A082970 -id:A082970 - cp's OEIS frontend

A062786 Centered 10-gonal numbers.

1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, 1201, 1361, 1531, 1711, 1901, 2101, 2311, 2531, 2761, 3001, 3251, 3511, 3781, 4061, 4351, 4651, 4961, 5281, 5611, 5951, 6301, 6661, 7031, 7411, 7801, 8201, 8611, 9031, 9461, 9901, 10351, 10811

Offset: 1

Jason Earls, Jul 19 2001

Deleting the least significant digit yields the (n-1)-st triangular number: a(n) = 10*A000217(n-1) + 1. - Amarnath Murthy, Dec 11 2003
All divisors of a(n) are congruent to 1 or -1, modulo 10; that is, they end in the decimal digit 1 or 9. Proof: If p is an odd prime different from 5 then 5n^2 - 5n + 1 == 0 (mod p) implies 25(2n - 1)^2 == 5 (mod p), whence p == 1 or -1 (mod 10). - Nick Hobson, Nov 13 2006
Centered decagonal numbers. - Omar E. Pol, Oct 03 2011
The partial sums of this sequence give A004466. - Leo Tavares, Oct 04 2021
The continued fraction expansion of sqrt(5*a(n)) is [5n-3; {2, 2n-2, 2, 10n-6}]. For n=1, this collapses to [2; {4}]. - Magus K. Chu, Sep 12 2022
Numbers m such that 20*m + 5 is a square. Also values of the Fibonacci polynomial y^2 - x*y - x^2 for x = n and y = 3*n - 1. This is a subsequence of A089270. - Klaus Purath, Oct 30 2022
All terms can be written as a difference of two consecutive squares a(n) = A005891(n-1)^2 - A028895(n-1)^2, and they can be represented by the forms (x^2 + 2mxy + (m^2-1)y^2) and (3x^2 + (6m-2)xy + (3m^2-2m)y^2), both of discriminant 4. - Klaus Purath, Oct 17 2023

T. D. Noe, Table of n, a(n) for n = 1..1000
Leo Tavares, Illustration: Pentagonal Stars.
Leo Tavares, Illustration: Mid-section Stars.
Leo Tavares, Illustration: Mid-section Star Pillars.
Leo Tavares, Illustration: Trapezoidal Rays.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001263, A124080, A101321, A028387, A016861, A003154, A005891, A000217, A004466, A144390, A000326, A060544.

Cf. also A016754, A002378, A069099, A045943, A003215, A046092, A001844, A028896, A005448, A024966, A082970.

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

[1+5*n*(n-1): n in [1..50]]; // G. C. Greubel, Mar 30 2019

FoldList[#1+#2 &, 1, 10Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
1+5*Pochhammer[Range[50]-1, 2] (* G. C. Greubel, Mar 30 2019 *)

j=[]; for(n=1,75,j=concat(j,(5*n*(n-1)+1))); j

for (n=1, 1000, write("b062786.txt", n, " ", 5*n*(n - 1) + 1) ) \\ Harry J. Smith, Aug 11 2009

def a(n): return(5*n**2-5*n+1) # Torlach Rush, May 10 2024

[1+5*rising_factorial(n-1, 2) for n in (1..50)] # G. C. Greubel, Mar 30 2019

a(n) = 5*n*(n-1) + 1.
From  Gary W. Adamson, Dec 29 2007: (Start)
Binomial transform of [1, 10, 10, 0, 0, 0, ...];
Narayana transform (A001263) of [1, 10, 0, 0, 0, ...]. (End)
G.f.: x*(1+8*x+x^2) / (1-x)^3. - R. J. Mathar, Feb 04 2011
a(n) = A124080(n-1) + 1. - Omar E. Pol, Oct 03 2011
a(n) = A101321(10,n-1). - R. J. Mathar, Jul 28 2016
a(n) = A028387(A016861(n-1))/5 for n > 0. - Art Baker, Mar 28 2019
E.g.f.: (1+5*x^2)*exp(x) - 1. - G. C. Greubel, Mar 30 2019
Sum_{n>=1} 1/a(n) = Pi * tan(Pi/(2*sqrt(5))) / sqrt(5). - Vaclav Kotesovec, Jul 23 2019
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=1} a(n)/n! = 6*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 6/e - 1. (End)
a(n) = A005891(n-1) + 5*A000217(n-1). - Leo Tavares, Jul 14 2021
a(n) = A003154(n) - 2*A000217(n-1). See Mid-section Stars illustration. - Leo Tavares, Sep 06 2021
From Leo Tavares, Oct 06 2021: (Start)
a(n) = A144390(n-1) + 2*A028387(n-1). See Mid-section Star Pillars illustration.
a(n) = A000326(n) + A000217(n) + 3*A000217(n-1). See Trapezoidal Rays illustration.
a(n) = A060544(n) + A000217(n-1). (End)
From Leo Tavares, Oct 31 2021: (Start)
a(n) = A016754(n-1) + 2*A000217(n-1).
a(n) = A016754(n-1) + A002378(n-1).
a(n) = A069099(n) + 3*A000217(n-1).
a(n) = A069099(n) + A045943(n-1).
a(n) = A003215(n-1) + 4*A000217(n-1).
a(n) = A003215(n-1) + A046092(n-1).
a(n) = A001844(n-1) + 6*A000217(n-1).
a(n) = A001844(n-1) + A028896(n-1).
a(n) = A005448(n) + 7*A000217(n).
a(n) = A005448(n) + A024966(n). (End)
From Klaus Purath, Oct 30 2022: (Start)
a(n) = a(n-2) + 10*(2*n-3).
a(n) = 2*a(n-1) - a(n-2) + 10.
a(n) = A135705(n-1) + n.
a(n) = A190816(n) - n.
a(n) = 2*A005891(n-1) - 1. (End)

Better description from Terrel Trotter, Jr., Apr 06 2002

A263771 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n and k occurrences of the pattern 312.

Original entry on oeis.org

1, 1, 2, 5, 1, 14, 5, 4, 1, 42, 21, 23, 14, 12, 5, 3, 132, 84, 107, 82, 96, 55, 64, 37, 29, 22, 10, 0, 2, 429, 330, 464, 410, 526, 394, 475, 365, 360, 298, 281, 175, 206, 126, 93, 55, 23, 14, 13, 1, 2, 1430, 1287, 1950, 1918, 2593, 2225, 2858, 2489, 2682, 2401

Offset: 0

Christian Stump, Oct 26 2015

Row sums give A000142.
First column gives A000108.
Also the number of permutations of n and k occurrences of either of the fixed pattern 132, 213, 231 (these are all connected by reverses and inverses).
Columns k=1-5 give: A002054(n-2) for n>=3, A082970, A082971, A138162, A138163. - Alois P. Heinz, Oct 27 2015

			Triangle begins:
    1;
    1;
    2;
    5,  1;
   14,  5,   4,  1;
   42, 21,  23, 14, 12,  5,  3;
  132, 84, 107, 82, 96, 55, 64, 37, 29, 22, 10, 0, 2;
  ...

Alois P. Heinz, Rows n = 0..10, flattened
Miklós Bóna, The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.
FindStat - Combinatorial Statistic Finder, The number of occurrences of the pattern 312 in a permutation, The number of occurrences of the pattern 213 in a permutation, The number of occurrences of the pattern 231 in a permutation, The number of occurrences of the pattern 132 in a permutation
T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, arXiv:math/0105073 [math.CO], 2001.

Cf. A000108, A000142, A001810, A002054, A082970, A082971, A138162, A138159, A138163.

Join@@Array[Table[Length@Select[Permutations@Range@#,Length@Select[Subsets[#,{3}],Ordering@Ordering@#=={3,1,2}&]==k&],{k,0,Binomial[#+1,3]}]//.{a__,0}:>{a}&,8,0]  (* Giorgos Kalogeropoulos, Mar 26 2021 *)

Sum_{k>0} k * T(n,k) = A001810(n). - Alois P. Heinz, Oct 27 2015

More terms from Alois P. Heinz, Oct 26 2015

A082971 Number of permutations of {1,2,...,n} containing exactly 3 occurrences of the 132 pattern.

Original entry on oeis.org

1, 14, 82, 410, 1918, 8657, 38225, 166322, 716170, 3059864, 12994936, 54924212, 231235054, 970347575, 4060697955, 16952812170, 70629116910, 293720506860, 1219498444500, 5055891511980, 20933654593020, 86571545598642, 357628915621698, 1475896409177780

Offset: 4

Benoit Cloitre, May 27 2003

			a(4)=1 because we have 1432 (the 132 occurrences are 143, 142 and 132).

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Miklós Bóna, The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.
Miklós Bóna, Permutations with one or two 132-subsequences, Discrete Math., 181 (1998) 267-274.
T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, arXiv:math/0105073 [math.CO], 2001.

Cf. A002054, A082970, A138162, A138163.

Column k=3 of A263771.

[1] cat [(n^6+51*n^5-407*n^4-99*n^3+7750*n^2-22416*n+20160)* Factorial(2*n-9)/(6*Factorial(n)*Factorial(n-5)): n in [5..30]]; // Vincenzo Librandi, Oct 30 2018

P:=2*x^3-5*x^2+7*x-2: Q:=-22*x^6-106*x^5+292*x^4-302*x^3+135*x^2-27*x+2: g:= (P+Q/(1-4*x)^(5/2))*1/2: gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=4..25); # Emeric Deutsch, Mar 27 2008

a[4] = 1; a[n_] := (n^6 + 51 n^5 - 407 n^4 - 99 n^3 + 7750 n^2 - 22416 n + 20160) (2 n - 9)!/(6 n! (n - 5)!);
Table[a[n], {n, 4, 25}] (* Jean-François Alcover, Oct 30 2018 *)

a(n)=(2*n-9)!/n!/6/(n-5)!*(n^6+51*n^5-407*n^4-99*n^3 +7750*n^2 -22416*n+20160)

a(n) = (2*n-9)!/n!/6/(n-5)! *(n^6+51*n^5-407*n^4-99*n^3 +7750*n^2 -22416*n +20160).

a(n) = (n^6 + 51*n^5 - 407*n^4 - 99*n^3 + 7750*n^2 - 22416*n + 20160)*(2*n-9)!/(6*n!*(n-5)!) for n>=5; a(4)=1. G.f.: (1/2)*(P(x) + Q(x)/(1-4*x)^(5/2)), where P(x) = 2*x^3 - 5*x^2 + 7*x - 2, Q(x) = -22*x^6 - 106*x^5 + 292*x^4 - 302*x^3 + 135*x^2 - 27*x + 2. - Emeric Deutsch, Mar 27 2008

Edited by N. J. A. Sloane, May 21 2008 at the suggestion of R. J. Mathar

A138162 Number of permutations of {1,2,...,n} containing exactly 4 occurrences of the 132 pattern.

Original entry on oeis.org

12, 96, 526, 2593, 12165, 55482, 248509, 1099255, 4817998, 20968680, 90747564, 390927869, 1677551078, 7174848666, 30598014925, 130155932685, 552386655300, 2339526458640, 9890067346740, 41737405295250, 175859194700958

Offset: 5

Emeric Deutsch, Mar 27 2008

nonn

			a(5)=12 because we have 12534, 12453, 14253, 14523, 13254, 13524, 15324, 14352, 31542, 21534, 21453 and 25143.

Miklós Bóna, The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.
Miklós Bóna, Permutations with one or two 132-subsequences, Discrete Math., 181 (1998) 267-274.
T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, arXiv:math/0105073 [math.CO], 2001.

Cf. A002054, A082970, A082971, A138163.

Column k=4 of A263771.

P:=5*x^4-7*x^3+2*x^2+8*x-3: Q:=2*x^9+218*x^8+1074*x^7-1754*x^6 +388*x^5 +1087*x^4-945*x^3+320*x^2-50*x+3: g:=(P+Q/(1-4*x)^(7/2))*1/2: gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=5..25);

a(n) = (n^9+102n^8-282n^7-12264n^6+32589n^5+891978n^4-7589428n^3 +25452024n^2-39821760n +23950080)(2n-12)!/[24n!(n-6)! ] for n>=6, a(5)=12.

G.f.: (1/2)[P(x) + Q(x)/(1-4x)^(7/2)], where P(x)=5x^4-7x^3+2x^2+8x-3, Q(x)=2x^9 +218x^8+1074x^7 -1754x^6 +388x^5 +1087x^4 -945x^3+320x^2-50x+3.

A138163 Number of permutations of {1,2,...,n} containing exactly 5 occurrences of the pattern 132.

Original entry on oeis.org

5, 55, 394, 2225, 11539, 57064, 273612, 1283621, 5924924, 27005978, 121861262, 545368160, 2423923480, 10710273856, 47085144255, 206085075295, 898489543020, 3903621095130, 16906888008960, 73018012573950, 314540265217362

Offset: 5

Emeric Deutsch, Mar 28 2008

nonn

			a(5)=5 because we have 13542, 14532, 15243, 15342 and 15423.

B. K. Nakamura, Computational methods in permutation patterns, PhD Dissertation, Rutgers University, May 2013.

Miklós Bóna, Permutations with one or two 132-subsequences, Discrete Math., 181 (1998) 267-274.
Miklós Bóna, The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.
T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, arXiv:math/0105073 [math.CO], 2001.
B. Nakamura, Approaches for enumerating permutations with a prescribed number of occurrences of patterns, PU. M. A. Vol. 24 2013 No 2, pp. 179-194.

Cf. A002054, A082970, A082971, A138162.

Column k=5 of A263771.

a:=proc(n) options operator, arrow: (1/120)*(n^12+170*n^11 +1861*n^10 -88090*n^9 -307617*n^8 +27882510*n^7 -348117457*n^6 +2119611370*n^5 -6970280884*n^4 +10530947320*n^3 +2614396896*n^2 -30327454080*n +29059430400) *factorial(2*n-15) / (factorial(n)*factorial(n-7)) end proc: 5, 55, 394, seq(a(n), n = 8 .. 25);

terms = 21; offset = 5;
P[x_] := 14 x^5 - 17 x^4 + x^3 - 16 x^2 + 14 x - 2;
Q[x_] := -50 x^11 - 2568 x^10 - 10826 x^9 + 16252 x^8 - 12466 x^7 + 16184 x^6 - 16480 x^5 + 9191 x^4 - 2893 x^3 + 520 x^2 - 50 x + 2;
Drop[CoefficientList[(1/2) (P[x] + Q[x]/(1 - 4 x)^(9/2)) + O[x]^(terms + offset), x], offset] (* Jean-François Alcover, Dec 13 2017 *)

a(n) = (n^12+170n^11+1861n^10-88090n^9-307617n^8+27882510n^7 -348117457n^6 +2119611370n^5 -6970280884n^4 +10530947320n^3 +2614396896n^2 -30327454080n +29059430400)(2n-15)!/[120 n!(n-7)! ] for n>=8; a(5)=5; a(6)=55; a(7)=394.

G.f.: (1/2)[P(x) + Q(x)/(1-4x)^(9/2)], where P(x) = 14x^5 - 17x^4 + x^3 - 16x^2 + 14x - 2, Q(x)= -50x^11 - 2568x^10 - 10826x^9 + 16252x^8 - 12466x^7 + 16184x^6 - 16480x^5 + 9191x^4 - 2893x^3 + 520x^2 - 50x + 2.

cp's OEIS Frontend

A062786 Centered 10-gonal numbers.

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A263771 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n and k occurrences of the pattern 312.

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A082971 Number of permutations of {1,2,...,n} containing exactly 3 occurrences of the 132 pattern.

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A138162 Number of permutations of {1,2,...,n} containing exactly 4 occurrences of the 132 pattern.

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A138163 Number of permutations of {1,2,...,n} containing exactly 5 occurrences of the pattern 132.

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