[2n, k]], k++]; Expand@g[2n, k], {n, 41}]

g(n,k) = ([0,1;2*n-1,2]^k*[0;1])[1,1]
a(n) = for(k=3,oo,if(ispseudoprime(g(n,k)),return(g(n,k)))) \\ Jason Yuen, Apr 12 2025

When 2*n + 3 = p is prime, a(n) = p.

cp's OEIS Frontend

A015541 Expansion of x/(1 - 5*x - 7*x^2).

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A015544 Lucas sequence U(5,-8): a(n+1) = 5*a(n) + 8*a(n-1), a(0)=0, a(1)=1.

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A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2*x-(k-1)*x^2).

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A109447 Binomial coefficients C(n,k) with n-k odd, read by rows.

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A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2*n))^k - (1 - sqrt(2*n))^k)/(2*sqrt(2*n)).

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Mathematica

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Formula

A015541 Expansion of x/(1 - 5x - 7x^2).

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2).

A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2n))^k - (1 - sqrt(2n))^k)/(2sqrt(2n)).