A056115 -id:A056115 - cp's OEIS frontend

A212427 a(n) = 17*n + A000217(n-1).

0, 17, 35, 54, 74, 95, 117, 140, 164, 189, 215, 242, 270, 299, 329, 360, 392, 425, 459, 494, 530, 567, 605, 644, 684, 725, 767, 810, 854, 899, 945, 992, 1040, 1089, 1139, 1190, 1242, 1295, 1349, 1404, 1460, 1517, 1575, 1634, 1694, 1755, 1817, 1880, 1944, 2009

Offset: 0

Jesse Han, May 16 2012

Generalization: T(n,i) = A000217(i-1+n) - A000217(i-1) = i*n + A000217(n-1); in this case is i=17. See also the comment in A212428.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000096, A001008, A002805, A055998-A056000, A056115, A056119, A056121, A056126, A051942, A101859, A132754-A132758, A212428.

For n > 22, T(n,17) matches A074170 but with opposite sign.

[n*(n+33)/2: n in [0..49]]; // Bruno Berselli, Jun 22 2012

Table[-17 (17 - 1)/2 + (17 + n) (16 + n)/2, {n, 0, 100}]

a(n)=n*(n+33)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (16+n)*(17+n)/2 - 16*17/2 = 17*n + (n-1)*n/2 = n*(n+33)/2.
G.f.: x*(17-16*x)/(1-x)^3. - Bruno Berselli, Jun 22 2012
a(n) = 17*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
From Amiram Eldar, Jan 11 2021: (Start)
Sum_{n>=1} 1/a(n) = 2*A001008(33)/(33*A002805(33)) = 53676090078349/216605329340400.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/33 - 14606816124167/340379803249200. (End)
From Elmo R. Oliveira, Dec 12 2024: (Start)
E.g.f.: exp(x)*x*(34 + x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A212428 a(n) = 18*n + A000217(n-1).

Original entry on oeis.org

0, 18, 37, 57, 78, 100, 123, 147, 172, 198, 225, 253, 282, 312, 343, 375, 408, 442, 477, 513, 550, 588, 627, 667, 708, 750, 793, 837, 882, 928, 975, 1023, 1072, 1122, 1173, 1225, 1278, 1332, 1387, 1443, 1500, 1558, 1617, 1677, 1738, 1800, 1863, 1927, 1992, 2058

Offset: 0

Jesse Han, May 16 2012

Generalization: T(n,i) = A000217(i-1+n) - A000217(i-1) = i*n + A000217(n-1) (corrected by Zak Seidov, Jun 21 2012); in this case is i=18.

For i = 11..16, Milan Janjic observed that if we define f(n,b,i) = Sum_{k=0..n-b} binomial(n,k)*Stirling1(n-k,b)*Product_{j=0..k-1} (-i - j), then T(n-1,i) = -f(n,n-1,i) for n >= 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000096, A001008, A002805, A055998-A056000, A056115, A056119, A056121, A056126, A051942, A101859, A132754-A132758, A212427.

[n*(n+35)/2: n in [0..48]]; // Bruno Berselli, Jun 21 2012

Table[-18 (18 - 1)/2 + (18 + n) (17 + n)/2, {n, 0, 100}]
LinearRecurrence[{3,-3,1},{0,18,37},60] (* Harvey P. Dale, Jun 09 2024 *)

a(n)=n*(n+35)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (17+n)*(18+n)/2 - 17*18/2 = 18*n + (n-1)*n/2 = n*(n+35)/2.
G.f.: x*(18-17*x)/(1-x)^3. - Bruno Berselli, Jun 21 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 10 2012
a(n) = 18*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
From Amiram Eldar, Jan 11 2021: (Start)
Sum_{n>=1} 1/a(n) = 2*A001008(35)/(35*A002805(35)) = 54437269998109/229732925058000.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/35 - 102126365345729/2527062175638000. (End)
E.g.f.: exp(x)*x*(36 + x)/2. - Elmo R. Oliveira, Dec 12 2024

A174183 a(n) is the period k such that binomial(m, n) (mod 10) = binomial(m + k, n) (mod 10).

Original entry on oeis.org

1, 10, 20, 60, 240, 1200, 7200, 50400, 403200, 3628800, 36288000, 399168000, 4790016000, 62270208000, 871782912000, 13076743680000, 209227898880000, 3556874280960000, 64023737057280000, 1216451004088320000

Offset: 0

Michel Lagneau, Mar 11 2010

a(n) is the period (mod 10) of the numbers in each column n of Pascal's triangle.

			x(0)= 0.C(1,0)C(2,0)C(3,0) ... = 0.11111111111... and p(0)=1 ;
x(1)= 0.C(1,1)C(2,1)C(3,1) ... = 0.12345678901234... and p(1) = 10 ;
x(2)= 0.C(2,2)C(3,2)C(4,2) ... = 0.13605186556815063100 13605186556815063100... and p(2)=20.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Harvey P. Dale, Table of n, a(n) for n = 0..449
Michel Lagneau, Proof
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

Cf. A000142, A002415, A007318, A002024, A000096, A000124, A002378, A000292, A000330, A055998, A055999, A056000, A056115, A056119, A056121, A056126, A051942, A101859, A001477.

Join[{1},Array[10#!&,20]] (* Harvey P. Dale, Feb 18 2018 *)

from math import factorial
def A174183(n): return 10*factorial(n) if n else 1 # Chai Wah Wu, Aug 07 2025

a(0)=1, and a(n) = 10 * n! for n >= 1.

Additional comments, and errors in examples corrected by Michel Lagneau, May 07 2010

A243957 Primes of the form 2n^2+66n+31.

Original entry on oeis.org

499, 787, 1471, 1867, 2767, 3271, 4999, 5647, 8599, 13099, 14107, 16231, 19687, 22171, 24799, 33547, 40099, 43591, 52951, 63211, 65371, 67567, 79087, 88951, 94099, 99391, 104827, 107599, 116131, 119047, 124987, 131071, 153499, 160231, 167107, 177691, 192307

Offset: 1

Vincenzo Librandi, Jun 16 2014

Primes of the form 36*A056115(k)+31.

Conjecture: 2^a(n)-1 is not prime; in other words, these primes are included in A054723.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A056115, A142110 (supersequence).

Cf. similar sequences listed in A243888.

[a: n in [1..500] | IsPrime(a) where a is 2*n^2+66*n+31];

Select[Table[2 n^2 + 66 n + 31, {n, 800}], PrimeQ]

A233334 a(1) = 1; for n > 1, a(n) is the smallest number > a(n-1) such that a(1) + a(2) + ... + a(n) is a composite number.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Michel Lagneau, Dec 18 2013

{a(n)} = {1, 3, 4, 6, 7} union {9, 10, 11, 12, ...} and the sum s(n) = a(1) + a(2) + ... + a(n) is always composite because s(1) = 1, s(2) = 4, s(3) = 8, s(4) = 14 and for n = 5,6,7,... s(n) = (n-2)*(n+9)/2 = 21, 30, 40, 51, ... = A056115(n) for n >= 3.

			The third term is 4 because 1+3+4=8 is composite.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A051935, A056115.

p=1; lst={p}; Do[If[!PrimeQ[p+n], AppendTo[lst, n]; p=p+n], {n, 3, 70}]; lst
nxt[{c_,a_}]:=Module[{k=a+1},While[!CompositeQ[c+k],k++];{c+k,k}]; NestList[nxt,{1,1},70][[;;,2]] (* Harvey P. Dale, Dec 05 2023 *)

From Chai Wah Wu, Jan 28 2024: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 7.
G.f.: x*(-x^6 + x^5 - x^4 + x^3 - x^2 + x + 1)/(x - 1)^2. (End)

A249224 Lpf (n(n+11)/2): least prime dividing n(n+11)/2.

Original entry on oeis.org

2, 13, 3, 2, 2, 3, 3, 2, 2, 3, 11, 2, 2, 5, 3, 2, 2, 3, 3, 2, 2, 3, 17, 2, 2, 13, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2, 2, 7, 3, 2, 2, 3, 3, 2, 2, 3, 29, 2, 2, 5, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2, 2, 31, 3, 2, 2, 3, 3, 2, 2, 3, 41, 2, 2, 5, 3, 2, 2, 3, 3, 2, 2, 3, 47, 2, 2, 43, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2, 2, 7, 3, 2

Offset: 1

Zak Seidov, Oct 23 2014

a(n) = 2 for n = {4,5,8,9} (mod 12) and a(n) = 3 for n = {3,6,7,10} (mod 12).

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A056115, A020639.

FactorInteger[#(11+#)/2][[1,1]]& /@ Range[100]

a(n) = my(f=factor(n*(n+11)/2)); f[1,1]; \\ Michel Marcus, Oct 23 2014

a(n) = A020639(n(n+11)/2).

cp's OEIS Frontend

A212427 a(n) = 17*n + A000217(n-1).

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A212428 a(n) = 18*n + A000217(n-1).

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A174183 a(n) is the period k such that binomial(m, n) (mod 10) = binomial(m + k, n) (mod 10).

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A243957 Primes of the form 2*n^2+66*n+31.

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A233334 a(1) = 1; for n > 1, a(n) is the smallest number > a(n-1) such that a(1) + a(2) + ... + a(n) is a composite number.

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A249224 Lpf (n(n+11)/2): least prime dividing n(n+11)/2.

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A243957 Primes of the form 2n^2+66n+31.