A016897 -id:A016897 - cp's OEIS frontend

A134491 a(n) = Fibonacci(5n+4).

3, 34, 377, 4181, 46368, 514229, 5702887, 63245986, 701408733, 7778742049, 86267571272, 956722026041, 10610209857723, 117669030460994, 1304969544928657, 14472334024676221, 160500643816367088

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (11,1)

Cf. A000045, A016897, A102312, A099100, A134490.

[Fibonacci(5*n +4): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Mathematica
```
Table[Fibonacci[5n + 4], {n, 0, 30}]
```

a(n)=fibonacci(5*n+4) \\ Charles R Greathouse IV, Oct 16 2015

G.f.: ( -3-x ) / ( -1+11*x+x^2 ). - R. J. Mathar, Apr 17 2011

a(n) = A000045(A016897(n)). - Michel Marcus, Nov 07 2013

A154684 Triangle read by rows where T(m,n)=2mn + m + n - 3, 1<=n<=m.

Original entry on oeis.org

1, 4, 9, 7, 14, 21, 10, 19, 28, 37, 13, 24, 35, 46, 57, 16, 29, 42, 55, 68, 81, 19, 34, 49, 64, 79, 94, 109, 22, 39, 56, 73, 90, 107, 124, 141, 25, 44, 63, 82, 101, 120, 139, 158, 177, 28, 49, 70, 91, 112, 133, 154, 175, 196, 217, 31, 54, 77, 100, 123, 146, 169

Offset: 1

Vincenzo Librandi, Jan 18 2009

2*T(m,n)+7 = (2n+1)*(2m+1) is not prime.

First column: A016777; second column: A016897; third column: A008589; fourth column: A017173. - Vincenzo Librandi, Nov 19 2012

			Triangle begins:
1;
4,  9;
7,  14, 21;
10, 19, 28, 37;
13, 24, 35, 46, 57;
16, 29, 42, 55, 68,  81;
19, 34, 49, 64, 79,  94,  109;
22, 39, 56, 73, 90,  107, 124, 141;
25, 44, 63, 82, 101, 120, 139, 158, 177;
28, 49, 70, 91, 112, 133, 154, 175, 196, 217; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153053, A105760, A162265 (row sums), A016777, A016897, A008589, A017173.

[(2*n*k + n + k - 3): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 19 2012

t[n_,k_]:=2 n*k + n + k - 3; Table[t[n, k], {n, 20}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 19 2012 *)

A163672 Triangle T(n,m) = 2mn + m + n + 7 read by rows.

Original entry on oeis.org

11, 14, 19, 17, 24, 31, 20, 29, 38, 47, 23, 34, 45, 56, 67, 26, 39, 52, 65, 78, 91, 29, 44, 59, 74, 89, 104, 119, 32, 49, 66, 83, 100, 117, 134, 151, 35, 54, 73, 92, 111, 130, 149, 168, 187, 38, 59, 80, 101, 122, 143, 164, 185, 206, 227, 41, 64, 87, 110, 133, 156, 179

Offset: 1

Vincenzo Librandi, Aug 03 2009

2*T(n,n) - 13 = (2n+1)^2.

The numbers 2*T(m,n)-13 =(2*n+1)*(2*m+1) are not prime. Also: first column: A016789; second column: A016897; third column: A017017; fourth column: A017185. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
  11;
  14,  19;
  17,  24,  31;
  20,  29,  38,  47;
  23,  34,  45,  56,  67;
  26,  39,  52,  65,  78,  91;
  29,  44,  59,  74,  89, 104, 119;
  32,  49,  66,  83, 100, 117, 134, 151;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A016789, A016897, A017017, A017185, A153049, A163674, A163657.

[2*n*k + n + k + 7: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k + 7; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

for(n=1,10, for(m=1,n, print1(2*m*n + m + n + 7, ", "))) \\ G. C. Greubel, Aug 02 2017

T(n,m) = A163674(n,m)-2 = A163657(n,m)-1.

Edited by R. J. Mathar, Oct 12 2009

A327753 Primes powers (A246655) congruent to 4 mod 5.

Original entry on oeis.org

4, 9, 19, 29, 49, 59, 64, 79, 89, 109, 139, 149, 169, 179, 199, 229, 239, 269, 289, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, 509, 529, 569, 599, 619, 659, 709, 719, 729, 739, 769, 809, 829, 839, 859, 919, 929, 1009, 1019, 1024, 1039, 1049, 1069, 1109, 1129, 1229, 1249

Offset: 1

Jianing Song, Sep 24 2019

nonn

Numbers k such that x^4 + x^3 + x^2 + x + 1 factors into two irreducible quadratic polynomials over GF(k).
Note that x^4 + x^3 + x^2 + x + 1 is reducible over GF(k) if and only if there exists some a in GF(k) such that a^2 - a - 1 = 0, and then x^4 + x^3 + x^2 + x + 1 = (x^2 + a*x + 1) * (x^2 + (1-a)*x + 1). There exists some a in GF(k) such that a^2 - a - 1 = 0 if and only if kronecker(k,5) = 1, or k == 1, 4 (mod 5). If k == 1 (mod 5), then x^4 + x^3 + x^2 + x + 1 can be further factored into four linear polynomials.
This sequence consists of numbers of the form p^(2e+1) where prime p == 4 (mod 5) and p^(4e+2) where prime p == 2, 3 (mod 5),

			k = 4: let GF(4) = GF(2)[w], w^2 + w + 1 = 0, then x^4 + x^3 + x^2 + x + 1 = (x^2 + w*x + 1)*(x^2 + (w+1)*x + 1);
k = 9: let GF(9) = GF(3)[i], i^2 = -1, then x^4 + x^3 + x^2 + x + 1 = (x^2 + (-1+i)*x + 1)*(x^2 + (-1-i)*x + 1);
k = 19: in GF(19), x^4 + x^3 + x^2 + x + 1 = (x^2 + 5x + 1)*(x^2 - 4x + 1).

Marius A. Burtea, Table of n, a(n) for n = 1..10000

Cf. A137827, A327752.

Intersection of A016897 and A246655.

[n:n in [2..1250]|IsPrimePower(n) and (n mod 5 eq 4)]; // Marius A. Burtea, Sep 26 2019

Select[Range@ 1250, And[PrimePowerQ@ #, Mod[#, 5] == 4] &] (* Michael De Vlieger, Sep 27 2019 *)

PARI
```
isok(n) = isprimepower(n) && (n%5==4)
```

A013825 a(n) = 2^(5*n + 4).

Original entry on oeis.org

16, 512, 16384, 524288, 16777216, 536870912, 17179869184, 549755813888, 17592186044416, 562949953421312, 18014398509481984, 576460752303423488, 18446744073709551616, 590295810358705651712, 18889465931478580854784, 604462909807314587353088, 19342813113834066795298816

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (32).

Cf. A000079 (2^n), A009976, A016897 (5*n+4).

[2^(5*n+4): n in [0..15]]; // Vincenzo Librandi, Jul 07 2011

2^(5*Range[0, 20] + 4) (* or *) NestList[32#&,16,20] (* Harvey P. Dale, Sep 27 2015 *)

From Philippe Deléham, Nov 24 2008: (Start)
a(n) = 32*a(n-1), n > 0; a(0)=16.
G.f.: 16/(1-32*x).
a(n) = 16*A009976(n). (End)
From Elmo R. Oliveira, Feb 20 2025: (Start)
E.g.f.: 16*exp(32*x).
a(n) = A000079(A016897(n)). (End)

A155705 Triangle read by rows where T(m,n) = 2mn + m + n + 2.

Original entry on oeis.org

6, 9, 14, 12, 19, 26, 15, 24, 33, 42, 18, 29, 40, 51, 62, 21, 34, 47, 60, 73, 86, 24, 39, 54, 69, 84, 99, 114, 27, 44, 61, 78, 95, 112, 129, 146, 30, 49, 68, 87, 106, 125, 144, 163, 182, 33, 54, 75, 96, 117, 138, 159, 180, 201, 222, 36, 59, 82, 105, 128, 151

Offset: 1

Vincenzo Librandi, Jan 25 2009

2*T(m,n)-3 = (2*m+1)*(2*n+1) is not prime, obviously. Also: first column: 3*A020725; second column: A016897; third column: A017041; fourth column: 3*A016789. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
6;
9,  14;
12, 19, 26;
15, 24, 33, 42;
18, 29, 40, 51, 62;
21, 34, 47, 60, 73,  86;
24, 39, 54, 69, 84,  99,  114;
27, 44, 61, 78, 95,  112, 129, 146;
30, 49, 68, 87, 106, 125, 144, 163, 182;
33, 54, 75, 96, 117, 138, 159, 180, 201, 222; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153043, A020725, A016897, A017041.

[2*n*k + n + k + 2: k in [1..n],  n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k + 2; Table[t[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

A182719 Numbers of the form 5k + 4, 7k + 5, or 11k + 6.

Original entry on oeis.org

4, 5, 6, 9, 12, 14, 17, 19, 24, 26, 28, 29, 33, 34, 39, 40, 44, 47, 49, 50, 54, 59, 61, 64, 68, 69, 72, 74, 75, 79, 82, 83, 84, 89, 94, 96, 99, 103, 104, 105, 109, 110, 114, 116, 117, 119, 124, 127, 129, 131, 134, 138, 139, 144, 145, 149, 152, 154, 159, 160, 164

Offset: 1

Omar E. Pol, Feb 08 2011

Numbers such that the Ramanujan congruences apply, making p(a(n)) divisible by at least one of 5, 7, or 11, where p is A000041.
Union of A016897, A017041 and A017461.
First differences are periodic with period length 145.

Wikipedia, Ramanujan's congruences
Index entries for linear recurrences with constant coefficients, order 146.

Cf. A000041, A016897, A017041, A017461.

IsA182719:=func< n | exists{ k: k in [0..n div 5] | n in [5*k+4, 7*k+5, 11*k+6] } >; [ n: n in [1..160] | IsA182719(n) ]; // Klaus Brockhaus, Feb 08 2011

Union[With[{no=30},Join[5Range[0,no]+4,7Range[0,no]+5,11Range[0,no]+6]]]  (* Harvey P. Dale, Feb 18 2011 *)

a(n) = a(n-145) + 385 = a(n-1) + a(n-145) - a(n-146).

Rewritten by Charles R Greathouse IV and Klaus Brockhaus, Feb 08 2011

A220485 spt(5n+4) where spt(n) = A092269(n).

Original entry on oeis.org

10, 80, 440, 1820, 6545, 20630, 59960, 161840, 412950, 1002435, 2335760, 5246120, 11416820, 24146510, 49795175, 100348950, 198063060, 383516700, 729726660, 1366124700, 2519441030, 4581865140, 8224627160, 14584074770, 25565740130, 44334556890, 76102268520

Offset: 0

Omar E. Pol, Jan 18 2013

nonn

a(n) is divisible by 5 (see A220505).

G. E. Andrews, The number of smallest parts in the partitions of n

Cf. A092269, A019897, A213260, A220502, A220503, A220505.

a(n) = A092269(A016897(n)).

A220505 a(n) = spt(5n+4)/5 where spt(n) = A092269(n).

Original entry on oeis.org

2, 16, 88, 364, 1309, 4126, 11992, 32368, 82590, 200487, 467152, 1049224, 2283364, 4829302, 9959035, 20069790, 39612612, 76703340, 145945332, 273224940, 503888206, 916373028, 1644925432, 2916814954, 5113148026, 8866911378, 15220453704

Offset: 0

Omar E. Pol, Jan 18 2013

nonn

That spt(5n+4) == 0 (mod 5) is one of the congruences stated by George E. Andrews. See theorem 2 in the Andrews' paper. See also A220507 and A220513.

G. E. Andrews, The number of smallest parts in the partitions of n
G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function
K. C. Garrett, C. McEachern, T. Frederick, O. Hall-Holt, Fast computation of Andrews' smallest part statistic and conjectured congruences, Discrete Applied Mathematics, 159 (2011), 1377-1380.
F. G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2010.
K. Ono, Congruences for the Andrews spt-function

Cf. A016897, A071734, A092269, A220485, A220507, A220513.

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r == 0, q, 0] + Sum[b[n - i j, i - 1], {j, 0, n/i}]];
spt[n_] := b[n, n];
a[n_] := spt[5n+4]/5;
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jan 30 2019, after Alois P. Heinz in A092269 *)

a(n) = A092269(A016897(n))/5 = A220485(n)/5.

A339087 Number of compositions (ordered partitions) of n into distinct parts congruent to 4 mod 5.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 1, 0, 0, 0, 4, 1, 0, 0, 6, 4, 1, 0, 0, 6, 6, 1, 0, 0, 12, 6, 1, 0, 0, 18, 8, 1, 0, 24, 24, 8, 1, 0, 24, 30, 10, 1, 0, 48, 42, 10, 1, 0, 72, 48, 12, 1, 0, 120, 60, 12, 1, 120, 144, 72, 14, 1, 120, 216, 84, 14, 1, 240

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

			a(27) = 6 because we have [14, 9, 4], [14, 4, 9], [9, 14, 4], [9, 4, 14], [4, 14, 9] and [4, 9, 14].

Index entries for sequences related to compositions

Cf. A016897, A017827, A032020, A032021, A109700, A281243, A337547, A337548, A339059, A339060, A339086, A339088, A339089.

nmax = 80; CoefficientList[Series[Sum[k! x^(k (5 k + 3)/2)/Product[1 - x^(5 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(5*k + 3)/2) / Product_{j=1..k} (1 - x^(5*j)).

cp's OEIS Frontend

A134491 a(n) = Fibonacci(5n+4).

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A154684 Triangle read by rows where T(m,n)=2mn + m + n - 3, 1<=n<=m.

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A163672 Triangle T(n,m) = 2mn + m + n + 7 read by rows.

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A327753 Primes powers (A246655) congruent to 4 mod 5.

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A013825 a(n) = 2^(5*n + 4).

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A155705 Triangle read by rows where T(m,n) = 2*m*n + m + n + 2.

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Magma

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A182719 Numbers of the form 5k + 4, 7k + 5, or 11k + 6.

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Magma

Mathematica

A155705 Triangle read by rows where T(m,n) = 2mn + m + n + 2.