Marco Piazzalunga - cp's OEIS frontend

A210524 a(n) = n - sum of even digits of n.

0, 1, 0, 3, 0, 5, 0, 7, 0, 9, 10, 11, 10, 13, 10, 15, 10, 17, 10, 19, 18, 19, 18, 21, 18, 23, 18, 25, 18, 27, 30, 31, 30, 33, 30, 35, 30, 37, 30, 39, 36, 37, 36, 39, 36, 41, 36, 43, 36, 45, 50, 51, 50, 53, 50, 55, 50, 57, 50, 59, 54, 55, 54, 57, 54, 59, 54, 61

Offset: 0

Marco Piazzalunga, Jan 27 2013

In even positions there are odd terms.

The difference between n and even digits of n and n has odd and even digits gives even terms finishing with 0.

			a(14) = 14 - 4 = 10.
a(28) = 28 - 2 - 8 = 18.

Cf. A066568.

Table[n-Total[Select[IntegerDigits[n],EvenQ]],{n,0,90}] (* Harvey P. Dale, May 20 2017 *)

a(n) = {digs = digits(n, 10); return (n - sum(i=1, #digs, digs[i]*(1 - (digs[i] % 2))));} \\ Michel Marcus, Jul 15 2013

a(n) ~ n. a(n) = n mod 2. - Charles R Greathouse IV, Jan 28 2013

A210497 a(n) = 2*prime(n+1) - prime(n).

Original entry on oeis.org

4, 7, 9, 15, 15, 21, 21, 27, 35, 33, 43, 45, 45, 51, 59, 65, 63, 73, 75, 75, 85, 87, 95, 105, 105, 105, 111, 111, 117, 141, 135, 143, 141, 159, 153, 163, 169, 171, 179, 185, 183, 201, 195, 201, 201, 223, 235, 231, 231, 237, 245, 243, 261, 263, 269, 275, 273, 283

Offset: 1

Marco Piazzalunga, Jan 24 2013

The subsequence of multiples of 3 begins: 9, 15, 15, 21, 21, 27, 33, 45.
The subsequence of primes begins: 7, 43, 73, 163, 179, 223.
Some terms, like a(3)=15 or a(5)=21, are repeated twice, other terms, like a(23)=105, are repeated three times.

			a(2) = 7 because prime(3) = 5, prime(2) = 3, and 2 * 5 - 3 = 7.
a(3) = 9 because prime(4) = 7, prime(3) = 5, and 2 * 7 - 5 = 9.
a(4) = 15 because prime(5) = 11, prime(4) = 7, and 2 * 11 - 7 = 15.

Bruno Berselli and Zak Seidov, Table of n, a(n) for n = 1..10000 (a(1)-a(1000) from Bruno Berselli).

Cf. A001223, A062234, A085704 (subsequence).

[2*NextPrime(p)-p: p in PrimesUpTo(300)]; // Bruno Berselli, Jan 24 2013

Table[2 Prime[n + 1] - Prime[n], {n, 50}] (* Vincenzo Librandi, May 03 2015 *)
ListConvolve[{2, -1}, Prime[Range[100]]] (* Paolo Xausa, Oct 29 2024 *)

a(n)=my(p=prime(n));2*nextprime(p+1)-p \\ Charles R Greathouse IV, Jan 24 2013

from sympy import prime, nextprime
def A210497(n): return -(p:=prime(n))+(nextprime(p)<<1) # Chai Wah Wu, Oct 29 2024

a(n) ~ n log n. - Charles R Greathouse IV, Jan 24 2013

A209294 a(n) = (7n^2 - 7n + 4)/2.

Original entry on oeis.org

2, 9, 23, 44, 72, 107, 149, 198, 254, 317, 387, 464, 548, 639, 737, 842, 954, 1073, 1199, 1332, 1472, 1619, 1773, 1934, 2102, 2277, 2459, 2648, 2844, 3047, 3257, 3474, 3698, 3929, 4167, 4412, 4664, 4923, 5189, 5462

Offset: 1

Marco Piazzalunga, Jan 17 2013

a(n) is the sum of the n-th centered triangular number and n-th centered square number.

Difference of consecutive terms gives A008589 (multiples of 7).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A005448, A001844, A008589.

[(7*n^2 - 7*n + 4)/2: n in [1..30]]; // G. C. Greubel, Jan 04 2018

Table[(7*n^2 - 7*n + 4)/2, {n, 1, 50}] (* G. C. Greubel, Jan 04 2018 *)
LinearRecurrence[{3,-3,1},{2,9,23},40] (* Harvey P. Dale, Nov 02 2020 *)

a(n)=7*n*(n-1)/2+2 \\ Charles R Greathouse IV, Jan 17 2013

a(n) = (7*n^2 - 7*n + 4) = 7*T(n) + 2 with T = A000217.
G.f.: x*(2+3*x+2*x^2)/(1-x)^3. - Bruno Berselli, Jan 18 2013
a(n) = a(-n+1) = 3*a(n-1)-3*a(n-2)+a(n-3).  - Bruno Berselli, Jan 18 2013
a(n) = 1 + A069099(n). - Omar E. Pol, Apr 27 2017
E.g.f.: ((7*x^2 + 4)*exp(x) - 4)/2. - G. C. Greubel, Jan 04 2018

A214731 a(n) = n^3 - 2*n^2 - 1.

Original entry on oeis.org

-2, -1, 8, 31, 74, 143, 244, 383, 566, 799, 1088, 1439, 1858, 2351, 2924, 3583, 4334, 5183, 6136, 7199, 8378, 9679, 11108, 12671, 14374, 16223, 18224, 20383, 22706, 25199, 27868, 30719, 33758, 36991, 40424, 44063, 47914, 51983, 56276, 60799, 65558, 70559

Offset: 1

Marco Piazzalunga, Jul 27 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A080859, A085490, A144390 (first differences), A152619.

Similar sequences: A152015 (of the type m^3+2m^2-1), A081437 (m^3-2m^2+1).

[n^3-2*n^2-1: n in [1..50]]; // Vincenzo Librandi, Jul 29 2012

A214731:=n->n^3-2*n^2-1: seq(A214731(n), n=1..60); # Wesley Ivan Hurt, Apr 18 2017

Table[n^3 - 2 n^2 - 1, {n, 50}] (* Vincenzo Librandi, Jul 29 2012 *)

a(n)=n^3-2*n^2-1 \\ Charles R Greathouse IV, Jul 27 2012

[n^2*(n-2)-1 for n in range(1,51)] # G. C. Greubel, Dec 31 2023

From Bruno Berselli, Jul 27 2012: (Start)
G.f.: -x*(2-7*x-x^3)/(1-x)^4.
a(n) = A085490(n-1) + 2.
a(n) = A152619(n-2) - 1 for n>1.
a(n) - a(n-2) = A080859(n-2) - 1 for n>2. (End)
E.g.f.: 1 - (1-x)*(1+x)^2*exp(x). - G. C. Greubel, Dec 31 2023

a(3) corrected by Charles R Greathouse IV, Jul 27 2012

A214446 n(n^2-2n-1).

Original entry on oeis.org

-2, -2, 6, 28, 70, 138, 238, 376, 558, 790, 1078, 1428, 1846, 2338, 2910, 3568, 4318, 5166, 6118, 7180, 8358, 9658, 11086, 12648, 14350, 16198, 18198, 20356, 22678, 25170, 27838, 30688, 33726, 36958, 40390, 44028, 47878

Offset: 1

Marco Piazzalunga, Jul 18 2012

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

a(n) = n*(n^2 - 2*n - 1)  \\ Michel Marcus, Jul 15 2013

a(n) = -2*A110427(n). G.f. 2*x*(-1+3*x+x^2) / (x-1)^4 . - R. J. Mathar, Jul 18 2012

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User: Marco Piazzalunga

A210524 a(n) = n - sum of even digits of n.

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A210497 a(n) = 2*prime(n+1) - prime(n).

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A209294 a(n) = (7*n^2 - 7*n + 4)/2.

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A214731 a(n) = n^3 - 2*n^2 - 1.

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A214446 n*(n^2-2*n-1).

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A209294 a(n) = (7n^2 - 7n + 4)/2.

A214446 n(n^2-2n-1).