A017305 -id:A017305 - cp's OEIS frontend

A030431 Primes of form 10n+3.

3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263, 283, 293, 313, 353, 373, 383, 433, 443, 463, 503, 523, 563, 593, 613, 643, 653, 673, 683, 733, 743, 773, 823, 853, 863, 883, 953, 983, 1013, 1033, 1063, 1093, 1103, 1123, 1153, 1163, 1193

Offset: 1

Warut Roonguthai

nonn

Also primes of form 5n+3.
Union of A132233, A132235, {3}. - Ray Chandler, Apr 07 2009
Primes p such that arithmetic mean of divisors of p^4 is an integer. There are 2 such sequences of primes, this one and A030430. - Ctibor O. Zizka, Oct 20 2009
5 is not quadratic residue of primes of this form. - Vincenzo Librandi, Jun 25 2014
Intersection of A000040 and A017305. - Iain Fox, Dec 30 2017

Michael B. Porter, Table of n, a(n) for n = 1..100000
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.

Cf. A045414, A049508, A053179, A102338.

Select[Prime@Range[200], Mod[ #, 10] == 3 &] (* Ray Chandler, Nov 07 2006 *)
Select[10 Range[0, 150] + 3, PrimeQ]  (* Harvey P. Dale, Apr 06 2011 *)

select(n->n%10==3, primes(500)) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = 10*A102338(n) + 3.

Extended by Ray Chandler, Nov 07 2006

A080851 Square array of pyramidal numbers, read by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 10, 10, 1, 6, 14, 20, 15, 1, 7, 18, 30, 35, 21, 1, 8, 22, 40, 55, 56, 28, 1, 9, 26, 50, 75, 91, 84, 36, 1, 10, 30, 60, 95, 126, 140, 120, 45, 1, 11, 34, 70, 115, 161, 196, 204, 165, 55, 1, 12, 38, 80, 135, 196, 252, 288, 285, 220, 66, 1, 13, 42, 90, 155, 231, 308, 372, 405, 385, 286, 78

Offset: 0

Paul Barry, Feb 21 2003

The first row contains the triangular numbers, which are really two-dimensional, but can be regarded as degenerate pyramidal numbers. - N. J. A. Sloane, Aug 28 2015

			Array begins (n>=0, k>=0):
1,  3,  6, 10,  15,  21,  28,  36,  45,   55, ... A000217
1,  4, 10, 20,  35,  56,  84, 120, 165,  220, ... A000292
1,  5, 14, 30,  55,  91, 140, 204, 285,  385, ... A000330
1,  6, 18, 40,  75, 126, 196, 288, 405,  550, ... A002411
1,  7, 22, 50,  95, 161, 252, 372, 525,  715, ... A002412
1,  8, 26, 60, 115, 196, 308, 456, 645,  880, ... A002413
1,  9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Numerous sequences in the database are to be found in the array. Rows include the pyramidal numbers A000217, A000292, A000330, A002411, A002412, A002413, A002414, A007584, A007585, A007586.
Columns include or are closely related to A017029, A017113, A017017, A017101, A016777, A017305. Diagonals include A006325, A006484, A002417.
Cf. A057145, A027660 (antidiagonal sums).
See A257199 for another version of this array.

vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^4,x,11),x,n),n,0,11),k,-1,10) VECTOR(VECTOR(comb(k+2,2)+comb(k+2,3)n, k, 0, 11), n, 0, 11)

A080851 := proc(n,k)
    binomial(k+3,3)+(n-1)*binomial(k+2,3) ;
end proc:
seq( seq(A080851(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Oct 01 2021

pyramidalFigurative[ ngon_, rank_] := (3 rank^2 + rank^3 (ngon - 2) - rank (ngon - 5))/6; Table[ pyramidalFigurative[n-k-1, k], {n, 4, 15}, {k, n-3}] // Flatten (* Robert G. Wilson v, Sep 15 2015 *)

T(n, k) = binomial(k+3, 3) + (n-1)*binomial(k+2, 3), corrected Oct 01 2021.
T(n, k) = T(n-1, k) + C(k+2, 3) = T(n-1, k) + k*(k+1)*(k+2)/6.
G.f. for rows: (1 + n*x)/(1-x)^4, n>=-1.
T(n,k) = sum_{j=1..k+1} A057145(n+2,j). - R. J. Mathar, Jul 28 2016

A075326 Anti-Fibonacci numbers: start with a(0) = 0, and extend by the rule that the next term is the sum of the two smallest numbers that are not in the sequence nor were used to form an earlier sum.

Original entry on oeis.org

0, 3, 9, 13, 18, 23, 29, 33, 39, 43, 49, 53, 58, 63, 69, 73, 78, 83, 89, 93, 98, 103, 109, 113, 119, 123, 129, 133, 138, 143, 149, 153, 159, 163, 169, 173, 178, 183, 189, 193, 199, 203, 209, 213, 218, 223, 229, 233, 238, 243, 249, 253, 258, 263, 269, 273, 279, 283

Offset: 0

Amarnath Murthy, Sep 16 2002

In more detail, the sequence is constructed as follows: Start with a(0) = 0. The missing numbers are 1 2 3 4 5 6 ... Add the first two, and we get 3, which is therefore a(1). Cross 1, 2, and 1+2=3 off the missing list. The first two missing numbers are now 4 and 5, so a(2) = 4+5 = 9. Cross off 4,5,9 from the missing list. Repeat.
In other words, this is the sum of consecutive pairs in the sequence 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, ..., (A249031) the complement to the present one in the natural numbers. For example, a(1)=1+2=3, a(2)=4+5=9, a(3)=6+7=13, ... - Philippe Lallouet (philip.lallouet(AT)orange.fr), May 08 2008
The new definition is due to Philippe Lalloue (philip.lallouet(AT)orange.fr), May 08 2008, while the name "anti-Fibonacci numbers" is due to D. R. Hofstadter, Oct 23 2014.
Original definition: second members of pairs in A075325.
If instead we take the sum of the last used non-term and the most recent (i.e., 1+2, 2+4, 4+5, 5+7, etc.), we get A008585. - Jon Perry, Nov 01 2014
The sequences a = A075325, b = A047215, and c = A075326 are the solutions of the system of complementary equations defined recursively as follows:
  a(n) = least new,
  b(n) = least new,
  c(n) = a(n) + b(n),
where "least new k" means the least positive integer not yet placed. For anti-tribonacci numbers, see A265389; for anti-tetranacci, see A299405. - Clark Kimberling, May 01 2018
We see the Fibonacci numbers 3, 13, 89 and 233 occur in this sequence of anti-Fibonacci numbers. Are there infinitely many Fibonacci numbers occurring in (a(n))? The answer is yes: at least 13% of the Fibonacci numbers occur in (a(n)). This follows from Thomas Zaslavsky's formula, which implies that the sequence A017305 = (10n+3) is a subsequence of (a(n)). The Fibonacci sequence A000045 modulo 10 equals A003893, and has period 60. In this period, the number 3 occurs 8 times. - Michel Dekking, Feb 14 2019
From Augusto Santi, Aug 16 2025: (Start)
If we apply the anti-Fibonacci algorithm to the set of natural numbers minus the multiples of 3, we get 5, 10, 20, 25, 35, 40, 50, ...; that is, all the multiples of 5 present in the restricted set used. It is quite curious that in this particular case the algorithm can be applied recursively to its own output, generating, at the generic step s, the subset of multiples of 5^s (see Mathematics StackExchange link).
Conjectures:
After the first 0, the residues (mod 5) all fall in the classes 3 and 4. More generally, for k-nacci sequences the residue classes (mod k^2+1) all fall in k consecutive ones, the first being ceiling((k^2+1)/2​).
It is known that the sequence contains the arithmetic progression 10k+3, 20k+9 and 40k+18. These three progressions cover, experimentally, the 87.5% = 7/8 of the entire sequence. The remaining terms all belong to two forms: 40k+38 and 40k+39.
The anti-Fibonacci sequence contains all the squares of the numbers of the form 10k+3 and 10k+7, and all the cubes of the numbers of the form 10k+7, for k>=0. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, Using Walnut to solve problems from the OEIS, arXiv:2503.04122 [math.NT], 2025. See pp. 7, 14.
Robbert Fokkink and Gandhar Joshi, Anti-recurrence sequences, arXiv:2506.13337 [math.NT], 2025. See pp. 2, 18.
D. R. Hofstadter, Anti-Fibonacci numbers, Oct 23 2014.
Augusto Santi, A conjecture on Anti-k-nacci numbers, Mathematics StackExchange, 2025.
Thomas Zaslavsky, Anti-Fibonacci Numbers: A Formula, Sep 26 2016

Cf. A008585, A075325, A075327, A249031, A249032 (first differences), A000045.

Cf. also A079523, A131323, A249031, A249032, A249406.

import Data.List ((\\))
a075326 n = a075326_list !! n
a075326_list = 0 : f [1..] where
   f ws@(u:v:_) = y : f (ws \\ [u, v, y]) where y = u + v
-- Reinhard Zumkeller, Oct 26 2014

# Maple code for M+1 terms of sequence, from N. J. A. Sloane, Oct 26 2014
c:=0; a:=[c]; t:=0; M:=100;
for n from 1 to M do
s:=t+1; if s in a then s:=s+1; fi;
t:=s+1; if t in a then t:=t+1; fi;
c:=s+t;
a:=[op(a),c];
od:
[seq(a[n],n=1..nops(a))];

(* Three sequences a,b,c as in Comments *)
z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {}; b = {}; c = {};
Do[AppendTo[a,
   mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]];
  AppendTo[b, mex[Flatten[{a, b, c}], Last[a]]];
  AppendTo[c, Last[a] + Last[b]], {z}];
Take[a, 100] (* A075425 *)
Take[b, 100] (* A047215 *)
Take[c, 100] (* A075326 *)
Grid[{Join[{"n"}, Range[0, 20]], Join[{"a(n)"}, Take[a, 21]],
  Join[{"b(n)"}, Take[b, 21]], Join[{"c(n)"}, Take[c, 21]]},
Alignment -> ".",
Dividers -> {{2 -> Red, -1 -> Blue}, {2 -> Red, -1 -> Blue}}]
(* Peter J. C. Moses, Apr 26 2018 *)
********
(* Sequence "a" via A035263 substitutions *)
Accumulate[Prepend[Flatten[Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 0}}] &, {0}, 7] /. Thread[{0, 1} -> {{5, 5}, {6, 4}}]], 3]]
(* Peter J. C. Moses, May 01 2018 *)
********
(* Sequence "a" via Hofstadter substitutions; see his 2014 link *)
morph = Rest[Nest[Flatten[#/.{1->{3},3->{1,1,3}}]&,{1},6]]
hoff = Accumulate[Prepend[Flatten[morph/.Thread[{1,3}->{{6,4,5,5},{6,4,6,4,6,4,5,5}}]],3]]
(* Peter J. C. Moses, May 01 2018 *)

def aupton(nn):
    alst, disallowed, mink = [0], {0}, 1
    for n in range(1, nn+1):
        nextk = mink + 1
        while nextk in disallowed: nextk += 1
        an = mink + nextk
        alst.append(an)
        disallowed.update([mink, nextk, an])
        mink = nextk + 1
        while mink in disallowed: mink += 1
    return alst
print(aupton(57)) # Michael S. Branicky, Jan 31 2022

def A075326(n): return 5*n-1-int((n|(~((m:=n-1>>1)+1)&m).bit_length())&1) if n else 0 # Chai Wah Wu, Sep 11 2024

See Zaslavsky (2016) link.

More terms from David Wasserman, Jan 16 2005

Entry revised (including the addition of an initial 0) by N. J. A. Sloane, Oct 26 2014 and Sep 26 2016 (following a suggestion from Thomas Zaslavsky)

A106621 a(n) = numerator of n/(n+20).

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 3, 7, 2, 9, 1, 11, 3, 13, 7, 3, 4, 17, 9, 19, 1, 21, 11, 23, 6, 5, 13, 27, 7, 29, 3, 31, 8, 33, 17, 7, 9, 37, 19, 39, 2, 41, 21, 43, 11, 9, 23, 47, 12, 49, 5, 51, 13, 53, 27, 11, 14, 57, 29, 59, 3, 61, 31, 63, 16, 13, 33, 67, 17, 69, 7, 71, 18, 73, 37, 15, 19, 77, 39, 79

Offset: 0

N. J. A. Sloane, May 15 2005

Contains as subsequences A026741, A017281, A017305, A005408, A017353, and A017377. - Luce ETIENNE, Nov 04 2018

Multiplicative and also a strong divisibility sequence: gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - Peter Bala, Feb 24 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A000010, A010879.

Cf. Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106620 (k = 13 thru 19).

List([0..80],n->NumeratorRat(n/(n+20))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+20)): n in [0..100]]; // Vincenzo Librandi, Mar 06 2018

seq(numer(n/(n+20)),n=0..80); # Muniru A Asiru, Feb 19 2019

f[n_]:=Numerator[n/(n+20)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)

a(n) = numerator(n/(n+20)); \\ Michel Marcus, Mar 07 2018

[lcm(20,n)/20 for n in range(0, 80)] # Zerinvary Lajos, Jun 12 2009

a(n) = lcm(20, n)/20. - Zerinvary Lajos, Jun 12 2009
a(n) = n/gcd(n, 20). - Andrew Howroyd, Jul 25 2018
From Luce ETIENNE, Nov 04 2018: (Start)
a(n) = 9*a(n-20) - 36*a(n-40) + 84*a(n-60) - 126*a(n-80) + 126*a(n-100) - 84*a(n-120) + 36*a(n-140) - 9*a(n-160) + a(n-180).
a(n) = (5*(119*m^9 - 4923*m^8 + 86250*m^7 - 832230*m^6 + 4807887*m^5 - 16882299*m^4 + 34770400*m^3 - 37855620m^2 + 16581744*m + 54432)*floor(n/10) + 72*m*(3*m^8 - 120*m^7 + 2030*m^6 - 18900*m^5 + 105329*m^4 - 356580*m^3 + 706220*m^2 - 733200*m + 300258) + ((19*m^9 - 855*m^8 + 15810*m^7 - 154350*m^6 + 849387*m^5 - 2597175*m^4 + 4037840*m^3 - 2600100*m^2 + 540144*m - 90720)*floor(n/10) - 72*m*(m^7 - 35*m^6 + 490*m^5 - 3500*m^4 + 13489*m^3 - 27335*m^2 + 26340*m - 9450))*(-1)^floor(n/10))/362880 where m = (n mod 10). (End)
From Peter Bala, Feb 24 2019: (Start)
a(n) = n/gcd(n,20) is a quasi-polynomial in n since gcd(n,20) is a purely periodic sequence of period 20.
O.g.f.: F(x) - F(x^2) - F(x^4) - 4*F(x^5) + 4*F(x^10) + 4*F(x^20), where F(x) = x/(1 - x)^2.
O.g.f. for reciprocals: Sum_{n >= 1} x^n/a(n) = Sum_{d divides 20} (phi(d)/d) * log(1/(1 - x^d)) = log(1/(1 - x)) + (1/2)*log(1/(1 - x^2)) + (2/4)*log(1/(1 - x^4)) + (4/5)*log(1/(1 - x^5)) + (4/10)*log(1/(1 - x^10)) + (8/20)*log(1/(1 - x^20)), where phi(n) denotes the Euler totient function A000010. (End)
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(2^e) = 2^max(0, e-2), a(5^e) = 5^max(0,e-1), and a(p^e) = p^e otherwise.
Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s - 1/4^s - 4/5^s + 4/10^s + 4/20^s).
Sum_{k=1..n} a(k) ~ (231/800) * n^2. (End)

Keyword:mult added by Andrew Howroyd, Jul 25 2018

A017317 a(n) = 10*n + 4.

Original entry on oeis.org

4, 14, 24, 34, 44, 54, 64, 74, 84, 94, 104, 114, 124, 134, 144, 154, 164, 174, 184, 194, 204, 214, 224, 234, 244, 254, 264, 274, 284, 294, 304, 314, 324, 334, 344, 354, 364, 374, 384, 394, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 504, 514, 524, 534

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(59).

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008592, A016873, A016897, A017305.

[10*n+4: n in [0..60]]; // Vincenzo Librandi, May 29 2011

Range[4, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

a(n)=10*n+4 \\ Charles R Greathouse IV, Jul 10 2016

a(n) = 10*n + 4; a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, May 29 2011
G.f.: 2*(2+3*x)/(x-1)^2. - R. J. Mathar, Mar 20 2018
From Elmo R. Oliveira, Apr 05 2025: (Start)
E.g.f.: 2*exp(x)*(2 + 5*x).
a(n) = 2*A016873(n) = A016897(2*n). (End)

A063226 Dimension of the space of weight 2n cuspidal newforms for Gamma_0(63).

Original entry on oeis.org

3, 7, 13, 17, 23, 27, 33, 37, 43, 47, 53, 57, 63, 67, 73, 77, 83, 87, 93, 97, 103, 107, 113, 117, 123, 127, 133, 137, 143, 147, 153, 157, 163, 167, 173, 177, 183, 187, 193, 197, 203, 207, 213, 217, 223, 227, 233, 237, 243, 247

Offset: 1

N. J. A. Sloane, Jul 10 2001

Also, dimension of the space of weight 2n cuspidal newforms for Gamma_0(88). - N. J. A. Sloane, Nov 24 2016
First differences are 4,6,4,6,4,6.... Also  values of k such that k^(10*n) mod 10 = 8*(n mod 2)+1. - Gary Detlefs, Jul 04 2014
In other words, numbers n such that n^(2+4*k) + 1 is divisible by 10, for k >= 0. - Altug Alkan, Mar 30 2016
The rational generating function, the periodic first differences and Greubel's closed form are an immediate consequence of the structure of formula given by [Martin]. - R. J. Mathar, Apr 09 2016
A quasipolynomial of order 2 and degree 1: a(n) = 5n - 3 if n is even and 5n - 2 if n is odd. - Charles R Greathouse IV, Nov 03 2021
Numbers that are congruent to {3, 7} mod 10. - Amiram Eldar, Nov 23 2024

G. C. Greubel, Table of n, a(n) for n = 1..10000
Greg Martin, Dimensions of the spaces of cusp forms and newforms on Gamma_0(N) and Gamma_1(N), J. Numb. Theory 112 (2005) 298-331, Theorem 1.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A017305 (bisection), A017353 (bisection), A019934, A182007.

Maple
```
# see A063195
```

Table[4 Floor[n/2] + 6 Floor[(n - 1)/2] + 3, {n, 50}] (* or *)
Table[SeriesCoefficient[3 x - x^2 (-7 - 6 x + 3 x^2)/((1 + x) (x - 1)^2), {x, 0, n}], {n, 50}] (* Michael De Vlieger, Mar 30 2016 *)
LinearRecurrence[{1, 1, -1}, {3, 7, 13}, 100] (* G. C. Greubel, Mar 30 2016 *)

my(x='x+O('x^99)); Vec(3*x-x^2*(-7-6*x+3*x^2)/((1+x)*(x-1)^2)) \\ Altug Alkan, Mar 31 2016

a(n)=5*n-3+n%2 \\ Charles R Greathouse IV, Mar 31 2016

a(n) = 4*floor(n/2) + 6*floor((n-1)/2) + 3. - Gary Detlefs, Jul 04 2014
G.f.: 3*x - x^2*(-7-6*x+3*x^2)/((1+x)*(x-1)^2). - R. J. Mathar, Jul 15 2015
From G. C. Greubel, Mar 30 2016: (Start)
a(n) = (1/2)*(10*n - 5 - (-1)^n).
E.g.f.: (5*x + 3)*cosh(x) + (5*x + 2)*sinh(x). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(5-2*sqrt(5))*Pi/10. - Amiram Eldar, Sep 26 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*sin(Pi/5) (A182007).
Product_{n>=1} (1 + (-1)^n/a(n)) = tan(Pi/5) (A019934). (End)

A147874 a(n) = (5n-7)(n-1).

Original entry on oeis.org

0, 3, 16, 39, 72, 115, 168, 231, 304, 387, 480, 583, 696, 819, 952, 1095, 1248, 1411, 1584, 1767, 1960, 2163, 2376, 2599, 2832, 3075, 3328, 3591, 3864, 4147, 4440, 4743, 5056, 5379, 5712, 6055, 6408, 6771, 7144, 7527, 7920, 8323, 8736, 9159, 9592, 10035

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 16 2008

Zero followed by partial sums of A017305.
Appears to be related to various other sequences: a(n) = A036666(2*n-2) for n>1; a(n) = A115006(2*n-3) for n>1; a(n) = A118015(5*n-6) for n>1; a(n) = A008738(5*n-7) for n>1.
Even dodecagonal numbers divided by 4. - Omar E. Pol, Aug 19 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Leo Tavares, Illustration: Triangular Badges
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A017305 (10n+3), A036666, A115006, A118015 (floor(n^2/5)), A008738 (floor((n^2+1)/5)), A294830.
Cf. A051624, A193872. - Omar E. Pol, Aug 19 2011
Cf. A014105, A002378.

List([1..50], n-> (5*n-7)*(n-1)); # G. C. Greubel, Jul 30 2019

[ 0 ] cat [ &+[ 10*k+3: k in [0..n-1] ]: n in [1..50] ]; // Klaus Brockhaus, Nov 17 2008

Magma
```
[ 5*n^2-2*n: n in [0..50] ];
```

s=0;lst={s};Do[s+=n++ +3;AppendTo[lst,s],{n,0,6!,10}];lst
Table[5n^2-12n+7,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{0,3,16},50] (* or *) PolygonalNumber[12,Range[0,100,2]]/4 (* Harvey P. Dale, Aug 08 2021 *)

{m=50; a=7; for(n=0, m, print1(a=a+10*(n-1)+3, ","))} \\ Klaus Brockhaus, Nov 17 2008

[(5*n-7)*(n-1) for n in (1..50)] # G. C. Greubel, Jul 30 2019

a(n) = Sum_{k=0..n-2} 10*k+3 = Sum_{k=0..n-2} A017305(k).
G.f.: x*(3 + 7*x)/(1-x)^3.
a(n) = 10*(n-2) + 3 + a(n-1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A193872(n-1)/4. - Omar E. Pol, Aug 19 2011
a(n+1) = A131242(10n+2). - Philippe Deléham, Mar 27 2013
E.g.f.: -7 + (7 - 7*x + 5*x^2)*exp(x). - G. C. Greubel, Jul 30 2019
Sum_{n>=2} 1/a(n) = A294830. - Amiram Eldar, Nov 15 2020
a(n) = A014105(n-1) + 3*A002378(n-2). - Leo Tavares, Mar 31 2025

Edited by R. J. Mathar and Klaus Brockhaus, Nov 17 2008, Nov 20 2008

A115006 Row 2 of array in A114999.

Original entry on oeis.org

0, 3, 8, 16, 26, 39, 54, 72, 92, 115, 140, 168, 198, 231, 266, 304, 344, 387, 432, 480, 530, 583, 638, 696, 756, 819, 884, 952, 1022, 1095, 1170, 1248, 1328, 1411, 1496, 1584, 1674, 1767, 1862, 1960, 2060, 2163, 2268, 2376, 2486, 2599, 2714, 2832, 2952, 3075, 3200

Offset: 0

N. J. A. Sloane, Feb 23 2006

Number of lattice points (x,y) in the region of the coordinate plane bounded by y < 3x+1, y > x/2 and x <= n. - Wesley Ivan Hurt, Oct 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A114999, A000217 (triangular numbers), A002620 (quarter-squares), A001859 (triangular numbers plus quarter-squares), A017305 (10n+3), A147874 (zero followed by partial sums of A017305).

Partial Sums of A047218.

[ n*(n+1) + (n+1)^2 div 4: n in [0..50] ];

A115006:=n->(10*n^2 + 12*n + 1 - (-1)^n)/8: seq(A115006(n), n=0..50); # Wesley Ivan Hurt, Oct 27 2014

Table[(10*n^2 + 12*n + 1 - (-1)^n)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Oct 27 2014 *)
LinearRecurrence[{2,0,-2,1},{0,3,8,16},60] (* Harvey P. Dale, Jan 13 2015 *)

{for(n=0, 50, print1(n*(n+1)+floor((n+1)^2/4), ","))}

a(n) = floor((n+1)^2/4)+n*(n+1).
G.f.: x*(2*x+3)/((1-x)^3*(1+x)).
From Wesley Ivan Hurt, Oct 27 2014: (Start)
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = (10*n^2 + 12*n + 1 - (-1)^n)/8.
a(n) = Sum_{i=1..n+1} (10*i + (-1)^i - 9)/4. (End)
E.g.f.: (x*(11 + 5*x)*cosh(x) + (1 + 11*x + 5*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023

Edited by Klaus Brockhaus, Nov 18 2008

A156677 a(n) = 81n^2 - 118n + 43.

Original entry on oeis.org

43, 6, 131, 418, 867, 1478, 2251, 3186, 4283, 5542, 6963, 8546, 10291, 12198, 14267, 16498, 18891, 21446, 24163, 27042, 30083, 33286, 36651, 40178, 43867, 47718, 51731, 55906, 60243, 64742, 69403, 74226, 79211, 84358, 89667, 95138, 100771, 106566, 112523, 118642

Offset: 0

Vincenzo Librandi, Feb 15 2009

The identity (6561*n^2 - 9558*n + 3482)^2 - (81*n^2 - 118*n + 43)*(729*n - 531)^2 = 1 can be written as A156773(n)^2 - a(n)*A156771(n)^2 = 1 for n > 0.

For n >= 1, the continued fraction expansion of sqrt(a(n)) is [9n-7; {2, 4, 9n-7, 4, 2, 18n-14}]. For n=1, this collapses to [2; {2, 4}]. - Magus K. Chu, Sep 09 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A017305, A156771, A156773, A171198.

I:=[43, 6, 131]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];

LinearRecurrence[{3,-3,1},{43,6,131},40]

a(n)=81*n^2-118*n+43 \\ Charles R Greathouse IV, Dec 23 2011

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (-43+123*x-242*x^2)/(x-1)^3.
For n > 1: a(n) = A171198(n-2) - A017305(n-2). - Reinhard Zumkeller, Jul 13 2010
E.g.f.: exp(x)*(43 - 37*x + 81*x^2). - Elmo R. Oliveira, Oct 19 2024

Edited by Charles R Greathouse IV, Jul 25 2010

A260181 Numbers whose last digit is prime.

Original entry on oeis.org

2, 3, 5, 7, 12, 13, 15, 17, 22, 23, 25, 27, 32, 33, 35, 37, 42, 43, 45, 47, 52, 53, 55, 57, 62, 63, 65, 67, 72, 73, 75, 77, 82, 83, 85, 87, 92, 93, 95, 97, 102, 103, 105, 107, 112, 113, 115, 117, 122, 123, 125, 127, 132, 133, 135, 137, 142, 143, 145, 147

Offset: 1

Wesley Ivan Hurt, Jul 17 2015

Numbers ending in 2, 3, 5 or 7.
The subsequence of primes is A042993. - Michel Marcus, Jul 19 2015
From Wesley Ivan Hurt, Aug 15 2015, Sep 26 2015: (Start)
Ceiling(a(n)/2) = A047201(n).
Complement of (A197652 Union A262389). (End)

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A042993, A047201, A092620, subset of A118950.
Union of A017293, A017305, A017329 and A017353.
First differences are [1,2,2,5,...] = A002522(A140081(n-1)).
Cf. A197652, A262389.

a:=n->(5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2; List([1..60],n->a(n)); # Muniru A Asiru, Feb 16 2018

[(5*n-4-(-1)^n+((3-(-1)^n) div 2)*(-1)^((2*n+5-(-1)^n) div 4))/2: n in [1..70]]; // Vincenzo Librandi, Jul 18 2015

A260181:=n->(5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2: seq(A260181(n), n=1..100);

CoefficientList[Series[(2 + x + 2 x^2 + 2 x^3 + 3 x^4)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x]
LinearRecurrence[{1, 0, 0, 1, -1}, {2, 3, 5, 7, 12}, 60] (* Vincenzo Librandi, Jul 18 2015 *)
Table[(5n - 4 - (-1)^n + ((3 - (-1)^n)/2)*(-1)^((2*n + 5 - (-1)^n)/4))/2, {n, 100}] (* Wesley Ivan Hurt, Aug 11 2015 *)

is(n)=my(m=digits(n));isprime(m[#m]) \\ Anders Hellström, Jul 19 2015

A260181(n)=(n--)\4*10+prime(n%4+1) \\ is(n)=isprime(n%10) is much more efficient than the above. - M. F. Hasler, Sep 16 2016

G.f.: x*(2+x+2*x^2+2*x^3+3*x^4) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1)+a(n-4)-a(n-5), n>5.
a(n) = (5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(5*sqrt(5+2*sqrt(5))) - 25*log(5) - 40*log(2) + 5*sqrt(5)*arccoth(843/2))/200. - Amiram Eldar, Jul 30 2024

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A156677 a(n) = 81n^2 - 118n + 43.